Assam Board Solutions for Chapter: Systems of Particles and Rotational Motion, Exercise 1: EXERCISES

Read the statement below carefully, and state, with reasons, if it is true or false:

During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.

Read the statement below carefully, and state, with reasons, if it is true or false:

The instantaneous speed of the point of contact during pure rolling is zero.

Read the statement below carefully, and state, with reasons, if it is true or false:

The instantaneous acceleration of the point of contact during rolling is zero.

Read the statement below carefully, and state, with reasons, if it is true or false:

For perfect rolling motion, work done against friction is zero.

Read the statement below carefully, and state, with reasons, if it is true or false:

A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

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

Show $K=K^{\text{'}}+\frac{1}{2}M{V}^2$
where $K$ is the total kinetic energy of the system of particles, $K\text{'}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $\frac{1}{2}M{V}^2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).

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

Show $L=L^{\text{'}}+R\times MV$
where $L^{\text{'}}=\Sigma r_i^{\text{'}}\times p_i^{\text{'}}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r_i^{\text{'}}=r_i-R$, $r_i$ is the position of $i^{th}$ particle with respect to origin and $R$ and $V$ is the position and velocity of centre of mass with respect to origin, respectively.

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

Show $\frac{d{L}^{\text{'}}}{dt}=\Sigma r_i^{\text{'}}\times \frac{d{p}^{\text{'}}}{dt}$
Further, show that
$\frac{d{L}^{\text{'}}}{dt}={\tau }_{ext}^{\text{'}}$
where ${\tau }_{ext}^{\text{'}}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)