An object with a mass of #3 kg# is hanging from an axle with a radius of #2 m#. If the wheel attached to the axle has a radius of #12 m#, how much force is needed to raise the object?

To calculate the force needed to raise the object, we can use the principle of torque. The torque applied to the wheel must overcome the torque due to the weight of the hanging object.

Torque (τ) = Force (F) × Distance (r)

The distance from the axle to where the force is applied (radius of the wheel) is 12 m. The distance from the axle to where the object is hanging is 2 m.

The torque due to the weight of the object is given by:

τ_weight = Mass (m) × Gravitational acceleration (g) × Distance (r)

Substituting the given values:

τ_weight = 3 kg × 9.8 m/s² × 2 m = 58.8 N·m

The torque needed to overcome the weight of the object is equal to the torque due to the weight of the object. So, the force needed to raise the object is:

F = τ_weight / r_wheel

Substituting the values:

F = 58.8 N·m / 12 m = 4.9 N

To raise the object, we need to overcome the force of gravity acting on it. The force needed to raise the object is equal to the gravitational force acting on it, which can be calculated using the formula:

Force = mass × gravitational acceleration

Gravitational acceleration (g) is approximately 9.8 m/s².

Given that the mass of the object is 3 kg, the force needed to raise it is:

Force = 3 kg × 9.8 m/s² = 29.4 N

Therefore, a force of 29.4 Newtons is needed to raise the object.