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NeTakaya
3 years ago
11

A solid ball of inertia m rolls without slipping down a ramp that makes an angle θ with the horizontal.

Physics
1 answer:
LekaFEV [45]3 years ago
8 0

Answer:

Part a)

f = \frac{2}{7}mgsin\theta

Part b)

\mu = \frac{2}{7} gtan\theta

Explanation:

Part a)

Force equation on the inclined plane is given as

mgsin\theta - f = ma

now for torque equation of the ball

fR = \frac{2}{5}mR^2 ( \frac{a}{R})

f = \frac{2}{5}ma

now from above two equations

mg sin\theta - \frac{2}{5}ma = ma

mg sin\theta = \frac{7}{5} ma

a = \frac{5}{7} gsin\theta

so frictional force is given as

f = \frac{2}{5}ma

f = \frac{2}{5}m(\frac{5}{7}gsin\theta)

f = \frac{2}{7}mgsin\theta

Part b)

Also we know that in the normal direction of the motion we have

F_n = mgcos\theta

so we have

f = \mu F_n

\frac{2}{7} mg sin\theta = \mu (mg cos\theta)

now we have

\mu = \frac{2}{7} gtan\theta

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In absence of friction, the total mechanical energy E remains constant. So we have:

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