Advertisements
Advertisements
Question
A solid sphere rolling on a rough horizontal surface with a linear speed ν collides elastically with a fixed, smooth, vertical wall. Find the speed of the sphere after it has started pure rolling in the backward direction.
Solution
Consider two cases:-
(a) Just after the collision with the wall, the sphere rebounds with velocity \[\nu\] towards left but it continues to rotate in the clockwise direction (as shown in figure)
Angular momentum of the sphere about centre,
\[L = mvR - I\omega\]
\[ = mvR - \frac{2}{5}m R^2 \times \frac{\nu}{R}\]
\[ = \frac{3}{5}mvR\]
(b) When pure rolling starts
Let the velocity be \[\nu'\] and the corresponding angular velocity be \[\frac{v'}{R}.\]
\[L' = mv'R + I\omega'\]
\[ = mv'R + \frac{2}{5}m R^2 \times \frac{v'}{R}\]
\[ = \frac{7}{5}mv'R\]
Angular momentum is constant.
Therefore, we have
\[L = L'\]
\[ \Rightarrow v' = \frac{3v}{7}\]
So, the sphere will move with velocity \[\frac{3v}{7}.\]
APPEARS IN
RELATED QUESTIONS
A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m2.
(a) What is his new angular speed? (Neglect friction.)
(b) Is kinetic energy conserved in the process? If not, from where does the change come about?
A ball is whirled in a circle by attaching it to a fixed point with a string. Is there an angular rotation of the ball about its centre? If yes, is this angular velocity equal to the angular velocity of the ball about the fixed point?
If the angular momentum of a body is found to be zero about a point, is it necessary that it will also be zero about a different point?
A body is uniformly rotating about an axis fixed in an inertial frame of reference. Let \[\overrightarrow A\] be a unit vector along the axis of rotation and \[\overrightarrow B\] be the unit vector along the resultant force on a particle P of the body away from the axis. The value of \[\overrightarrow A.\overrightarrow B\] is _________.
A particle moves with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin ____________.
A person sitting firmly over a rotating stool has his arms stretched. If he folds his arms, his angular momentum about the axis of rotation ___________ .
A particle moves on a straight line with a uniform velocity. Its angular momentum __________ .
(a) is always zero
(b) is zero about a point on the straight line
(c) is not zero about a point away from the straight line
(d) about any given point remains constant.
If there is no external force acting on a nonrigid body, which of the following quantities must remain constant?
(a) angular momentum
(b) linear momentum
(c) kinetic energy
(d) moment of inertia.
A wheel rotating with uniform angular acceleration covers 50 revolutions in the first five seconds after the start. Find the angular acceleration and the angular velocity at the end of five seconds.
A disc rotates about its axis with a constant angular acceleration of 4 rad/s2. Find the radial and tangential accelerations of a particle at a distance of 1 cm from the axis at the end of the first second after the disc starts rotating.
Two particles of masses m1 and m2 are joined by a light rigid rod of length r. The system rotates at an angular speed ω about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is \[L = \mu r^2 \omega\] where \[\mu\] is the reduced mass of the system defined as \[\mu = \frac{m_1 + m_2}{m_1 + m_2}\]
A uniform rod of mass m and length l is struck at an end by a force F perpendicular to the rod for a short time interval t. Calculate (a) the speed of the centre of mass, (b) the angular speed of the rod about the centre of mass, (c) the kinetic energy of the rod and (d) the angular momentum of the rod about the centre of mass after the force has stopped to act. Assume that t is so small that the rod does not appreciably change its direction while the force acts.
Two small balls A and B, each of mass m, are joined rigidly by a light horizontal rod of length L. The rod is clamped at the centre in such a way that it can rotate freely about a vertical axis through its centre. The system is rotated with an angular speed ω about the axis. A particle P of mass m kept at rest sticks to the ball A as the ball collides with it. Find the new angular speed of the rod.
Suppose the rod with the balls A and B of the previous problem is clamped at the centre in such a way that it can rotate freely about a horizontal axis through the clamp. The system is kept at rest in the horizontal position. A particle P of the same mass m is dropped from a height h on the ball B. The particle collides with B and sticks to it. (a) Find the angular momentum and the angular speed of the system just after the collision. (b) What should be the minimum value of h so that the system makes a full rotation after the collision.
A solid sphere is set into motion on a rough horizontal surface with a linear speed ν in the forward direction and an angular speed ν/R in the anticlockwise directions as shown in the following figure. Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.
