Questions: A torque of 50.244 Nm in the counterclockwise direction is applied to a wheel on a fixed axle. A frictional torque of 10.372 Nm also acts on the wheel. If the wheel's angular acceleration is 2.0957 rad / s^2, what is the moment of inertia of the wheel? - Answer: 19.??? kgm^2

A torque of 50.244 Nm in the counterclockwise direction is applied to a wheel on a fixed axle. A frictional torque of 10.372 Nm also acts on the wheel. If the wheel's angular acceleration is 2.0957 rad / s^2, what is the moment of inertia of the wheel?
- Answer: 19.??? kgm^2

Transcript text: 3. A torque of 50.244 Nm in the counterclockwise direction is applied to a wheel on a fixed axle. A frictional torque of 10.372 Nm also acts on the wheel. If the whel's angular acceleration is $2.0957 \mathrm{rad} / \mathrm{s}^{2}$, What is the moment of inertia of the wheel? - Answer: 19.??? $\mathrm{kgm}^{2}$

Solution

Solution Steps

Step 1: Understand the Problem

We are given a wheel with a counterclockwise torque of $50.244 \, \text{Nm}$ and a frictional torque of $10.372 \, \text{Nm}$ acting on it. The wheel's angular acceleration is $2.0957 \, \text{rad/s}^2$. We need to find the moment of inertia of the wheel.

Step 2: Calculate the Net Torque

The net torque ($\tau_{\text{net}}$) acting on the wheel is the difference between the applied torque and the frictional torque. Since the frictional torque opposes the applied torque, we subtract it:

\[ \tau_{\text{net}} = 50.244 \, \text{Nm} - 10.372 \, \text{Nm} = 39.872 \, \text{Nm} \]

Step 3: Use the Relationship Between Torque, Moment of Inertia, and Angular Acceleration

The relationship between torque ($\tau$), moment of inertia ($I$), and angular acceleration ($\alpha$) is given by:

\[ \tau_{\text{net}} = I \cdot \alpha \]

We can rearrange this formula to solve for the moment of inertia ($I$):

\[ I = \frac{\tau_{\text{net}}}{\alpha} \]

Step 4: Calculate the Moment of Inertia

Substitute the values of $\tau_{\text{net}}$ and $\alpha$ into the equation:

\[ I = \frac{39.872 \, \text{Nm}}{2.0957 \, \text{rad/s}^2} = 19.025 \, \text{kgm}^2 \]