An object with a mass of #7 kg# is hanging from an axle with a radius of #15 cm#. If the wheel attached to the axle has a radius of #125 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?

The work required is 16,343.24 joules.

To calculate the work required to turn the wheel a length equal to the circumference of the axle, we need to determine the force required to overcome the gravitational force acting on the hanging object, and then use this force to calculate the work done.

First, let's find the gravitational force acting on the object: [ F_{\text{gravity}} = m \cdot g ] where: ( m = 7 ) kg (mass of the object) ( g = 9.81 , \text{m/s}^2 ) (acceleration due to gravity)

[ F_{\text{gravity}} = 7 , \text{kg} \times 9.81 , \text{m/s}^2 ] [ F_{\text{gravity}} = 68.67 , \text{N} ]

Next, we need to find the torque required to turn the wheel. Torque (( \tau )) is given by: [ \tau = F \cdot r ] where: ( F ) is the force applied ( r ) is the radius

Since the force required to turn the wheel is equal to the gravitational force acting on the object, we use ( F_{\text{gravity}} = 68.67 , \text{N} ). We'll use the radius of the axle, ( r_{\text{axle}} = 0.15 ) m.

[ \tau = 68.67 , \text{N} \times 0.15 , \text{m} ] [ \tau = 10.3 , \text{Nm} ]

Finally, we calculate the work done using the formula: [ \text{Work} = \tau \cdot \theta ] where: ( \tau ) is the torque ( \theta ) is the angle turned in radians

Since the length equal to the circumference of the axle corresponds to one full revolution, ( \theta = 2\pi ).

[ \text{Work} = 10.3 , \text{Nm} \times 2\pi ] [ \text{Work} \approx 64.6 , \text{J} ]

Therefore, it would take approximately 64.6 joules of work to turn the wheel a length equal to the circumference of the axle.