Question

A wheel rotates without friction about a stationary horizontal axis at the center of the wheel. A constant tangential force equal to 80.0 N is applied to the rim of the wheel. The wheel has radius 0.120 m. Starting from rest, the wheel has an angular speed of 12.0 rev/s after 2.00 s. What is the moment of inertia of the wheel?

Short answer

The moment of inertia of the wheel is approximately 0.254 kg m².

Step-by-step solution

Identify Given Information

We know that a tangential force of \( F = 80.0 \text{ N} \) is applied to a wheel with radius \( r = 0.120 \text{ m} \). The final angular speed is \( \omega = 12.0 \text{ rev/s} \), which needs to be converted to radians per second. Since 1 revolution is \( 2\pi \) radians, \( \omega = 12.0 \times 2\pi = 24\pi \text{ rad/s} \). The wheel accelerates from rest, so initial angular speed \( \omega_0 = 0 \text{ rad/s} \), in a time \( t = 2.00 \text{ s} \).

Calculate Angular Acceleration

The angular acceleration \( \alpha \) can be calculated using the formula \( \alpha = \frac{\omega - \omega_0}{t} \). Substituting the known values, \( \alpha = \frac{24\pi - 0}{2} = 12\pi \text{ rad/s}^2 \).

Relate Force to Torque

Torque (\( \tau \)) is related to force by the equation \( \tau = rF \). Substituting the known values: \( \tau = 0.120 \times 80.0 = 9.6 \text{ Nm} \).

Relate Torque to Angular Acceleration

Torque is also related to angular acceleration and the moment of inertia \( I \) by the equation \( \tau = I\alpha \). Therefore, the moment of inertia \( I \) can be calculated as \( I = \frac{\tau}{\alpha} \).

Calculate the Moment of Inertia

Substitute the known values of torque and angular acceleration into the equation \( I = \frac{\tau}{\alpha} \): \( I = \frac{9.6}{12\pi} \). This simplifies to \( I \approx 0.254 \text{ kg m}^2 \).

Key concepts

Angular Acceleration

Angular acceleration measures how quickly an object is speeding up or slowing down its rotation, similar to how linear acceleration describes changes in velocity for a moving object. In our example, the wheel starts from rest and reaches a particular angular speed in a given time.
To calculate angular acceleration, use the formula:

• \( \alpha = \frac{\omega - \omega_0}{t} \)

Here:

• \( \omega \) = final angular speed
• \( \omega_0 \) = initial angular speed
• \( t \) = time duration

In our problem, \( \omega_0 = 0 \) rad/s, \( \omega = 24\pi \) rad/s, and \( t = 2 \) s. So, the angular acceleration is \( 12\pi \) rad/s².
Understanding angular acceleration helps determine how forces impact rotational motion, which is crucial for solving dynamic problems.

Torque

Torque is a measure of the rotational force applied to an object. It's the product of force and the radius at which this force is applied. Torque is what causes objects to rotate.
The formula for torque is:

• \( \tau = rF \)

Where:

• \( \tau \) = torque
• \( r \) = radius
• \( F \) = force applied

In the given exercise, a tangential force of 80 N is applied at the wheel’s rim of radius 0.120 m.
The torque is calculated as \( 9.6 \) Nm.
Torque plays a pivotal role in determining the moment of inertia, linking directly to how much an object resists angular acceleration when a torque is applied.

Angular Speed

Angular speed refers to how quickly an object is rotating, described by how many radians it moves through per second. Unlike linear speed, which deals with straight-line paths, angular speed is about rotation around a point or axis.
To convert revolutions per second to radians per second, use the conversion:

• 1 revolution = \( 2\pi \) radians

So, if the wheel rotates at 12 revolutions per second:

• Angular speed \( \omega = 12 \times 2\pi = 24\pi \) rad/s

Angular speed is vital to determining many other properties of rotating objects, including kinetic energy and moment of inertia. It’s also foundational in understanding rotational motion dynamics.

Physics Problem Solving

Approaching physics problems systematically helps simplify even the toughest challenges. This includes recognizing what the problem is asking for, identifying given data, and applying the correct physics principles and formulas.
Here are some steps to tackle physics problems:

• Identify and list all given values and required findings from the problem statement.
• Translate any units to ensure consistency (like converting revolutions per second to radians per second).
• Use the relevant physics equations to connect the known variables and solve for the unknowns.
• Keep track of units throughout the calculations to maintain accuracy.

By breaking down each calculation step-by-step, we can better understand the overall concepts, like how force impacts angular motion or how angular speed and acceleration are connected. Practicing these steps regularly enhances problem-solving skills and deepens our comprehension of physical phenomena.