An object with a mass of #8 kg# is hanging from an axle with a radius of #15 cm#. If the wheel attached to the axle has a radius of #75 cm#, how much force must be applied to the wheel to keep the object from falling?

To keep the object from falling, the force applied to the wheel must counteract the gravitational force acting on the object. The gravitational force can be calculated using the formula F = mg, where F is the force, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s²). Therefore, the gravitational force acting on the object is:

F = (8 kg) * (9.8 m/s²) = 78.4 N.

To find the force required to keep the object from falling, we need to consider the torque exerted by the weight of the object. Torque (τ) is calculated as the product of force (F) and the distance (r) from the point of rotation (the axle) to the point where the force is applied (the rim of the wheel). The torque due to the weight of the object is:

τ = F * r = (78.4 N) * (0.15 m) = 11.76 Nm.

Since the radius of the wheel is 75 cm (or 0.75 m), the force required to counteract the torque is:

F = τ / r = (11.76 Nm) / (0.75 m) = 15.68 N.

Therefore, approximately 15.68 Newtons of force must be applied to the wheel to keep the object from falling.