An object with a mass of #5 kg# is hanging from an axle with a radius of #8 cm#. If the wheel attached to the axle has a radius of #32 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?

To calculate the work required to turn the wheel a length equal to the circumference of the axle, we first need to determine the distance the wheel needs to be turned. Since the circumference of a circle is equal to (2\pi \times \text{radius}), the distance the wheel needs to be turned is (2\pi \times 8 , \text{cm}). Once we have the distance, we can use the formula for work done in rotating an object, which is given by ( \text{work} = \text{torque} \times \text{angle of rotation}). The torque can be calculated by multiplying the force applied (due to the hanging object) by the radius of the wheel. Since the force is equal to the weight of the object (mass times gravitational acceleration), we have all the necessary information to compute the work. We can then plug in the values into the formula and calculate the work.

To find the work required to turn the wheel a length equal to the circumference of the axle, you can use the formula for work, which is given by the product of force and displacement. In this case, the force exerted is the tension in the rope supporting the object, and the displacement is the distance the rope moves.

First, calculate the circumference of the axle using its radius, which is 8 cm.

Circumference of the axle = 2π(radius of axle) Circumference of the axle = 2π(8 cm) Circumference of the axle ≈ 50.27 cm

Now, calculate the work done using the formula:

Work = Force × Distance

The force required to turn the wheel is the tension in the rope, which is equal to the weight of the object hanging from it. The weight of the object can be found using the formula:

Weight = mass × gravitational acceleration

Given that the mass of the object is 5 kg and gravitational acceleration is approximately 9.8 m/s², we have:

Weight = 5 kg × 9.8 m/s² Weight ≈ 49 N

To convert Newtons to joules (the unit of work), we need to multiply by the distance moved. The distance moved is equal to the circumference of the axle, which we calculated earlier.

Work = Force × Distance Work = 49 N × 50.27 cm Work ≈ 2451.23 N·cm or 24.51 J