You place a solid cylinder of mass M on a ramp that is inclined at an angle β to the horizontal. The coefficient of static friction for the cylinder on the ramp is μs. If the cylinder rolls downhill without slipping, what is the magnitude of the friction force that the ramp exerts on the cylinder? Express your answer in terms of the variables β, M and acceleration due to gravity g.

Let the frictional force required be f.

frictional force is responsible in creating rotational motion in the cylinder.

torque created by frictional force = f R

if angular acceleration be α

α = f R / I, I is moment of inertia of cylinder.

α = a / R, a is linear acceleration.

f R / I = a / R

a = f R² / I

linear acceleration a of cylinder down the slope

ma = mgsinθ - f, (f force is acting upwards and mgsinθ is acting downwards)

mf R² / I = mgsinθ - f

f (m R² / I + 1) = mgsinθ

f = mgsinθ / (m R² / I + 1)

= mgsinθ / (m R² / mk² + 1), k is radius of gyration of cylinder.

= mgsinθ / (R² / k² + 1)

Putting the given values

f = Mgsinβ / (R² / k² + 1)

for cylinder, R² / k² = 2

f = Mgsinβ / 3