NCERT Solutions for Chapter: Systems of Particles and Rotational Motion, Exercise 5: LA

Find the centre of mass of a uniform $\left(a\right)$ half-disc, $\left(b\right)$ quarter-disc.

A disc of radius $R$ is rotating with an angular speed ${\omega }_0$ about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is ${\mu }_k$.

What condition should be satisfied for rolling to begin?

A disc of radius $R$ is rotating with an angular speed ${\omega }_0$ about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is ${\mu }_k$.

Calculate the time taken for the rolling to begin.

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

Show the frictional forces just after contact.

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

Identify forces and torques external to the system just after contact.

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

What would be the ratio of final angular velocities when friction ceases?

A uniform square plate $S$ (side $c$ ) and a uniform rectangular plate $R$ (sides $b, a$) have identical areas and masses (Figure).

Show that
$\left(a\right)\frac{I_{xR}}{I_{xS}}<1$
$\left(b\right)\frac{I_{yR}}{I_{yS}}>1$
$\left(c\right)\frac{I_{zR}}{I_{zS}}>1$

A uniform disc of radius $R$, is resting on a table on its rim. The coefficient of friction between disc and table is $\mu$ (Figure).

Now the disc is pulled with a force $\overrightarrow{F}$ as shown in the figure. What is the maximum value of $F$ for which the dise rolls without slipping?