Question $6(25$ marks$)$

A uniform hollowed sphere of mass $M$ and radius $R$ is set into rotation with angular speed ${}_0$ and linear speed

$u($with $u>{}_{0}R),$ and is then placed gently at time $t=0$ on a rough inclined plane at an angle $$ to the

horizontal as shown in Figure $6.$ Suppose that the coefficient of $($dynamic$)$ friction between the sphere and the

plane is $.$ The moment of inertia of the uniform hollowed sphere about the axis passing through its centre is

$I_{C.G.}=M{R}^2$ with $>0.$ Neglect the air resistance.

Let $x=x(t)$ and $=(t)$ be the magnitude of the displacement of the centre of the sphere and the angle rotated

through by the hollowed sphere measured from its initial position at time $t.$ If necessary, let $+$i and $+j$ be the

unit vectors along the up$-$slope of and the upward direction perpendicular to the inclined plane respectively.

$($a$)(3$ marks$)$ Draw a force diagram showing all the external forces acting on the hollowed sphere. Please give

the name of each force and briefly define each newly introduced symbol if it exists on the force diagram

that you draw.

$($b$)(4$ marks$)$ Find the equation of the linear motion of the centre of mass of the hollowed sphere in terms of

$g,,.$

$($c$)(2$ marks$)$ Find the equation of the rotational motion of the hollowed sphere about its centre of mass in

terms of $g,,,R,.$

$($d$)(4$ marks$)$ Write down all the initial conditions of $x$ and $$ and their time derivatives. Hence, show that

when the sphere is still rolling with slipping,

$x(t)=ut-\frac{1}{2}g{t}^2[sin()+cos()]$

$(t)={}_{0}t+\frac{g{t}^2}{2R}cos()$

$($e$)(2$ marks$)$ Show that the hollowed sphere first begins to roll on the inclined plane without slipping after

a time

$t=\frac{(u-{}_{0}R)}{g[(+1)cos()+sin()]}.$

has elapsed.

$($f$)(2$ marks$)$ Using the previous result$($s$),$ or otherwise, find the magnitude of the linear displacement that

the sphere slips, and the angular speed of the sphere at the moment of rolling without slipping.

$($g$)(6$ marks$)$ Show that the magnitude of the maximum linear displacement of the centre of mass of the

sphere measured from its initial position is

$x_{\text{m}\text{a}\text{x}}=\frac{\{(R^2{}_0^2+u^2)+cot()({}_{0}R+u{)}^2\}}{2gsin()[(+1)cot()+]}.$

Also, find the corresponding time of attaining that maximum displacement $($measured from its initial

moment$).$

$($h$)(2$ marks$)$ Show that the total energy loss of the sphere during the process of rotation with slipping is

$\frac{M(u-{}_{0}R{)}^2}{2[(+1)+tan()]}.$