the wheel on a vehicle has a rotational inertia of 2.0 kg⋅m2. at the instant the wheel has a counterclockwise angular velocity of 6.0 rad/s, an average counterclockwise torque of 5.0 n⋅m is applied, and continues for 4.0 s. what is the change in angular momentum of the wheel?
Calculating Change in Angular Momentum
Question: What is the change in angular momentum of the wheel?
Answer:
To calculate the change in angular momentum of the wheel, we first need to determine the initial angular momentum and final angular momentum, then find the difference between them.
Given:
- Initial angular velocity, \omega_{i} = 6.0 \: \text{rad/s}
- Rotational inertia, I = 2.0 \: \text{kg} \cdot \text{m}^2
- Torque, \tau = 5.0 \: \text{N} \cdot \text{m}
- Time, t = 4.0 \: \text{s}
First, calculate the initial angular momentum, L_{i} :
L_{i} = I \cdot \omega_{i}
L_{i} = 2.0 \: \text{kg} \cdot \text{m}^2 \times 6.0 \: \text{rad/s}
L_{i} = 12 \: \text{kg} \cdot \text{m}^2/\text{s}
Next, calculate the final angular momentum, L_{f} :
L_{f} = L_{i} + \tau \cdot t
L_{f} = 12 \: \text{kg} \cdot \text{m}^2 + 5.0 \: \text{N} \cdot \text{m} \times 4.0 \: \text{s}
L_{f} = 12 \: \text{kg} \cdot \text{m}^2 + 20 \: \text{kg} \cdot \text{m}^2
L_{f} = 32 \: \text{kg} \cdot \text{m}^2/\text{s}
Finally, calculate the change in angular momentum, \Delta L :
\Delta L = L_{f} - L_{i}
\Delta L = 32 \: \text{kg} \cdot \text{m}^2/\text{s} - 12 \: \text{kg} \cdot \text{m}^2
\Delta L = 20 \: \text{kg} \cdot \text{m}^2/\text{s}
Therefore, the change in angular momentum of the wheel is 20 kg⋅m²/s.