An object with a mass of #8 kg# is hanging from an axle with a radius of #30 m#. If the wheel attached to the axle has a radius of #95 m#, how much force must be applied to the wheel to keep the object from falling?

To calculate the force required to keep the object from falling, we can use the equation for torque:

Torque = Force × Distance

First, we need to find the torque exerted by the object hanging from the axle. The distance from the axle to the object is the radius of the axle, which is 30 m. The force exerted by the object is its weight, which can be calculated using the formula:

Weight = mass × gravity

where gravity is approximately 9.8 m/s^2.

Weight = 8 kg × 9.8 m/s^2 = 78.4 N

Now, we can calculate the torque exerted by the object:

Torque = Force × Distance = 78.4 N × 30 m = 2352 N·m

To counteract this torque and keep the object from falling, an equal and opposite torque must be applied to the wheel. The distance from the axle to the point where the force is applied is the radius of the wheel, which is 95 m. Therefore, we can rearrange the torque equation to solve for the force:

Force = Torque / Distance = 2352 N·m / 95 m ≈ 24.76 N

So, approximately 24.76 Newtons of force must be applied to the wheel to keep the object from falling.