A wheel turns through an angle of 188 rad in 8.0 s, and its angular speed at the end of the period is 40 rad/s. If the angular acceleration is constant, what was the angular speed of the wheel at the beginning of the 8.0 s interval?

Options

A. 4.8 rad/s
B. 9.1 rad/s
C. 7.0 rad/s
D. 23.5 rad/s

Answer

To solve the problem step by step, we will use the equations of motion for rotational dynamics. Given that the angular acceleration is constant, we can use the following equations:

1. **Given Data:**
- Angular displacement (θ) = 188 rad
- Time (t) = 8.0 s
- Final angular velocity (ω_f) = 40 rad/s
- Initial angular velocity (ω_i) = ? (This is what we need to find)
- Angular acceleration (α) = ? (This will also be derived)

2. **Using the First Equation of Motion:**
The first equation of motion for rotational dynamics is:
\[ \omega_f = \omega_i + \alpha t \]
Rearranging this gives:
\[ \omega_i = \omega_f - \alpha t \]
(1)

3. **Using the Second Equation of Motion:**
The second equation of motion for rotational dynamics is:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Rearranging this gives:
\[ \alpha = \frac{2(\theta - \omega_i t)}{t^2} \]
(2)

4. **Substituting Equation (1) into Equation (2):**
We can substitute ω_i from equation (1) into equation (2):
\[ \alpha = \frac{2\left(188 - (\omega_f - \alpha t)t\right)}{t^2} \]
Plugging in the known values:
\[ \alpha = \frac{2\left(188 - (40 - \alpha \cdot 8) \cdot 8\right)}{8^2} \]
Simplifying this gives:
\[ \alpha = \frac{2\left(188 - (40 \cdot 8 - 8\alpha \cdot 8)\right)}{64} \]
\[ \alpha = \frac{2\left(188 - 320 + 64\alpha\right)}{64} \]
\[ \alpha = \frac{2(-132 + 64\alpha)}{64} \]
\[ 64\alpha = -264 + 128\alpha \]
Rearranging gives:
\[ 64\alpha = 264 \]
\[ \alpha = \frac{264}{64} = 4.125 \text{ rad/s}^2 \]

5. **Finding Initial Angular Velocity (ω_i):**
Now that we have α, we can substitute it back into equation (1):
\[ \omega_i = 40 - 4.125 \cdot 8 \]
\[ \omega_i = 40 - 33 = 7 \text{ rad/s} \]

Thus, the initial angular speed of the wheel at the beginning of the 8.0 s interval is **7 rad/s**.