The angular position of a line of a disc of radius r = 6cm is given by `theta=10 - 5t +4t^2` rad , the average angular velocity between 1 and 3s is

Options

A. (a)`pi "rad "s ^(-1)`
B. (b)`11 "rad "s ^(-1)`
C. (c)`22 "rad "s ^(-1)`
D. (d)`5.5 "rad "s ^(-1)`

Answer

To find the average angular velocity between 1 and 3 seconds for the given angular position function \(\theta(t) = 10 - 5t + 4t^2\), we will follow these steps:

### Step 1: Identify the formula for average angular velocity
The average angular velocity \(\omega_{avg}\) is defined as the change in angular position \(\Delta \theta\) over the change in time \(\Delta t\):
\[ \omega_{avg} = \frac{\Delta \theta}{\Delta t} = \frac{\theta_f - \theta_i}{t_f - t_i} \]

### Step 2: Determine the initial and final times
We are given:
- Initial time \(t_i = 1 \, \text{s}\)
- Final time \(t_f = 3 \, \text{s}\)

### Step 3: Calculate the initial angular position \(\theta_i\)
Substituting \(t = 1\) into the angular position equation:
\[ \theta_i = \theta(1) = 10 - 5(1) + 4(1)^2 \]
Calculating this:
\[ \theta_i = 10 - 5 + 4 = 9 \, \text{radians} \]

### Step 4: Calculate the final angular position \(\theta_f\)
Substituting \(t = 3\) into the angular position equation:
\[ \theta_f = \theta(3) = 10 - 5(3) + 4(3)^2 \]
Calculating this:
\[ \theta_f = 10 - 15 + 4(9) = 10 - 15 + 36 = 31 \, \text{radians} \]

### Step 5: Calculate the change in angular position \(\Delta \theta\)
\[ \Delta \theta = \theta_f - \theta_i = 31 - 9 = 22 \, \text{radians} \]

### Step 6: Calculate the change in time \(\Delta t\)
\[ \Delta t = t_f - t_i = 3 - 1 = 2 \, \text{s} \]

### Step 7: Calculate the average angular velocity \(\omega_{avg}\)
Now we can substitute \(\Delta \theta\) and \(\Delta t\) into the average angular velocity formula:
\[ \omega_{avg} = \frac{\Delta \theta}{\Delta t} = \frac{22 \, \text{radians}}{2 \, \text{s}} = 11 \, \text{radians/s} \]

### Final Answer
The average angular velocity between 1 and 3 seconds is \(11 \, \text{radians/s}\).

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