Rotational Statics- Physics

Question

A wheel of diameter 70 cm has an axle of diameter 8 cm. A force of 130 N is exerted along the rim of the wheel. What force should be applied to the outside of the axle in order to prevent the wheel from rotating? Assume the direction of this force to be tangential to the axle. Answer in units of N.

Answer 1

τ = rFsinθ

Since forces are acting tangential to the rim and axle, θ = 90°. We know that sin(90°) = 1.

Στ = 0 = .35m (130N) sin(90°) + .04m (F_axle) sin(90°)

F_axle = - (.35m * 130N) / .04m = -1137.5N

Answer 2

In order to prevent the wheel from rotating, the net torque must be zero, so the torque applied to the outside of the axle must equal the torque applied to the rim of the wheel. The wheel has radius 0.35 meters, so the torque applied to the rim of the wheel is 130 * 0.35 = 45.5 Newton meters. The axle has radius 0.04 meters, so the torque applied is 0.04 * x where x is the magnitude of the force. Set the two torques equal to each other, 45.5 = 0.04 * x, so x = 1137.5 Newtons.