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 #45 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 find the circumference of the axle. The formula for the circumference of a circle is C = 2 * π * r, where C is the circumference and r is the radius. Substituting the given radius of the axle (5 cm) into the formula, we find that the circumference of the axle is 2 * π * 5 = 10π cm.

Next, we need to calculate the force required to turn the wheel. The force required is equal to the torque divided by the radius of the wheel. Torque (τ) is calculated by multiplying the force (F) by the radius of the axle (5 cm), τ = F * r. The weight of the object hanging from the axle provides the force. Since weight (W) is equal to mass (m) multiplied by gravity (g), and gravity is approximately 9.8 m/s^2, the force (F) is equal to the weight (m * g). Therefore, F = 8 kg * 9.8 m/s^2.

Now, we can calculate the torque (τ) using the formula τ = F * r, where F is the force and r is the radius of the axle. Substituting the force (F) and the radius of the axle (5 cm) into the formula, we find that the torque is equal to 8 kg * 9.8 m/s^2 * 0.05 m = 3.92 Nm.

Finally, we can calculate the work (W) required to turn the wheel a length equal to the circumference of the axle using the formula W = τ * θ, where τ is the torque and θ is the angle through which the wheel is turned. Since the wheel is turned a length equal to the circumference of the axle, the angle through which it is turned is 360 degrees (or 2π radians). Therefore, substituting the torque (3.92 Nm) and the angle (2π radians) into the formula, we find that the work required is approximately 3.92 Nm * 2π = 24.66 Nm.