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.)