A 31.4-kg wheel with radius 1.21 m is rotating at 283 rev/min. It must be brought to a stop in 14.8 s. Find the required average power. Assume the wheel to be a thin hoop.

To determine the average power required to stop a rotating wheel, we need to calculate the wheel's initial kinetic energy and the work done to bring it to a stop. Let's break this down step by step.

Step 1: Understanding the Wheel's Properties

The wheel is given as a thin hoop, which means we can use the formula for the moment of inertia of a hoop. The moment of inertia (I) for a thin hoop is calculated as:

• I = m * r²

Where:

• m = mass of the wheel = 31.4 kg
• r = radius of the wheel = 1.21 m

Substituting in the values:

• I = 31.4 kg * (1.21 m)²
• I = 31.4 kg * 1.4641 m²
• I ≈ 46.05 kg·m²

Step 2: Calculating Angular Velocity

Next, we need to convert the wheel's rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). The formula for this conversion is:

• ω = (rev/min) * (2π rad/rev) * (1 min/60 s)

Plugging in the given rotational speed:

• ω = 283 rev/min * (2π rad/rev) * (1 min/60 s)
• ω ≈ 29.57 rad/s

Step 3: Finding the Initial Kinetic Energy

The kinetic energy (KE) of the rotating wheel can be calculated using the formula:

• KE = 0.5 * I * ω²

Substituting the moment of inertia and angular velocity:

• KE = 0.5 * 46.05 kg·m² * (29.57 rad/s)²
• KE ≈ 0.5 * 46.05 * 874.6
• KE ≈ 20153.66 J

Step 4: Calculating Average Power

Power is defined as the rate of doing work or the rate of energy transfer. To find the average power (P) required to stop the wheel, we can use the formula:

• P = Work done / Time

In this case, the work done is equal to the initial kinetic energy of the wheel since we are bringing it to a stop:

• P = KE / t

Given that the time (t) to stop the wheel is 14.8 seconds:

• P = 20153.66 J / 14.8 s
• P ≈ 1361.64 W

Final Calculation

Thus, the average power required to bring the wheel to a stop is approximately:

This means you would need to exert an average power of about 1362 watts to stop the wheel in the given time frame. This example illustrates how we can connect mass, radius, rotational speed, and time to find power in a rotational system.