###### Advertisements

###### Advertisements

A disc rotating about its axis with angular speed ω_{o}is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is *R*. What are the linear velocities of the points A, B and C on the disc shown in Figure? Will the disc roll in the direction indicated?

###### Advertisements

#### Solution 1

*v*_{A} = *R*ω_{o}; *v*_{B} = *R*ω_{o}; `v_c = (R/2)omega_o`

The disc will not roll Angular speed of the disc = ω_{o}

Radius of the disc = *R*

Using the relation for linear velocity, *v* = ω_{o}*R*

__For point A:__

*v*_{A} = *R*ω_{o}; in the direction tangential to the right

__For point B:__

*v*_{B} = *R*ω_{o}; in the direction tangential to the left

__For point C:__

`v_c = (R/2)omega_o` in the direction same as that of *v*_{A}

The directions of motion of points A, B, and C on the disc are shown in the following figure

Since the disc is placed on a frictionless table, it will not roll. This is because the presence of friction is essential for the rolling of a body.

#### Solution 2

Since `v = romega`

For Point A, `v_A = Romega_0` in the direction of arrow

For point B, `v_B = Romega_0` in the opposite direction of arrow

For point C, `v_C = R/2omega_0` in the direction of arrow

The disc will not roll in the given direction because friction is necessary for the same

#### APPEARS IN

#### RELATED QUESTIONS

The moon rotates about the earth in such a way that only one hemisphere of the moon faces the earth (see the following figure). Can we ever see the "other face" of the moon from the earth? Can a person on the moon ever see all the faces of the earth?

A sphere rolls on a horizontal surface. If there any point of the sphere which has a vertical velocity?

A body is in pure rotation. The linear speed \[\nu\] of a particle, the distance *r* of the particle from the axis and the angular velocity \[\omega\] of the body are related as \[\omega = \frac{\nu}{r}.\] Thus

The following figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting A and B do not slip on the wheels. If x and y be the distance travelled by A and B in the same time interval, then _________ .

The angular velocity of the engine (and hence of the wheel) of a scooter is proportional to the petrol input per second. The scooter is moving on a frictionless road with uniform velocity. If the petrol input is increased by 10%, the linear velocity of the scooter is increased by ___________ .

A sphere is rolled on a rough horizontal surface. If gradually slows down and stops. The force of friction tries to

(a) decrease the linear velocity

(b) increase the angular velocity

(c) increase the linear momentum

(d) decrease the angular velocity.

A disc of radius 10 cm is rotating about its axis at an angular speed of 20 rad/s. Find the linear speed of the middle point of a radius.

A block hangs from a string wrapped on a disc of radius 20 cm free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is 10 rad/s at some instant, with what speed is the block going down at that instant?

A wheel rotating at a speed of 600 rpm (revolutions per minute) about its axis is brought to rest by applying a constant torque for 10 seconds. Find the angular deceleration and the angular velocity 5 seconds after the application of the torque.

A wheel of mass 10 kg and radius 20 cm is rotating at an angular speed of 100 rev/min when the motor is turned off. Neglecting the friction at the axle, calculate the force that must be applied tangentially to the wheel to bring it to rest in 10 revolutions.

A cylinder rotating at an angular speed of 50 rev/s is brought in contact with an identical stationary cylinder. Because of the kinetic friction, torques act on the two cylinders accelerating the stationary one and decelerating the moving one. If the common magnitude of the acceleration and deceleration be one revolution per second square, how long will it take before the two cylinders have equal angular speed?

A dumb-bell consists of two identical small balls of mass 1/2 kg each connected to the two ends of a 50 cm long light rod. The dumb-bell is rotating about a fixed axis thorough the centre of the rod and perpendicular to it at an angular speed of 10 rad/s. An impulsive force of average magnitude 5⋅0 N acts on one of the masses in the direction of its velocity for 0⋅10 s. Find the new angular velocity of the system.

A boy is standing on a platform which is free to rotate about its axis. The boy holds an open umbrella in his hands. The axis of the umbrella coincides with that of the platform. The moment of inertia of "the platform plus the boy system" is 3⋅0 × 10^{−3} kg-m^{2} and that of the umbrella is 2⋅0 × 10^{−3} kg-m^{2}. The boy starts spinning the umbrella about the axis at an angular speed of 2⋅0 rev/s with respect to himself. Find the angular velocity imparted to the platform.

Suppose the platform of the previous problem is brought to rest with the ball in the hand of the kid standing on the rim. The kid throws the ball horizontally to his friend in a direction tangential to the rim with a speed \[\nu\] as seen by his friend. Find the angular velocity with which the platform will start rotating.

A ball falls on the ground from a height of 2.0 m and rebounds up to a height of 1.5 m. Find the coefficient of restitution.

The variation of angular position θ, of a point on a rotating rigid body, with time t is shown in figure. Is the body rotating clock-wise or anti-clockwise?

Two cylindrical hollow drums of radii R and 2R, and of a common height h, are rotating with angular velocities ω(anti-clockwise) and ω(clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by (3R + δ). They are now brought in contact (δ → 0).

- Show the frictional forces just after contact.
- Identify forces and torques external to the system just after contact.
- What would be the ratio of final angular velocities when friction ceases?