Rotational Dynamics
A uniform sphere of mass $$500g$$ rolls without slipping on a plane horizontal surface with its centre moving at a speed of $$5.00cm/s$$. Its kinetic energy is:
the distance covered by the particle while the force acted.
Again from the equation $$\vec{F}=m\vec{w}$$
$$\vec{at}(\tau-t)=m\dfrac{d\vec{v}}{dt}$$
or, $$\vec{a}(t\tau-t^{2})dt=md\vec{v}$$
Integrating within the limits for $$\vec{v}(t)$$,
$$\displaystyle\int_{0}^{t}\vec{a}(t\tau-t^{2})dt=m\int_{0}^{\vec{v}}d\vec{v}$$
or, $$\vec{v}=\dfrac{\vec{a}}{m}\left(\dfrac{\tau ^{2}}{3}-\dfrac{t^{3}}{3}\right)=\dfrac{\vec{a}t^{2}}{m}\left(\dfrac{\tau}{2}-\dfrac{t}{3}\right)$$
Thus $$v=\dfrac{at^{2}}{m}\left(\dfrac{\tau}{2}-\dfrac{t}{3}\right)$$ for $$t\le \tau$$
Hence distance covered during the time interval $$t=\tau$$,
$$s=\displaystyle\int_{0}^{\tau}v\ dt$$
$$=\displaystyle\int_{0}^{\tau}\dfrac{a\ t^{2}}{m}\left(\dfrac{\tau}{2}-\dfrac{t}{3}\right)dt=\dfrac{a}{m}\dfrac{\tau^{4}}{12}$$
A uniform sphere of mass $$500g$$ rolls without slipping on a plane horizontal surface with its centre moving at a speed of $$5.00cm/s$$. Its kinetic energy is:
A hoop of radius $$r$$ and mass $$m$$ rotating with an angular velocity $$\omega_{o}$$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?
A bowler throws a bowling a lane. The ball slides on the lane with initial speed $$v_{com.0}=8.5\ m/s$$ and initial angular speed $$\omega _0 =0$$. The coefficient of kinetic friction between the ball and the lane is $$0.21$$. The kinetic friction force $$\vec f_{k}$$ acting on the ball causes an angular acceleration of the ball. When speed $$v_{com}$$ has decreases enough and angular speed $$\omega$$ has increased enough, the ball stops sliding and then rolls smoothly. What then is $$v_{com} $$ in term of $$\omega$$? During the sliding, what are the ball's.
A bowler throws a bowling a lane. The ball slides on the lane with initial speed $$v_{com.0}=8.5\ m/s$$ and initial angular speed $$\omega _0 =0$$. The coefficient of kinetic friction between the ball and the lane is $$0.21$$. The kinetic friction force $$\vec f_{k}$$ acting on the ball causes an angular acceleration of the ball. When speed $$v_{com}$$ has decreases enough and angular speed $$\omega$$ has increased enough, the ball stops sliding and then rolls smoothly. Linear acceleration
A uniform wheel of mass $$10.0\ kg$$ and radius $$0.400\ m$$ is mounted rigidly on a massless axle through its center. The radius of the axle is $$0.200\ m$$, and the rotational inertia of the wheel-axle combination about its central axis is $$0.600\ kg.m^2$$.The wheel is initially at rest at the top of a surface that is inclined at angle $$\theta =30.0^{o}$$ with the horizontal; the axle rests on the surface while he wheel extends into a groove in the surface without touching the surface. Once released, the axle rolls down along he surface smoothly and without slipping. When the wheel-axle combination has moved down the surface by $$2.00\ m$$, what areits translational kinetic energy?
A solid homogeneous sphere is moving on a rough horizontal surface, partially sliding. During this kind of motion of sphere
A thin spherical shell of radius $$R$$ lying on a rough horizontal surface is hit sharply and horizontally by a cue. At what height from the ground should it be hit so that the shell does not slip on the surface.
A particle of mass m comes down on a smooth inclined plane from point B at a height of h from rest. The magnitude of change in momentum of the particle between position A(just before arriving on horizontal surface) and C(assuming the angle of inclination of the plane as $$\theta$$ with respect to the horizontal) is?
A bullet of mass $$m$$ moving horizontally with velocity $$u$$ sticks to the top of a solid cylinder of mass $$M$$ and radius $$R$$ resulting on a rough horizontal surface as shown. (Assume cylinder rolls without slipping). Angular velocity of cylinder is
Two rigid bodies $$A$$ and $$B$$ rotate with angular momenta $${L}_{A}$$ and $${L}_{B}$$ respectively. The moments of inertia of $$A$$ and $$B$$ about the axes of rotation are $${I}_{A}$$ and $${I}_{B}$$ respectively. If $${ I }_{ A }={ I }_{ B }/4$$ and $${ L }_{ A }=5{ L }_{ B }$$. Then the ratio of rotational kinetic energy $${K}_{A}$$ of $$A$$ to the rotational energy $${K}_{B}$$ of $$B$$ is given by