Motion of System of Particles and Rigid Body

A constant torque of 1000 N-m turns a wheel of moment of inertia 200 $k g - m^{2}$ about an axis through its centre.Its angular velocity after 3 seconds is [2001]
1. 1 rad/s
2. 5 rad/s
3. 10 rad/s
4. 15 rad/s
A wheel of radius 1 m rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is [2002]
1. π
2. 2π
3. $\sqrt{2} π$
4. $\sqrt{(π^{2} + 4)}$
If the linear density (mass per unit length) of a rod of length 3 m is proportional to x, where x is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is
[2002]
1. 2.5m
2. 1 m
3. 1.5m
4. 2m
A composite disc is to be made using equal masses of aluminium and iron so that it has as high a moment of inertia as possible. This is possible when
[2002]
1. The surfaces of the discs are made of iron with aluminium inside
2. The whole of aluminium is kept in the core and the iron at the outer rim of the disc
3. The whole of the iron is kept in the core and the aluminium at the outer rim of the disc
4. The whole disc is made with thin alternate sheets of iron and aluminium
A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is K .If radius of the ball be R, then the fraction of total energy associated with its rotational energy will be
[2003]
1. $\frac{K^{2}}{R^{2}}$
2. $\frac{K^{2}}{K^{2} + R^{2}}$
3. $\frac{R^{2}}{K^{2} + R^{2}}$
4. $\frac{K^{2} + R^{2}}{R^{2}}$
The angular velocity af Second's hand of a watch will be
[2003]
1. $\frac{π}{60} \frac{r a d}{s e c}$
2. $\frac{π}{30} \frac{r a d}{s e c}$
3. $60 π \frac{r a d}{s e c}$
4. $30 π \frac{r a d}{s e c}$
A solid cylinder of mass M and radius R rolls without wipping down an inclined plane of length L and height h. What in the speed of its centre of mass when the cylinder reaches its bottom
[2003]
1. $\sqrt{\frac{3}{4} g h}$
2. $\sqrt{\frac{4}{3} g h}$
3. $\sqrt{4 g h}$
4. $\sqrt{2 g h}$
A wheel having moment of inertia 2 $k g - m^{2}$ about its vertical axis, rotates at the rate of 60 rpm ahout this axis. The torque which can stop the wheel's rotatin in one minute would be
[2004]
1. $\frac{2 π}{15} N - m$
2. $\frac{π}{12} N - m$
3. $\frac{π}{15} N - m$
4. $\frac{π}{18} N - m$
Consider a system of two particles having masses $m_{1}$ and $m_{2}$ .If the particle of mass $m_{1}$ , is pushed towards the mass centre of particles through a distance d. by what distance would the particle of mass $m_{2}$ move so as to keep the mass centre of particles at the original position ?
[2004]
1. $\frac{m_{1}}{(m_{1} + m_{2})} d$
2. $\frac{m_{1}}{m_{2}}$
3. d
4. $\frac{m_{2}}{m_{1}} d$
A round disc of moment of inertia $I_{2}$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $I_{2}$ rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is
[2004]
1. $\frac{I_{2} ω}{(I_{1} + I_{2})}$
2. ω
3. $\frac{I_{1} ω}{(I_{1} + I_{2})}$
4. $\frac{(I_{1} + I_{2}) ω}{I_{1}}$
Three particles, each of mass m grams situated at the vertices of an equilateral triangle ABC of side I cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC, in
$g r a m - c m^{2}$ units will be

[2004]
1. $\frac{3}{4} m l^{2}$
2. $2 m l^{2}$
3. $\frac{5}{4} m l^{2}$
4. $\frac{3}{2} m l^{2}$
The moment of inertia of a uniform circular disc of radius R and mass M about an axis passing from the edge of the disc and normal to the disc is
[2005]
1. $\frac{1}{2} M R^{2}$
2. $M R^{2}$
3. $\frac{7}{2} M R^{2}$
4. $\frac{3}{2} M R^{2}$
Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the
[2005]
1. 1:2
2. √2 : 1
3. 2:1
4. 1 : √2
The moment of inertia of a uniform circular disc of radius R and mass M about an axis touching the disc at its diameter and normal to the disc is
[2006]
1. $M R^{2}$
2. $\frac{2}{5} M R^{2}$
3. $\frac{3}{2} M R^{2}$
4. $\frac{1}{2} M R^{2}$
A wheel has angular acceleration of 3.0 ${\frac{r a d}{s}}^{6}$ and an initial angular speed of 2.00 rad/s. In a tine of 2 s it has rotated through an angle (in radian) of
[2007]
1. 6
2. 10
3. 12
4. 4
A uniform rod AB of length I and mass m is free to rotate about point A. The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about A is $m l^{\frac{2}{3}}$ , the Initial angular acceleration of the rod will be:

[2007]
1. $\frac{2 g}{3 l}$
2. $m g \frac{I}{2}$
3. $\frac{3}{2} g l$
4. $\frac{3 g}{2 l}$
A particle of mass m moves in the XY plane with a velocity v along the straight line AB. If the angular momentum of the particle with respect to origin O is $L_{A}$ , when it is at A and $L_{B}$ when it is at B, then

[2007]
1. $L_{A} > L_{B}$
2. $L_{A} = L_{B}$
3. the relationship between LA and $L_{B}$ depends upon the slope of the line AB
4. $L_{A} < L_{B}$
The ratio of the radii of gyration of a circular disc of a circular ring, each of same mass and radius, around their respective axes is
[2008]
1. √3:√2
2. 1:√2
3. √2:1
4. √2:√3
A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is 90°. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is
[2008]
1. $\frac{M L^{2}}{24}$
2. $\frac{M L^{2}}{12}$
3. $\frac{M L^{2}}{6}$
4. $\frac{\sqrt{2} M L^{2}}{24}$
A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity,
[2009]
1. $\frac{ω (M - 2 m)}{M + 2 m}$
2. $\frac{ω M}{M + 2 m}$
3. $\frac{ω (M + 2 m)}{M}$
4. $\frac{ω M}{M + m}$
If $\vec{F}$ is the force acting on a particle having position vector $\vec{τ}$ and $\vec{r}$ be the torque of this force about the origin,then
[2009]
1. $\vec{r} . \vec{τ} \neq 0 a n d \vec{F .} \vec{τ} = 0$
2. $\vec{r} . \vec{τ} > 0 a n d \vec{F .} \vec{τ} < 0$
3. $\vec{r} . \vec{τ} = 0 a n d \vec{F .} \vec{τ} = 0$
4. $\vec{r} . \vec{τ} = 0 a n d \vec{F .} \vec{τ} \neq 0$
Four identical thin rods each of mass M and length I form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is
[2009]
1. $\frac{4}{3} M l^{2}$
2. $\frac{2}{3} M l^{2}$
3. $\frac{13}{3} M l^{2}$
4. $\frac{1}{3} M l^{2}$
Two bodies of mass 1 kg and 3 kg have position vectors $\vec{i} + 2 \vec{j} + \vec{k}$ and $- 3 \vec{i} - 2 \vec{j} + \vec{k}$ , respectively. The centre of mass of this system has a position vector
[2009]
1. $- 2 \vec{i} + 2 \vec{k}$
2. $- 2 \vec{i} - \vec{j} + \vec{k}$
3. $2 \vec{i} - \vec{j} - 2 \vec{k}$
4. $- \vec{i} + \vec{j} + \vec{k}$
A circular disk of moment of inertia It, is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed ωi. Another disk of moment of inertia Ib, is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed ωf. The energy lost by the initially rotating disc due to friction is:
[2010]
1. $\frac{1}{2} \frac{{I^{2}}_{b}}{(I_{t} + I_{b})} {ω^{2}}_{i}$
2. $\frac{1}{2} \frac{{I^{2}}_{t}}{(I_{t} + I_{b})} {ω^{2}}_{i}$
3. $\frac{1}{2} \frac{I_{b} - I_{t}}{(I_{t} + I_{b})} {ω^{2}}_{i}$
4. $\frac{1}{2} \frac{I_{b} I_{t}}{(I_{t} + I_{b})} {ω^{2}}_{i}$
Two particles which are initially at rest move towards each other under the action of their internal attraction. If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be
[2010]
1. 2 v
2. 0
3. 1.5 v
4. v
A man of 50 kg mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 $m s^{1}$ . When the stone reaches the floor, the distance of the man above the floor will be

[2010]
1. 9.9 m
2. 10.1 m
3. 10 m
4. 20 m
The instantaneous angular position of a point on a rotating wheel is given by the equation $θ (t) = 2 t^{3} \times 6 t^{2}$
[2011]
1. t = 0.5 s
2. t = 0.25 s
3. t = 2 s
4. t = 1 s
The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its mid-point and perpendicular to its length is lo. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is
[2011]
1. $l_{0} + \frac{M L^{2}}{4}$
2. $l_{0} + 2 M L^{2}$
3. $l_{0} + M L^{2}$
4. $l_{0} + \frac{M L^{2}}{2}$
A smaIl mass attached to a string rotates on frictionless top as shown. If the tension of the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2, the kinetic energy of the
mass will be

[2011]
1. remain constant
2. increase by a factor of 2
3. increase by a factor of 4
4. decrease by a factor of 2
When a mass is rotating in a plane about a fixed angular momentum is directed along.
[2012]
1. the tangent to the orbit
2. a line perpendicular to the plane of rotation
3. the line making an angle of 45° to the plane of rotation
4. the radius
Two persons of masses 55 kg and 65 kg respectively are at the opposite ends of a boat.The length of the boat is 3.0 m and weighs 100 kg.The 55 kg man walks up to the 65 kg man and sits with him.If the boat is in still water the centre of mass of the system shifts by point, its
[2012]
1. 0.75m
2. 3.0m
3. 23m
4. zero
ABC is an equilateral triangle with O as its centre. $F_{1}$ , $F_{2}$ and $F_{3}$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of F3 is,

[2012]
1. $2 (F_{1} + F_{2})$
2. $(F_{1} + F_{2})$
3. $(F_{1} - F_{2})$
4. $\frac{(F_{1} + F_{2})}{2}$
A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 mand its moment of inertia about the axle is 200 kg m?. tis initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 $m s^{- 1}$ relative to the ground. Time taken by the man to complete one revolution is:
[2012]
1. π sec
2. (3π/2)sec
3. 2π sec
4. (π/2) sec
The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through

[2012]
1. B
2. C
3. D
4. A
Three masses are placed on the x-axis: 300 g at origin, 500 g at x = 40 cm and 400 g at x = 70 cm. The distance of the centre of mass from the origin is
[2012]
1. 40 cm
2. 45cm
3. 50 cm
4. 30cm
A solid cylinder of mass 3kg is rolling on a horizontal surface with velocity $4 m s^{- 1}$ . It collides with a horizontal spring of force constant $200 m s^{- 1}$ . The maximum compression produced in the spring will be
[2012]
1. 0.2 m
2. 0.5 m
3. 0.6m
4. 0.7 m
A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of $\frac{3 v^{2}}{4 g}$ with respect to the initial position. The object is
[2013]
1. Ring
2. Solid sphere
3. Hollow sphere
4. Disc
A rod PQ of mass Mand length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in the figure. When string is cut, the initial angular acceleration of the rod is

[2013]
1. (3g/2L)
2. (g/L)
3. (2g/L)
4. (2g/3L)
A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions $s^{- 2}$ is

[]2014]
1. 25N
2. 50 N
3. 78.5 N
4. 157 N
The ratio of the accelerations for a solid sphere(mass m and radius R) rolling down an incline angle of 'θ' without slipping down the incline without rolling is
[2014]
1. 5:7
2. 2:3
3. 2:5
4. 157 N
A mass m moves in a circle on a smooth horizontal plane with velocity Vo at a radius Ro. The mass is attached to a string which passes through a smooth hole in the plane as shown.

The tension in the string is increased gradually and finally m moves in a circle of radius Ro/2.
The final value of kinetic energy is
[2015]
1. $m {v^{2}}_{0}$
2. $(1 / 4) m {v^{2}}_{0}$
3. $2 m {v^{2}}_{0}$
4. $(1 / 2) m {v^{2}}_{0}$
A rod of weight Wis supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other.The centre of mass of the rod is at distance x from A.The normal reaction on A is
[2015]
1. Wx/d
2. Wd/x
3. (W(d-x)/x)
4. (W(d-x)/d)
Three identical spherical shells, each of mass m and radius r are placed as shown in the figure. Consider an axis XX" which is touching to two shells and passing
through diameter to third shell.
Moment of inertia of the system consisting of these three spherical shells about XX" axis is

[2015]
1. $\frac{11}{5} m r^{2}$
2. $\frac{16}{5} m r^{2}$
3. $3 m r^{2}$
4. $4 m r^{2}$
An automobile moves on a road with a speed of $54 k m h^{- 1}$ .The radius of its wheels is 0.45 on and the moment of inertia of the wheel about its axis of rotation is $3 k g m^{2}$ . If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is
[2015Re]
1. $2.86 k g m^{2} s^{- 2}$
2. $6.66 k g m^{2} s^{- 2}$
3. $8.58 k g m^{2} s^{- 2}$
4. $10.86 k g m^{2} s^{- 2}$
Point masses m1 and m2 are placed at the opposite ends of a rigid rod of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rotating with angular velocity ω0, is minimum is
given by

[2015Re]
1. $x = m_{2} L / (m_{1} + m_{2})$
2. $x = m_{1} L / (m_{1} + m_{2})$
3. $x = L (m_{1} / m_{2})$
4. $x = L (m_{2} / m_{1})$
A force F= ai + 3j +6k is acting at a point r =2i- 6j -12k. The value of a for which angular momentum about origin is conserved is
[2015Re]
1. 1
2. -1
3. 2
4. zero
From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about perpendicular axis, passing through centre?

[2016]
1. $3 \frac{M R^{2}}{32}$
2. $15 \frac{M R^{2}}{32}$
3. $13 \frac{M R^{2}}{32}$
4. $11 \frac{M R^{2}}{32}$
A uniform circular disc of radius 50 cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of $2.0 r a d s^{- 2}$ . Its net acceleration in $m s^{- 2}$ at the end of 2.0s is approximately:
[2016]
1. 8.0
2. 7.0
3. 6.0
4. 3.0
A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
[2016]
1. Disk
2. Sphere
3. Both reach at the same time
4. Depends on their masses
A metal ball of mass 2 kg moving with a velocity of 36 km/h has a head on collision with a stationary ball of mass 3 kg. If after the collision, the two balls move together, the loss in kinetic energy due to collision is
[2001]
1. 40 J
2. 60 J
3. 100 J
4. 140 J
A constant torque of 31.4 N-m is exerted on a pivoted wheel. If the angular acceleration of the wheel is $47 r a d / s^{2}$ , then the moment of inertia willbe
[2001]
1. $2.5 k g - m^{2}$
2. $5.8 k g - m^{2}$
3. $4.5 k g - m^{2}$
4. $5.6 k g - m^{2}$
We have two spheres, one of which is hollow and the other solid. They have identical masses and moment of inertia about their respective diameters. The ratio of their radii is
[2002]
1. 5:7
2. 3:5
3. √3 : √5
4. √3 : √7
The direction of the angular velocity vector is along

[2002]
1. the tangent to the circular path
2. the axis of rotation
3. the inward radius
4. the outward radius
A gun fires a bullet of mass 50 g with a velocity of 30 m/s. Due to this, the gun is pushed back with a velocity of 1 m/s. then the mass of the gun is
[2003]
1. 5.5 kg
2. 1.5 kg
3. 0.5 kg
4. 3.5 kg
In an orbital motion, the angular momentum vector is
[2003]
1. perpendicular to the orbital plane
2. along the radius vector
3. parallel to the linear momentum
4. in the orbital plane
A neutron makes a head-on elastic collision with a stationary deuteron. The fractional energy loss of the neutron in the collision is
[2004]
1. 16/81
2. 8/9
3. 8/27
4. 2/3
O is the centre of equilateral triangle ABC $F_{1}$ , $F_{2}$ , and $F_{3}$ , are the three forces acting along the sides AB, BC and AC respectively. What should be the value
of F3 so that the total torque about O is zero?

[2004]
1. $2 (F_{1} + F_{2})$
2. $\frac{(F_{1} + F_{2})}{2}$
3. $(F_{1} - F_{2})$
4. $(F_{1} + F_{2})$
A horizontal platform js rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass m is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period
[2005]
1. decreases continuously
2. 2gh
3. ≥ √(10/7)gh
4. (10/7)gh
A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity v mys. If it is to climb the inclined surface then y should be

[2005]
1. ≥ √2gh
2. 2gh
3. ≥ √(10/7)gh
4. (10/7)gh
A particle of mass m moving with a velocity u makes an elastic one- dimensional collision with a stationary particle of mass m establishing a contact with it for extremely small time T.Their force of contact increases from zero to F, linearly
in time T/4, remains constant for a further time T/2 and decreases linearly from F0 to zero in further time T/4 as shown. The magnitude possessed by F0 is

[2006]
1. mu/T
2. 2mu/T
3. 4mu/3T
4. 3mu/4T
If a street light of mass M is suspended from the end of a uniform rod of length L in different possible patterns a shown in figure, then

[2006]
1. pattern B is more sturdy
2. pattern C is more sturdy
3. pattern A is more sturdy
4. all will have same sturdiness
Figure shows a thin metallic triangular sheet ABC. The sides AB and BC are of
equal lengths /. The mass of the sheet is M. What is its moment of inertia
about AC?

[2006]
1. $M I^{\frac{2}{18}}$
2. $M I^{\frac{2}{12}}$
3. $M I^{\frac{2}{6}}$
4. $M I^{\frac{2}{4}}$
Two equal masses m1 and m2 moving along the same straight line with velocities +3 m/s and -5 m/s respectively collide elastically. Their velocities after the collision will be respectively
[2007]
1. +4 m/s for both
2. -3 m/s and +5 m/s
3. -4 m/s and +4 m/s
4. -5 m/s and +3 m/s
A bomb of mass 3.0 kg explodes in air into two pieces of masses 2.0 kg and 1.0 kg. The smaller mass goes at a speed of 80 m/s. The total energy imparted to the two fragments is
[2008]
1. 1.07 kJ
2. 2.14 kJ
3. 4.8 kJ
4. 2.4 kJ
For the given uniform square lamina ABCD, whose centre is O

[2008]
1. Iac =Ief
2. √2 Iac =Ief
3. √4 Iac =Ief
4. None of these
For inelastic collision between two spherical rigid bodies
[2009]
1. the total kinetic energy is conserved
2. the total mechanical energy is not conserved
3. the linear momentum is not conserved
4. the linear momentum is conserved
A particle of mass m moving with velocity v strikes a stationary particle of mass 2m and sticks to it. The speed of the system will be
[2010]
1. v/2
2. 2v
3. v/3
4. 3v
A wheel has angular acceleration of $3.0 r a d / s^{2}$ and an initial angular speed of 2.00 rad/s. In a time of 2 s it has rotated through an angle (in radian) of
[2010]
1. 6
2. 2v
3. v/3
4. 3v
A ball of mass m moving with velocity V , makes a head on elastic collision with a ball of the same mass moving with velocity 2V towards it. Taking direction of as positive velocities of the two balls after collision are
[2011]
1. -V and 2V
2. 2V and -V
3. V and - 2V
4. -2V and V
In the figure shown, a cylinder A is initially rolling with velocity v on the horizontal surface of the wedge B (of same mass as A). All surfaces are smooth and B has no initial velocity. Then maximum height reached by cylinder on the wedge will be

[2012]
1. $\frac{v^{2}}{4 g}$
2. $\frac{v^{2}}{g}$
3. $\frac{v^{2}}{2 g}$
4. $\frac{v^{2}}{8}$
A solid iron sphere A rolls down an inclined plane, while an identical hollow sphere B of same mass slides down the plane in a frictionless manner. At the bottom of the inclined plane, the total kinetic energy of sphere A is
[2013]
1. less than that of B
2. equal to that of B
3. more than that of B
4. sometimes more and sometimes less
A particle of mass 0.1 kg is subjected to a force which varies with distance as shown in the figure. If it starts its journey from rest at x = 0, its velocity at x =12m
is

[2013]
1. 0 m/s
2. 20√2 m/s
3. 20√3 m/s
4. 40 m/s
A ball of mass m and radius r rolls inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure. The angular velocity of centre of the ball in position B about the centre of the shell is

[2015]
1. $2 \sqrt{g / 5 (R - r)}$
2. $2 \sqrt{g / 7 (R - r)}$
3. $\sqrt{2 g / 7 (R - r)}$
4. $\sqrt{5 g / 2 (R - r)}$
Assertion: In an elastic collision of two billiard balls, the total KE is conserved during the short time of collision of the balls (i.e., when they are in contact).
Reason: Energy spent against friction does not follow the law of conservation of energy.
[2002]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
Assertion: The earth is slowing down and as a result the moon is coming nearer to it.
Reason: The angular momentum of the earth-moon system is not conserved.
[2003]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
Assertion: There are very small sporadic change speed of rotation of the earth.
Reason: Shifting of large air masses in the earth's atmosphere produce a change in the moment of inertia of the earth causing its speed of rotation to change.
[2004]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
Assertion: For system of particles under central field, the total angular momentum is conserved.
Reason: The torque acting on such a system is zero
[2006]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
Assertion: A judo fighter in order to throw his opponent on to the mat tries to initially bend his opponent and then rotate him around his hip.
Reason: As the mass of the opponent is brought closer to the fighter’s hip, the force required to throw the opponent is reduced.
[2007]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false
Assertion : A quick collision between two bodies is more violent than a slow collision;even when the initial and final velocities are idential.
Reason : The momdentum is greater in first case.
[2008]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false
Assertion : In an elastic collision of two bodies, the momentum and energy of each body is conserved.
Reason : If two bodies stick to each other, after colliding the collision is said to be perfectly elastic.
[2014]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false
Assertion : In an elastic collision of two bodies, the momentum and energy of each body is conserved.
Reason : If two bodies stick to each other , after colliding the collision is said to be perfectly elastic.
[2014]
1. If both assertion and reason are true and reason is a true explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false
A carpenter has constructed a toy as shown in the figure. If the density of the material of the sphere is 12 times that of cone, the position of the centre of mass of the toy is given by:
1. at a distance of 2R from O
2. at a distance of 3R from O
3. at a distance of 4R from O
4. at a distance of 5R from O
From a given sample of uniform wire, two circular loops P and Q are made, P of radius r and Q of radius nr. If the M.I of Q about its axis is 4 times that of P about it axis (assuming wire diameter much smaller than either
radius), the value of n is:
1. $4^{\frac{2}{3}}$
2. $4^{\frac{1}{3}}$
3. $4^{\frac{1}{2}}$
4. $4^{\frac{1}{4}}$
A particle of mass m is moving in a plane along a circular path of radius r. Its angular momentum about the axis of rotation is L. The centripetal force acting on the particle is
1. $\frac{L^{2}}{m r}$
2. $\frac{L^{2} m}{r}$
3. $\frac{L^{2}}{m r^{3}}$
4. $\frac{L^{2}}{m r^{2}}$
A mass is whirled in a circular path with a constant anglar velocity and its angular momentum is L. If the string is now halved keeping the angular velocity same, the angular momentum is
1. L/4
2. L
3. 2L
4. L/2
A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. In both cases
1. the speed and time of descend will be same
2. the speed will be same but time of descend will be different.
3. the speed will be different but time of descend will be same
4. speed and time of descend both are different
What should be the minimum coefficient of static friction between the plane and the cylinder, for the cylinder not to slip on an inclined plane?
1. (1/3)sinƟ
2. (1/3)tanƟ
3. (2/3)sinƟ
4. (2/3)tanƟ
The kinetic energy of an object rotating about a fixed axis with angular momentum L = Iω can be written as
1. $K = ({\frac{L}{2 I}}^{2})$
2. $\frac{K = 2 L^{2}}{I}$
3. $K = \frac{L^{2}}{I}$
4. $K = \frac{\sqrt{2} L^{2}}{I}$
An ice skater starts a spin with her arms stretched out to the sides. She balances on the tip of one skate to turn without friction. She then pulls her arms in so that moment of inertia decreases by a factor of 2. In the process of her doing so, what happens to her kinetic energy?
1. It increases by a factor of 4.
2. It increases by a factor of 2.
3. It remains constant
4. It decreases by a factor of 2
The x,y coordinates of the centre of mass of a uniform L-shaped lamina of mass 3 kg is
1. ((5/6) m,(5/6) m)
2. (1 m,1 m)
3. ((6/5) m,(6/5) m)
4. (2m ,2m)
Find the position of centre of mass of a uniform disc of radius R from which a hole of radius r is cut out. The centre of the hole is at a distance R/2 from the centre of the disc
1. $(\frac{R r^{2}}{2 (R r^{2} -} r^{2}))$ towards right of O
2. $(\frac{R r^{2}}{2 (R r^{2} -} r^{2}))$ towards left of O
3. $(\frac{2 R r^{2}}{(R r^{2} -} r^{2}))$ towards right of O
4. $(\frac{2 R r^{2}}{(R r^{2} -} r^{2}))$ towards left of O
Half of the rectangular plate shown in figure is made of a material of density $ρ_{1}$ and the other half of density $ρ_{2}$ . The length of the plate is L. The centre of mass of the system from O is
1. L/2
2. $(\frac{(ρ_{1} + ρ_{2}) L}{4 (ρ_{1} +} ρ_{2}))$
3. $(\frac{2 R r^{2}}{(R r^{2} -} r^{2}))$ towards right of O
4. $(\frac{2 R r^{2}}{(R r^{2} -} r^{2}))$ towards left of O
With reference to figure of a cube of edge a and mass m, Which of the following is the incorrect statement?
(O is the centre of the cube)
1. The moment of inertia of cube about z’ is
  
  $I' z = I z + (\frac{m a^{2}}{2)}$
2. The moment of inertia of cube about z’' is
  
  $I'' = I z + (\frac{m a^{2}}{2)}$
3. $I_{x =} I_{y}$
4. None of these
The moment of inertia of a uniform thin rod of mass m and length / about two axis PQ and RS passing through centre of rod C and in the plane of the rod are Ipg and Irs respectively. Then Ipg + Irs is equal to
1. $\frac{m l^{2}}{3}$
2. $\frac{m l^{2}}{2}$
3. $\frac{m l^{2}}{4}$
4. $\frac{m l^{2}}{12}$
Wheels A and B in Figure are connected by a belt that does not slip. The radius of B is 3.00 times the radius of A. What would be the ratio of the rotational inertias Ia/Ib ,if the two wheels had same angular momentum about their
central axes.
1. 3/4
2. 2/3
3. 1/3
4. 1/2
A disc of radius R and mass M is rolling horizontally without slipping with speed v. It then moves up an incline as shown in figure. The maximum height upto which it can reach is
1. $\frac{V^{2}}{g}$
2. $\frac{V^{2}}{2 g}$
3. $\frac{V^{2}}{3 g}$
4. $\frac{3 V^{2}}{4 g}$
A sphere of outer radius R having some cavity inside is allowed to roll down on an incline without slipping and it reaches a speed $v_{0}$ at the bottom of the incline. The incline is then made smooth by waxing and the sphere is allowed to slide without rolling and now the speed attained is (5/4) $v_{0}$ . What is the radius of gyration of the sphere about an axis passing through its centre?
1. $\sqrt{\frac{2}{5}} R$
2. $\sqrt{\frac{2}{3}} R$
3. $\sqrt{\frac{4}{5}} R$
4. $(3 / 4) R$
In a bicycle the radius of rear wheel is twice the radius of front wheel. If $v_{f}$ and $v_{r}$ are the speeds of top most points of front and rear wheels respectively, then
1. $v_{r} = 2 v_{f}$
2. $v_{f} = 2 v_{r}$
3. $v_{f} = v_{r}$
4. $v_{f} > v_{r}$
A cord is wound round the circumference of wheel of radius 7. The axis of the wheel is horizontal and fixed and moment of inertia about it is /. A weight mg is attached to the end of the cord and falls from rest. After falling through
a distance h, the angular velocity of the wheel will be
1. $\sqrt{\frac{2 g h}{I + m r}}$
2. ${[\frac{2 m g h}{I + m r^{2}}]}^{1 / 2}$
3. ${[\frac{2 m g h}{I + 2 m}]}^{1 / 2}$
4. $\sqrt{2 g h}$
Two identical cylinders roll from rest on two identical planes of slant lengths s and 2s but of the same height h. Then, the velocities, $v_{1}$ and $v_{2}$ acquired by the cylinders when they reach the bottom of the incline are related as
1. $v_{1}$ = $v_{2}$
2. ${[\frac{2 m g h}{I + m r^{2}}]}^{1 / 2}$
3. ${[\frac{2 m g h}{I + 2 m}]}^{1 / 2}$
4. $\sqrt{2 g h}$
A hollow cylinder and a solid cylinder of the same mass and radius are released simultaneously from rest at the top of the same inclined plane, which will reach the ground first?
1. solid cylinder
2. hollow cylinder
3. both will take the same time
4. it cannot be predicted
A solid sphere and a disc of same radii are falling along an inclined plane without slip. One reaches earlier than the other due to
1. different radius of gyration
2. different size
3. different friction
4. different moment of inertia
A homogeneous ball is placed on a plane making an angle Ɵ with the “horizontal.
At what values of the coefficient of friction µ will the ball roll down the plane without slipping?
1. ≥ (2/7)tan Ɵ
2. different size
3. different friction
4. different moment of inertia
A solid cylinder of mass M and radius R roll down an inclined plane of height h without slipping, The speed of its centre when it reaches the bottom is
1. $\sqrt{2 g h}$
2. ≥(2/5)tan Ɵ
3. ≥(2/3)tan Ɵ
4. ≥ - tan
A body of mass m slides down an incline(without friction) and reaches the bottom with a velocity v. If the same mass were in the form of a ring which rolls down(without slipping) this incline, the velocity of the ring at bottom would have been
1. v
2. √2 v
3. v/√2
4. $\sqrt{\frac{2 v}{5}}$
The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height A from rest without slipping will be
1. $\sqrt{\frac{10 g h}{7}}$
2. $\sqrt{g h}$
3. $\sqrt{\frac{6 g h}{5}}$
4. $\sqrt{\frac{4 g h}{3}}$
Consider a rod of mass M and length L pivoted at its centre is free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass M moving horizontally at a speed v strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be
1. v/L
2. 2v/L
3. 3v/2L
4. 6v/L
Match the options of the following columns:
Column-I Colums-II
i. Angular variables of a rotating p. will depend on relative position of rigid body the points.
ii. The corresponding linear variables q. are same for all points
of angular variables of rotating
rigid body
iii. If centripetal accelleration is zero r. uniform circular
and tangential acceleration is zero,
then motion is
iv.If centripetal accelleration is non-zero s. accelerated translatory
and tangential acceleration is zero,
then motion is
Now match the given columns and selection from the codes given below.
1. a)i-s,ii-q,iii-p,iv-s
2. b)i-q,ii-s,iii-r,iv-p
3. c)i-r,ii-p,iii-s,iv-q
4. d)i-q,ii-p,iii-s,iv-r
Assertion: The centre of gravity of a body coincides with its centre of mass only if the gravitational field does not vary from one part of the body to the other.
Reason: Centre of gravity is independent of the gravitational field.
1. If both assertion and reason are true and reason is the correct explanation of assertion
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false
4. If both assertion and reason are false.
Assertion: A rigid body not fixed in some way can have either pure translation or a combination of translation and rotation.
Reason: In rotation about a fixed axis, every particle of the rigid body moves in a circle which lies in a plane perpendicular to the axis and has its centre on the axis.
1. If both assertion and reason are true and reason is the correct explanation of assertion
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
Assertion: If there are no external forces, the centre of mass of a double star moves like a free particle.
Reason: If we go to the centre of mass frame, then we find that the two stars are moving in a circle about the centre of mass, which is at rest.
1. If both assertion and reason are true and reason is the correct explanation of assertion
2. If both assertion and reason are true but reason is not the correct explanation of assertion
3. If assertion is true but reason is false.
4. If both assertion and reason are false.
The motion of the centre of mass of a system of two particles is Unaffected by their internal forces
1. irrespective of the actual directions of the internal forces,
2. only if they are along the line joining the particles
3. only if they are at right angles to the line joining the particles
4. only if they are obliquely inclined to the line joining the particles.
Two bodies A and B initially at rest are attracted towards each other dye to gravitation. Given that A is much heavier than B. Which of the followings correctly describes the relative motion of the centre of mass of the bodies?
1. it moves towards A
2. it moves towards B
3. it moves Perpendicular to the line joining the particles
4. it remains at rest
The position of centre of mass of system of particles at any moment does not depend on
1. masses of the particles
2. forces on the particles
3. positions of the particles
4. relative distances between the particles
Two bodies A and B have masses M and m respectively where M > mm and they are at a distance A apart. Equal force is applied to each of them so that they approach each other. The position where they hit each other is:
1. nearer to B
2. nearer to A
3. at equal distance from A and B
4. cannot be decided
Three identical particles each of mass 1 kg are placed with their centres on a straight line.Their centres are marked A, B and C respectively. The distance of centre of mass of the system from A is
1. (AB+AC+BC)/3
2. (AB+AC)/3
3. (AB+AC)/3
4. (AB+BC)/3
Two particles of equal masses have velocity $\vec{v_{1}} = 2 \hat{i} m / s$ and $\vec{v_{1}} = 2 \hat{j} m / s$ . The first particle has an acceleration of the other particle is zero.The center of mass of the two particles moves in a:
1. circle
2. parabola
3. straight line
4. ellipse
The centre of mass of a system-of two particles of masses $m_{1}$ and $m_{2}$ is at a distance $d_{1}$ from $m_{1}$ and at a $d_{2}$
from mass $m_{2}$ such that
1. $\frac{d_{1}}{d_{2}} = \frac{m_{2}}{m_{1}}$
2. parabola
3. straight line
4. ellipse
The velocity of centre of mass of the system remains constant, if the total external force acting on the system is
1. minimum,
2. maximum
3. unity
4. zero
A system of particles is free from any external force .If $\vec{v}$ and $\vec{a}$ be the velocity and acceleration of the centre of mass, then it necessarily folows that:
1. $\vec{v} = 0; \vec{a} = 0$
2. $\vec{v} \neq 0; \vec{a} = 0$
3. $\vec{v} = 0; \vec{a} \neq 0$
4. None of these
A child is sitting at one end of a long trolley moving with a uniform speed v on a smooth horizontal track. If the child starts running towards the other end of the trolley with a speed u (W.r.t trolley), the speed of the center of mass of the system will .
1. u + v
2. v - u
3. v
4. none
When an explosive shell travelling in a parabolic path under the effect of gravity explodes in the mid air, the centre of mass of the fragments will move
1. vertically downwards
2. along the original parabolic path
3. vertically upwards and then vertically downwards
4. horizontally followed by parabolic path
The velocity of the CM of a system changes from $\vec{v_{1}} = 4 \hat{i} m / s$ to $\vec{v_{2}} = 3 \vec{j} m / s$ during time $∆ t = 2 s$ , If the mass of the system is m = 10 kg, the constant force acting on the system is
1. 25 N
2. 20 N
3. 50 N
4. 5 N
An insulated particle of mass m is moving in a horizontal plane (x-y) along the X-axis. At a certain height above the ground, it suddenly explodes into two fragments of masses m/4 and 3m/4. An instant later, the smaller fragment is at Y=+15 cm. The larger fragment at this instant at:
1. Y = -5 cm
2. Y = +20cm
3. Y = +5 cm
4. Y = -20 cm
Consider a large block placed on a smooth horizontal surface, with a man standing at one end of the block. The man walks to the other end, relative to the block.The distances(absolute) moved by the man and the block are :
1. In the inverse ratio of their masses
2. In the direction ratio of their masses
3. Independent of their masses
4. Dependent both on their mases and speeds.
A body A of mass M whil falling vertically downward under gravity breaks into two parts ; a body B of mass M/3 and a body C of mass 2M/3 . The centre of mass of bodies B and C taken together shifts compared to that of body A towards :
1. Body C
2. Body B
3. Depends on height of breaking
4. Does not shift
A boy of mass m is standing on a block of mass M kept on a rough surface. When the boy walks from left to right on the block, the centre of mass (boy + block) of system:
1. remains stationary
2. shifts towards left
3. shift towards right
4. shifts towards right if M>m and towards left if M <m.
A10 kg boy standing in a 40 kg boat floating on water is 20 m away from the shore of the river. If the boy moves 8 m on the boat towards the shore, then how far is he from the shore? (Assume no friction between boat and water)
1. 12.0 m
2. 13.6 m
3. 12.8 m
4. 11.6 m
In a gravity free space, a man of mass M standing at a height h above the floor, throws a ball of mass m straight down with a speed u. When the ball reaches the floor, the distance of the man above the floor will be
1. $h (1 + m / M)$
2. $h (2 - m / M)$
3. 2h
4. a function of m, M,h and u
Two particles A and B initially at rest. Move towards each other under a mutual force of attraction. At the instant, when the speed of A is v and the speed of B is 2v, the velocity of centre of the system is
1. 0
2. v
3. 1.5v
4. 3 v
Three balls of different masses are thrown at different instants up against gravity. While all the three balls are in air, the centre of mass of the system of three balls has an acceleration:
1. Equal to 'g'
2. Which depends on the direction of motion and speeds of different balls
3. Which depends on the velocities , heights, and massess of the balls
4. Which depends on the direction of motions , speeds and masses of the ball
A particle of mass 200 g is dropped from a height of 50 m and another particle of mass 100 g is simultaneously projected up from the ground along the same lime, with a speed of 100 m/s. the acceleration of the centre of mass after 1 sec is
1. $10 m / s^{2}$
2. $\frac{10}{3} m / s^{2}$
3. 0
4. None of these
In the above problem the velocity of the centre of mass after 1 sec will be:
1. 20/3 m/s vertically down
2. 20/3 m/s vertically up
3. 70/3 m/s vertically down
4. 70/3 m/s vertically up
Analogue of massin rotational motion is
1. moment of inertia
2. torque
3. radius of gyration
4. angular momentum
A person is standing on a rotating table with metal spheres in his hands. If he withdraws his hands to his chest , then the effect on his angular velocity will be
1. increase
2. decrease
3. remain same
4. can't say
The moment of inertia of a body depends upon
1. mass of the body
2. axis of rotation of the body
3. shape and size of the body
4. all of these
Two masses each are attached to the end of a rigid massless rod of length L. The moment of inertia of the system about an axis passing centre of mass and
pendicular to its length is
1. $M L^{2} / 4$
2. $M L^{2} / 2$
3. $M L^{2}$
4. $2 M L^{2}$
Four spheres each having mass m and radius r are placed with their centres on the four corners of a square of side a.Then the moment of inertia of the system axis along one of the sides of the square is
1. $\frac{8}{5} m r^{2}$
2. $\frac{8}{5} m r^{2} + m a^{2}$
3. $\frac{8}{5} m r^{2} + 2 m a^{2}$
4. $\frac{4}{5} m r^{2} + 4 m a^{2}$
The radius of gyration of an uniform rod of length l about an axis passing through one of its ends and perpendicular to its length is
1. $l / \sqrt{2}$
2. l/3
3. $\frac{l}{\sqrt{3}}$
4. $l / 2$
Moment of inertia of a thin rod of mass M and length L about an axis passing through its centre is $\frac{M L^{2}}{12}$ . Its moment of inertia about a parallel axis at distance of L/4 from this axis is given by
1. $\frac{M L^{2}}{48}$
2. $\frac{M L^{2}}{48}$
3. $\frac{M L^{2}}{12}$
4. $\frac{7 M L^{2}}{48}$
A flywheel rotating at 420 rpm slows down at a constant rate of 2 rad $s^{- 2}$ .The time required to stop the flywheel is
1. 22 s
2. 11 s
3. 44 s
4. 12 s
An athlete throws a discus from rest to a final angular velocity of $15 r a d s^{- 2}$ in 0.270 s before
releasing it. During acceleration, discuss moves a circular arc of radius 0.810 m.Acceleration of discus before it is released is
1. $45 m s^{- 2}$
2. $180 m s^{- 2}$
3. 1.0 km
4. 1.5 km
A cyclist rides a bicycle with a wheel radius of 0.500 m across campus. A piece of plastic on the front rim makes a clicking sound every time it passes through the fork. If the cyclist counts 320 clicks between her apartment and the cafeteria, how far has she travelled?
1. 0.50 km
2. 0.80 km
3. 1.0 km
4. 1.5 km
A wheel is rotating about a fixed axis with constant angular acceleration $3 \frac{r a d}{s^{2}}$ .At different moments, its angular speed is -2 rad/s, 0, and +2 rad/s. For a point on the rim of the wheel, consider at these moments the magnitude of the tangetial component of acceleration and the magnitude of the radial component of acceleration
(m) $|a_{t}| w h e n ω = - 2 r a d / s$
(n) $|a_{r}| w h e n ω = - 2 r a d / s$
(o) $|a_{r}| w h e n ω = 0$
(p) $|a_{t}| w h e n ω = 2 r a d / s, a n d$
(q) $|a_{r}| w h e n ω = 2 r a d / s$
1. n = q > m = p > o = 0
2. m = p > n = q > o = 0
3. n = o > m = p > q = 0
4. n = q > o = p > m = 0
A grindstone increases in angular speed from 4.00 rad/s to 12.00 rad/s in 4.00 s. Through what angle does it turn during that time intervsl if the angularacceleration is constant?
1. 8.00 rad
2. 12.0 rad
3. 16.0 rad
4. 32.0 rad
Suppose a car's standard times are replaced with tires 1.50 times larger in diameter. Wil the car's speedometer reading be
1. 2.25 times too high
2. 1.50 times too high
3. 1.50 times too low
4. 2.25 times too low
In previous problem, the car's fuel economy in miles per gallon or km/L appear to be
1. 1.50 times better
2. 2.25 time better
3. 1.50 time worse
4. 2.25 time worse
A couple produces
1. purely translational motion
2. purely rotational motion
3. both translational and rotational motion
4. no motion
When a torque acting upon a system is zero, which of the following will be constant?
1. Force
2. Linear impulse
3. Linear momentum
4. Angular momentum
When a steady torque (net force is zero) is acting on a body, the body
1. rotates at a constant speed
2. gets both linear and angular acceleration
3. gets no angular acceleration
4. centre of the body continues in its state of rest or uniform motion along a straight line
The force $7 \hat{i} + 3 \hat{j} - 5 \hat{k}$ acts on a particle whose position vector is $\hat{i} - \hat{j} + \hat{k}$ . What is the torque of a given force about the origin?
1. $2 \hat{i} + 12 \hat{j} + 10 \hat{k}$
2. $2 \hat{i} + 10 \hat{j} + 12 \hat{k}$
3. $2 \hat{i} + 10 \hat{j} + 10 \hat{k}$
4. $10 \hat{i} + 2 \hat{j} + \hat{k}$
A wheel of radius 20 cm has four forces applied to it as shown in fig. Then, the torque produced by ih these forces about O is
1. 5.4 Nm anticlockwise
2. 1.8 Nm clockwise
3. 1.8 Nm anticlockwise
4. 5.4 Nm clockwise
Where must a 800 N weight be hung on 4 uniform horizontal 100 N pole of length L so that a body supports one-third as much as a man at the other end?
1. at a distance of 0.22L from man
2. at a distance of 0.22L from boy
3. at a distance of 0..33L from man
4. in the middle
A uniform horizontal 200 N beam AB of length has two weights hanging from it, 300 N at L/3 from end A and 400 N at 31/4 from the same end. What single additional force acting on the beam will produced equilibrium?
1. 900 N, in the middle
2. 900 N, at 0.4L from A
3. 900 N, at 0.46L from A
4. 900 N, at 0.56L from A
A weightless rod is acted upon by two Upward parallel forces of 2 N and 4 N at ends A and B respectively.The total length of the rod AB = 3 m. To keep the rod in equilibrium a force of 6 N should act in the following manner:
1. Downwards at any point between A and B
2. Downwards at the mid point of AB
3. Downwards at a point C such that AC =1m
4. Downwards at a point D such that BD = 1m
A false balance has equal arms. An object weighs $W_{1}$ , when placed in one pan and $W_{2}$ , when placed in the other pan. The true weight W of the object is
1. $\sqrt{W_{1} W_{2}}$
2. $\sqrt{{W_{1}}^{2} + {W_{2}}^{2}}$
3. $\frac{W_{1} + W_{2}}{2}$
4. $\frac{2 W_{1} W_{2}}{W_{1} + W_{2}}$
The beam and pans of a balance have negligible mass. An object weighs $W_{1}$ , when placed in one pan and $W_{2}$ , when placed in the other pan. The weights W of the object is
1. $\sqrt{W_{1} W_{2}}$
2. $\sqrt{{W_{1}}^{2} + {W_{2}}^{2}}$
3. $\frac{W_{1} + W_{2}}{2}$
4. $\frac{2 W_{1} W_{2}}{W_{1} + W_{2}}$
In first figure a meter stick, half of which is wood and the other half steel is pivoted at the wooden end at A and a force is applied at the steel end at O. In second figure the stick is pivoted at the steel end at O and the same force is
applied at the wooden end at A. The angular acceleration

F F
0 AO
Steel Wood Steel Wood
1. in first is greater than in second
2. equal in both first and second
3. in second is greater than in first
4. None of the above
A ring starts from rest and acquires an angular speed of $10 r a d s^{- 1}$ in 2 s. The mass of the ring is 500 gm and its radius is 20 cm. The torque on the ring is
1. 0.02Nm
2. 0.20 Nm
3. 0.10 Nm
4. 0.01 Nm
A wheel of moment of inertia $2.0 \times 10^{3} k g m^{2}$ torque at uniform angular speed of $4 r a d s^{- 1}$ . What is the torque required to stop it in one second.
1. $0.5 \times 10^{3} N m$
2. $8.0 \times 10^{3} N m$
3. $2.0 \times 10^{3} N m$
4. none of these
The moment of inertia of an angular wheel shown in figure is $3200 k g m^{2}$ . If the inner radius is 5 cm and the outer radius is 20 cm, and the wheel is acted upon by the forces shown, then the angular acceleration of the wheel is A
1. $10^{- 1} r a d / s^{2}$
2. $10^{- 2} r a d / s^{2}$
3. $10^{- 3} r a d / s^{2}$
4. $10^{- 4} r a d / s^{2}$
A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. The angular acceleration of the disc is
1. T/MR
2. MR/T
3. 2T/MR
4. MR/2T
In a uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. the tangential acceleration of a point on the rim is:
1. T/M
2. $M R^{2} / T$
3. 2T/M
4. $M R^{2} / 2 T$
In a uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord.If we hang a body of mass m from the cord, the tangential acceleration of the disc is
1. $\frac{m g}{M + m}$
2. $\frac{m g}{M + 2 m}$
3. $\frac{2 m g}{M + 2 m}$
4. $\frac{M + 2 m}{2 m g}$
In a uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord,the tension in the cord in the above problem is:
1. $\frac{m g}{M + m}$
2. $\frac{M m g}{M + 2 m}$
3. $\frac{M + 2 m}{M m g}$
4. none of these
A particle is made to move in circular path with decreasing speed.Which of the following is correct?
1. Angular momentum is constant
2. Only the direction of $\vec{L}$ is constant
3. Acceleration is always directed towards the centre
4. Particle moves in helical path
A body of mass m and radius r is released from the rest along a smooth inclined plane of angle of inclination dpkw.The angular momentum of the body about the instatntaneous point of contact after a time t from the instant of release is equal to:
1. $m g r t \cos θ$
2. $m g r t \sin θ$
3. $(3 / 2) m g r t \sin θ$
4. none of these
Consider an isolated system moving through empty space.The space consists of objects that interact with each other and can change location with respect to one another.Which of the folloiwing quantities can change in time?
1. The angular momentum of the system
2. The linear momentum of the system
3. Both the angular momentum and linear momentum of the system
4. Neither the angular momentum nor linear momentum of the system
For a particle of a mass 100 gm, position and velocity at any instant are given as $10 \hat{i} + 6 \hat{j}$ cm and $\hat{v} = 5 \hat{i} c m / s$ .Calculate the angular momentum about the point (1, 1) cm.
1. $25 \times 10^{- 5} (- \hat{k}) k g m^{2} s^{- 1}$
2. $30 \times 10^{- 5} (- \hat{k}) k g m^{2} s^{- 1}$
3. $3 \times 10^{- 3} (- \hat{k}) k g m^{2} s^{- 1}$
4. $25 \times 10^{- 4} (- \hat{k}) k g m^{2} s^{- 1}$
A particle of mass m moves in the xy-plane with a velocity of $\vec{v} = v_{x} \hat{i} + v_{y} \hat{j}$ . When its position vector is $\vec{r} = x \hat{i} + y \hat{j}$ . the angular momentum of the particle about the origin is
1. $m (x v_{y} + y v_{x}) \hat{k}$
2. $- m (x v_{y} + y v_{x}) \hat{k}$
3. $m (y v_{x} - x v_{y}) \hat{k}$
4. $m (x v_{y} - y v_{x}) \hat{k}$
A hollow straight tube of length / and mass m can turn freely about its centre (fixed) on a smooth horizontal table. Another smooth uniform rod of same length and mass is fitted into the tube so that their centres coincide.
The system is set in motion with an initial angular velocity $ω$ ), The angular velocity of the rod at an instant when the rod slips out of the tube is:
1. $ω_{0} / 3$
2. $ω_{0} / 3$
3. $ω_{0} / 4$
4. $ω_{0} / 7$
A patrticle of mass 1 kg is moving along the line y=x+2 (here xand y are in m) with speed 2 ms. The magnitude of angular momentum of the particle about origin is:
1. $4 k g m^{2} s^{- 1}$
2. $2 \sqrt{2} k g m^{2} s^{- 1}$
3. $4 \sqrt{2} k g m^{2} s^{- 1}$
4. $2 k g m^{2} s^{- 1}$
A circular platform is mounted on a vertical frictionless axle. Its radius is = 2 m and its moment of inertia I = $200 k g m^{2}$ . It is initially at rest. A 70 kg man Stands on the edge of the platform and begins to walk along the edge at speed $v_{0}$ = $1 m s^{- 1}$ relative to the ground .The angular velocity of the platform is
1. $1.2 r a d s^{- 1}$
2. $0.4 r a d s^{- 1}$
3. $0.7 r a d s^{- 1}$
4. $2 r a d s^{- 1}$
A solid cylinder of mass 20 kg and radius 20 cm rotates about its axis with a angular speed $100 r a d s^{- 1}$ The angular momentum of the cylinder about its axis is
1. 40 J s
2. 400 J s
3. 20 J s
4. 200 J s
A child is standing with his two arms outstretched at the centre of a turnable that is rotating about its central axis with an angular speed $ω_{0}$ .Now ,the child folds his hands back so that moment of inertia becomes 3 times the initial value. The new angular speed is d folds his hands
1. $ω_{0}$
2. $ω_{0} / 3$
3. $6 ω_{0}$
4. $ω_{0} / 6$
Two discs of moments of inertia $I_{1}$ and $I_{2}$ about thier respective axes, rotating with angular frequencies $ω_{1}$ and $ω_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be A
1. $\frac{I_{1} ω_{1} + I_{2} ω_{2}}{I_{1} + I_{2}}$
2. $\frac{I_{2} ω_{1} + I_{1} ω_{2}}{I_{1} + I_{2}}$
3. $\frac{I_{1} ω_{1} + I_{2} ω_{2}}{I_{1} - I_{2}}$
4. $\frac{I_{2} ω_{1} + I_{1} ω_{2}}{I_{1} - I_{2}}$
A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with angular velocity of rad /s .When she stretches her arms fully, the spinning speed decrease in rad/s.If I is the initial moment of inertia of the dancer,the new moment of inertia is
1. 21
2. 31
3. 1/2
4. 1/3
A man stands on a rotating platform with his arms stretched holding a 5 kg weight in each hand. The angular speed of the platform is $1.2 r e v s^{- 1}$ . The moment of inertia of the man together with the platform may be taken to be
constant and equal to $6 k g m^{2}$ . If the man brings his arms close to his chest with the distance n each weight from the axis changing from 100 cm to 20 cm. The new angular speed of the platform is
1. $2 r e v s^{- 1}$
2. $3 r e v s^{- 1}$
3. $5 r e v s^{- 1}$
4. $6 r e v s^{- 1}$
A girl of mass M stands on the rim of a frictionless merry-go-round of radius R and rotational inertia I that is notmoving.She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry-go-round.The speed of the rock, relative to the ground, is v. Afterward, the linear speed of the girl is
1. $\frac{m v R^{2}}{1 + M R^{2}}$
2. $\frac{(m + M) v R^{2}}{1 + M R^{2}}$
3. $\frac{m v R^{2}}{1 + (M + m) R^{2}}$
4. $\frac{m v R^{2}}{1 + (M - m) R^{2}}$
A basketball rolls a ramp sloping upward without slipping, with its centre of mass moving at a certain initial speed. A block of ice of the same mass is set sliding up the ramp with the same speed along a parallel line. Which object will travel farther up the ramp?
1. basketball
2. the ice block
3. They will travel equally far up the ramp
4. cannot be decided
A loop rolls down on an inclined plane. The fraction of its kinetic energy that is associated with only the rotational motion is
1. 1:2
2. 1:3
3. 1:4
4. 2:3
When a sphere rolls without slipping the ratio of its kinetic energy of translation to its total kinetic energy is
1. 1:7
2. 1:2
3. 1:1
4. 5:7
The moment of inertia of a body about a given axis is $1.2 K g m^{2}$ . Initially the body is at rest. In order to produce a rotational KE of 1500 joule an angular acceleration of $25 K g m^{2}$ must be applied about the axis for a duration of
1. 4 s
2. 2 s
3. 8 s
4. 10 s
A loop and a disc have same mass and roll without slipping with the same linear velocity v. If the total kinetic energy of the loop is 8 J, the kinetic energy of the disc must be
1. 8 J
2. 16 J
3. 6 J
4. 4 J
Two bodies with moment of inertia $I_{1}$ and $I_{2}$ ( $I_{1} > I_{2}$ ) have equal angular momenta. If their kinetic energy of rotation are $E_{1}$ and $E_{2}$ respectively, then
1. $E_{1} = E_{2}$
2. $E_{1} < E_{2}$
3. $E_{1} > E_{2}$
4. $E_{1} E_{2}$
A uniform rod of length / is free to rotate in a vertical plane about a fixed
horizontal axis through O. The rod is allowed to rotate from rest from its unstable vertical position. Then, the angular velocity of the rod when it has
turned through an angle $θ$ is
1. $\sqrt{3 \frac{g}{l}} \sin (θ / 2)$
2. $\sqrt{6 \frac{g}{l}} \sin (θ / 2)$
3. $\sqrt{3 \frac{g}{l}} \cos (θ / 2)$
4. $\sqrt{6 \frac{g}{l}} \cos (θ / 2)$
A rod of length / whose lower end is fixed on a horizontal plane, starts toppling from the vertical position. The velocity of the upper end when it hits the ground is
1. $\sqrt{\frac{g}{l}}$
2. $\sqrt{3 g l}$
3. $3 \sqrt{\frac{g}{l}}$
4. $\sqrt{3 \frac{g}{l}}$
A body rolls without slipping. The radius of gyration of the body about an axis passing through its centre of mass is K. The radius of the body is R. The ratio of rotational kinetic energy to translational kinetic energy is
1. $\frac{K^{2}}{R^{2}}$
2. $\frac{R^{2}}{K^{2} + R^{2}}$
3. $\frac{K^{2}}{K^{2} + R^{2}}$
4. $K^{2} + R^{2}$
A horizontal 90 kg merry-go-round is a solid disk of radius 1.50 m and is started from rest by a constant horizontal force of 50.0 N applied tangentially to the edge of the disk. The kinetic energy of the disk after 3.00 s is
1. 125 J
2. 500 J
3. 250 J
4. 150 J
A solid sphere of mass 10 kg is placed on a rough surface having coefficient of friction $μ$ = 0.1. A constant force F = 7 N is applied along a line passing through the centre of the sphere as shown in the figure. The value of frictional force on the sphere is
1. 1 N
2. 2 N
3. 3 N
4. 7 N
When a body rolls without sliding up an inclined plane, the frictional force is:
1. directed up the plane
2. directed down the plane
3. zero
4. dependent on its velocity
Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are identical. Which of the bodies reaches the ground with maximum
velocity?
1. Ring
2. Solid cylinder
3. Solid sphere
4. All reach the ground with same velocity
A uniform rod of mass m and length $l_{0}$ , is rotating with a constant angular speed $ω$ about a vertical axis passing through its point of suspenison.The moment of inertia of the rod about the axis of rotation if it makes an angle $θ$ to the vertical (axis of rotation) is
1. $\frac{m {l^{2}}_{0} \sin^{2} θ}{12}$
2. $\frac{m {l^{2}}_{0} \sin^{2} θ}{6}$
3. $\frac{2 m {l^{2}}_{0} \sin^{2} θ}{3}$
4. $\frac{m {l^{2}}_{0} \sin^{2} θ}{3}$
Two points A and B on a disc have velocities $v_{1}$ and $v_{2}$ , respectively at some moment. Their directions make angles 60° and 30°, respectively, with the line of separation as shown in figure. The angular velocity of disc is:
1. $\frac{\sqrt{3} v_{1}}{d}$
2. $\frac{v_{2}}{\sqrt{3} d}$
3. $\frac{v_{2} - v_{1}}{d}$
4. $\frac{v_{2}}{d}$
Two steel ball of equal diameter are connected by a rigid bar of negliible weight as shown and are dropped in the horizontal position from height h above the heavy steel and brass base plates. If the coefficient of restitution between the ball and steel base is 0.6 and that between the other ball and the brass base is 0.4. The angular velocity of the bar immediately after
rebound is. (Assume the two impacts are simultaneous).
1. $\frac{2}{5} \frac{r a d}{s e c}$
2. $\frac{2}{5} \frac{r a d}{s e c}$
3. $\frac{3}{5} \frac{r a d}{s e c}$
4. $\frac{1}{4} \frac{r a d}{s e c}$
A brick of length L is placed on the horizontal floor. The bricks of same length and size are placed on this brick, one above the other by providing a margin of L from the edge of the brick placed just below, in the same direction.
Find the correct option:
1. Fifth brick will fall down
2. Sixth brick alone will fall down
3. Sixth brick along with fifth brick will fall down
4. Fifth brick along with fourth brick will fall down.
A car moves with speed v on a horizontal circular track of radius R. A head-on view of the car is shown in figure. The height of the car’s centre of mass above the ground is h, and the separation between its inner and outer wheels
is d. The road is dry, and the car does not skid. The maximum speed the car can have without overturning is.
1. $\sqrt{\frac{g R d}{2 h}}$
2. $\sqrt{\frac{g R d}{h}}$
3. $\sqrt{\frac{2 g R d}{h}}$
4. $\sqrt{\frac{2 g R d}{3 h}}$
A thin horizontal uniform rod AB of mass m and length l can rotate freely about a vertical axis passing through its end A. At a certain moment, the end B starts experiencing a constant force F which is always perpendicular to
the original position of the stationary rod and directed in a horizontal plane. The angular velocity of the rod as a function of its rotation angle $θ$ measured relative to the initial position should be
1. $\sqrt{\frac{6 F \sin θ}{m l}}$
2. $\sqrt{\frac{2 F \sin θ}{m l}}$
3. $\sqrt{\frac{3 F \sin θ}{m l}}$
4. $\sqrt{\frac{5 F \sin θ}{m l}}$
A uniform disc of mass m is fitted (pivoted smoothly) with a rod of mass m/2. If the bottom of the rod is pulled with a velocity v, it moves without changing
its angle of orientation and the disc rolls without sliding. The kinetic energy of the system (rod + disc) is.
1. $\frac{m v^{2}}{4}$
2. $\frac{m v^{2}}{2}$
3. $\frac{5 m v^{2}}{16}$
4. $m v^{2}$