Question

Example2 25k A solid sphere (mass M, radius R) is released from rest at the top of an inclined plane (angle ?). There is sufficient friction between the incline and the sphere to allow it to roll without slipping. (a) Draw and FBD for the sphere. (b) Find the linear acceleration of the sphere (c) Find the magnitude of the frictional force acting on the sphere. (d) Find the minimum required coefficient of friction to keep the sphere from slipping. (e) Work done by friction after it descends a length L along the track?

Answer 1

let the frictional force= f, Radius = R, mass = M ,

b) Mg sin $\theta$ - f = mA ( where A is the linera accleration)------eq(1)

Rf = I $\alpha$ (two expression of torques)

Rf = I ( A/R)

f = I ( A/R^2)-----eq(2)

usinf eq(1) and eq(2)

Mg sin $\theta$ - ( IA/ R^2) = MA

Mgsin$\theta$ = A ( M + I/R^2)

A = Mg sin $\theta$ / ( M + I/R^2)= Mgsin$\theta$ R^2 / ( MR^2 + I)

I = moment of inertia of sphere= 2/5 MR^2

A= 5/7 g sin $\theta$

c) frictional force= $\mu$ Mg cso $\theta$

d) for the ball to be kept away from slipping,

torque due to frictional force should be greater than or equal to tottal torque while rollinh

R ($\mu$Mg cso $\theta$ ) > =  I ( A/R)

$\mu$Mg cos $\theta$ > = ( 2/5 MR^2 ) ( A/R^2)

$\mu$g cos$\theta$) > = ( 2/5) ( A)

plugging, A= 5/7 g sin $\theta$

$\mu$g cos$\theta$ >= ( 2/5) ( 5/7) g sin $\theta$

$\mu$ cos $\theta$ > = ( 2/7) sin $\theta$

$\mu$> = ( 2/7) tan $\theta$

e) Work done= L $\mu$ Mg cos $\theta$