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A cylinder of mass $M$ is placed on top of a block of mass $2 M$ . Neglecting the friction between the block and the horizontal plane. But friction coefficient between the cylinder and the top surface of block is $μ$ . A small body of mass $M$ travelling with a velocity $v_{0}$ hits the block horizontally. Collision is completely elastic. If the maximum value of $v_{0}$ for which the cylinder starts pure rolling before it topples from the top of the block is ${(v_{0})}_{\max} = n \sqrt{m μ g d}$ . The value of $m \times n$ is ____

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Important Questions on Systems of Particles and Rotational Motion

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

An automobile moves on a road with a speed of

54 k m h^{- 1}

. The radius of its wheels is

0.45 m

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:

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

Point masses $m_{1}$ and $m_{2}$ are placed at the opposite ends of 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 $L$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $ω_{0}$ is minimum, is given by:
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HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

Consider a body of mass

1.0 kg

at rest at the origin at time

t = 0

. A force

\vec{F} = (α t \hat{i} + β \hat{j})

is applied on the body, where

α = 1.0 {Ns}^{- 1}

and

β = 1.0 N

. The torque acting on the body about the origin at time

= 1.0 s

\vec{τ}

. Which of the following statements is (are) true?

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A rope is wound around a hollow cylinder of mass

3 kg

and radius

40 cm.

What is the angular acceleration of the cylinder if the rope is pulled with a force of

30 N

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

The rope of a kite is wound around a hollow cylinder of mass

200 g

and radius

5 cm

. If the kite experiences a pull of

F = 1 N

, what will be the linear acceleration of the rope?

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

Which of the following statements are correct?

(i) Centre of mass of a body always coincides with the centre of gravity of the body.

(ii) Centre of mass of a body is the point at which the total gravitational torque on the body is zero

(iii) A couple on a body produce both translational and rotational motion in a body.

(iv) Mechanical advantage greater than one means that small effort can be used to lift a large load.

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

To mop-clean a floor, a cleaning machine presses a circular mop of radius

R

vertically down with a total force

F

and rotates it with a constant angular speed about its axis. If the force

F

is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is

μ,

the torque, applied by the machine on the mop is:

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A solid cylinder of mass

50 k g

and radius

0.5 m

is free to rotate about 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}

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A pulley of radius

1.5 m

is rotated about its axis by a force

F = (12 t - 3 t^{2}) N

applied tangentially (while

t

is measured in seconds). If moment of inertia of the pulley about its axis of rotation is

4.5 kg m^{2}

, the number of rotations made by the pulley before its direction of motion is reversed, will be

\frac{K}{π}

. The value of

K

is _____ .

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

The moment of the force,

\vec{F} = 4 \hat{i} + 5 \hat{j} - 6 \hat{k}

(2, 0, - 3),

about the point

(2, - 2, - 2),

is given by

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A rigid uniform bar

A B

of length

L

is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is

θ

. Which of the following statements about its motion is correct?

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HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A uniform disc of radius

R

is spinned to an angular velocity

ω_{o}

and then carefully placed on a horizontal surface. If pressure exerted by the disc on surface is regarded as uniform and the coefficient of friction is equal to

k

, the rotation of the disc will come to a stop in time

t

. The value of

t

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A wheel of radius

R

and mass

M

is placed at the bottom of a fixed step of height

R

as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque

τ

about an axis normal to the plane of the paper passing through the point

Q

. Which of the following options is/are correct?

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MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A slender uniform rod of mass

M

and length

l

is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle

θ

with the vertical is:

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MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A solid cylinder of mass

2 k g

and radius

4 c m

is rotating about its axis at the rate of

3 rpm

. The torque required to stop after

2 π

revolutions is

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A particle of mass

m

is moving along a trajectory given by

x = x_{0} + a c o s ω_{1} t

y = y_{0} + b s i n ω_{2} t

The torque, acting on the particle about the origin, at

t = 0

is:

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A metal coin of mass

5 g

and radius

1 c m

is fixed to a thin stick

A B

of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about

A B

25

rotations per second in

5 s

, is close to:
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MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A force of

40 N

acts on a point

B

at the end of an L-shaped object as shown in the figure. The angle

θ

that will produce the maximum moment of the force about point

A

is given by:
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EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A turbine fan in a jet engine has moment of inertia

2.5 kg m^{2}

about its axis of rotation. Its angular velocity is

40 t^{2}

. The net torque is

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Pure Rotational Motion

A uniform rod of length $l$ and mass $m$ is free to rotate in a vertical plane about $A$ . The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about $A$ is $\frac{m l^{2}}{3}$ )

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Important Questions on Systems of Particles and Rotational Motion

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