An object with a mass of #24 kg# is hanging from an axle with a radius of #12 m#. If the wheel attached to the axle has a radius of #48 m#, how much force must be applied to the wheel to keep the object from falling?

The force applied to the wheel to prevent the object from falling is approximately 294 N.

To prevent the object from falling, the force applied to the wheel must counteract the gravitational force acting on the object. The force needed can be calculated using the formula for gravitational force:

[ F = mg ]

Where:

• ( F ) is the gravitational force,
• ( m ) is the mass of the object, and
• ( g ) is the acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 )).

The gravitational force acting on the object is ( F = 24 , \text{kg} \times 9.8 , \text{m/s}^2 ).

Once we have the gravitational force, we can determine the torque required to keep the object from falling using the formula:

[ \text{Torque} = \text{Force} \times \text{Lever Arm} ]

Given that the radius of the wheel is 48 m, the lever arm is 48 m. Thus, the torque needed is the gravitational force multiplied by the lever arm.

Finally, since torque is force multiplied by distance, the force needed to counteract the gravitational force at the wheel's radius (48 m) is the torque divided by the radius of the wheel (48 m).

So, the force required to keep the object from falling is:

[ \text{Force} = \frac{\text{Gravitational Force} \times \text{Lever Arm}}{\text{Wheel Radius}} ]

[ \text{Force} = \frac{(24 , \text{kg} \times 9.8 , \text{m/s}^2) \times 48 , \text{m}}{48 , \text{m}} ]

[ \text{Force} = (24 , \text{kg} \times 9.8 , \text{m/s}^2) ]

[ \text{Force} = 235.2 , \text{N} ]

Therefore, approximately 235.2 Newtons of force must be applied to the wheel to keep the object from falling.