AA spool of thread consists of a cylinder of radius R1 with end caps of radius R

Question

AA spool of thread consists of a cylinder of radius R1 with end caps of radius R2 as depicted in the end view shown in the figure below. The mass of the spool, including the thread, ism, and its moment of inertia about an axis through its center is I. The spool is placed on a rough, horizontal surface so that it rolls without slipping when a force acting to the right is applied to the free end of the thread. (a) Show that the magnitude of the friction force exerted by the surface on the spool is given by f =

Explanation / Answer

Here T is Acting Towards Right

The Frictional Force will act towards Left

ByConsidering the Linear Motion

f - T = ma

Therfore

a = (f - T)/m ------------(1)

By Considering the Torques in Rotation

T*R1 - f*R2 = I*alpha

As there is pure Rooling

Therefore

a = alpha*R2

Therefore

alpha = a/R2

So we get

T*R1 - f*R2 = I*(a/R2)

Now Put the Va;lue of a from Eq(1) ,we get

T*R1 - f*R2 = (I/R2)*((f - T)/m)

mTR1R2/I - mfR2^2/I = f - T

mTR1R2/I + T = f(1 + mR2^2/I)

T*(mR1R2 + I) = f*(I + mR^2)

f = ((I + mR1R2)/(I + mR2^2))*T

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