An object with a mass of #8 kg# is hanging from an axle with a radius of #5 cm#. If the wheel attached to the axle has a radius of #15 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 can use the formula for work done by rotational force:

Work = Torque * Angle

The torque (τ) exerted on the wheel is equal to the force applied (T) multiplied by the radius of the wheel (r_wheel):

τ = T * r_wheel

The force applied (T) can be found using the formula for torque:

τ = F * r_axle

Where F is the force applied at the edge of the wheel, and r_axle is the radius of the axle.

Given that the object hanging from the axle creates a force due to its weight, we can find the force using:

F = m * g

Where m is the mass of the object and g is the acceleration due to gravity.

Finally, we need to find the angle through which the wheel is turned, which is given by:

Angle = Distance / Radius_wheel

Given that the distance is the circumference of the axle, and the radius of the axle is provided, we can find the angle.

Putting it all together:

Work = (m * g * r_axle) * r_wheel * (Circumference_axle / r_wheel)

Where Circumference_axle = 2 * π * r_axle

Plugging in the values:

Work = (m * g * r_axle) * r_wheel * (2 * π * r_axle / r_wheel)

Now, we can substitute the given values and solve for the work.