A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?

Separation of Motion of a system of particles into motion of the center of mass and motion about the center of mass:

A. Show

Where p_i is the momentum of the i^th particle (of mass m_i) and Note is the velocity of the i^th particle relative to the canter of mass.

Also, prove using the definition of the centre of mass

B. Show

Where K is the total kinetic energy of the system of particles, K’ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and MV²/2 is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.

C. Show

Where is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember; rest of the notation is the standard notation used in the chapter. Note and MR × V can be said to be angular momenta, respectively, about and of the canter of mass of the system of particles.

D. Show

Further, show that

Where is the sum of all external torques acting on the system about the canter of mass.

(Hint: Use the definition of canter of mass and Newton’s Third Law. Assume the internal forces between any two particles act along the line joining the particles.)

Find the ratio of the linear momenta of two particles of masses 1.0 kg and 4.0 kg if their kinetic energies are equal.

A man of mass 50 kg starts moving on the earth and acquires a speed of 1.8 m/s. With what speed does the earth recoil? Mass of earth = 6 × 10²⁴ kg.

Light in certain cases may be considered as a stream of particles called photons. Each photon has a linear momentum h/λ where h is the Planck’s constant and λ is the wavelength of the light. A beam of light of wavelength λ is incident on a plane mirror at an angle of incidence θ. Calculate the change in the linear momentum of a photon as the beam is reflected by the mirror.

A ball of mass m is dropped onto a floor from a certain height. The collision is perfectly elastic and the ball rebounds to the same height and again falls. Find the average force exerted by the ball on the floor during a long time interval.

A neutron initially at rest, decays into a proton, all electron and an antineutrino. The ejected electron has a momentum of 14 × 10^-28 kg-m/s and the antineutrino
6.4 × 10^-27 kg-m/s. Find the recoil speed of the proton (a) if the electron and the antineutrino are ejected along the same direction and (b) if they are ejected along perpendicular directions. Mass of the proton = 1.67 × X 10^-27 kg.

A ball of mass 0.50 kg moving at a speed of 5.0 m/s collides with another ball of mass 1.0 kg. After the collision the balls stick together and remain motionless. What was the velocity of the 1.0 kg block before the collision?

A block at rest explodes into three equal parts. Two parts start moving along X and Y axes respectively with equal speeds of 10 m/s. Find the initial velocity of the third part.

A railroad car of mass M is at rest on frictionless rails when a man of mass in starts moving on the car towards the engine. If the car recoils with a speed v backward on the rails, with what velocity is the man approaching the engine?

A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence 45^o. The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate (a) the magnitude of the change in momentum of the ball (b) the change in the magnitude of the momentum of the ball.