An object with a mass of #8 kg# is hanging from an axle with a radius of #25 cm#. If the wheel attached to the axle has a radius of #35 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?

The work done to turn the wheel a length equal to the circumference of the axle is ( W = F \times d ), where ( F ) is the force applied and ( d ) is the distance moved in the direction of the force. The force required to turn the wheel is equal to the weight of the object hanging from the axle, which is ( F = mg ), where ( m ) is the mass and ( g ) is the acceleration due to gravity. Given ( m = 8 , \text{kg} ) and ( g = 9.8 , \text{m/s}^2 ), the force is ( F = 8 , \text{kg} \times 9.8 , \text{m/s}^2 = 78.4 , \text{N} ). The distance moved is equal to the circumference of the axle, which is ( d = 2\pi r ), where ( r ) is the radius of the axle. Given ( r = 25 , \text{cm} = 0.25 , \text{m} ), the distance moved is ( d = 2\pi \times 0.25 , \text{m} = 1.57 , \text{m} ). Therefore, the work done is ( W = 78.4 , \text{N} \times 1.57 , \text{m} = 123.088 , \text{J} ).