During the spin-dry cycle of a washing machine, the motor slows from 95 rad/s to 30 rad/s while the turning the drum through an angle of 402 rad. What is the magnitude of the angular acceleration of the motor?

Options

A. `64rad//s^(2)`
B. `10rad//s^(2)`
C. `32rad//s^(2)`
D. `20rad//s^(2)`

Answer

To find the magnitude of the angular acceleration of the motor, we can use the following kinematic equation for rotational motion:

\[ \omega_f^2 = \omega_i^2 + 2\alpha \theta \]

Where:
- \(\omega_f\) = final angular velocity (30 rad/s)
- \(\omega_i\) = initial angular velocity (95 rad/s)
- \(\alpha\) = angular acceleration (unknown)
- \(\theta\) = angular displacement (402 rad)

### Step 1: Write down the known values
- \(\omega_i = 95 \, \text{rad/s}\)
- \(\omega_f = 30 \, \text{rad/s}\)
- \(\theta = 402 \, \text{rad}\)

### Step 2: Substitute the known values into the equation
We can rearrange the equation to solve for \(\alpha\):

\[ \omega_f^2 - \omega_i^2 = 2\alpha \theta \]

Substituting the known values:

\[ (30)^2 - (95)^2 = 2\alpha (402) \]

### Step 3: Calculate \(\omega_f^2\) and \(\omega_i^2\)
Calculating the squares:

\[ 900 - 9025 = 2\alpha (402) \]

### Step 4: Simplify the equation
\[ -8125 = 804\alpha \]

### Step 5: Solve for \(\alpha\)
Now, divide both sides by \(804\):

\[ \alpha = \frac{-8125}{804} \]

Calculating this gives:

\[ \alpha \approx -10.10 \, \text{rad/s}^2 \]

### Step 6: Find the magnitude of angular acceleration
Since we are interested in the magnitude, we take the absolute value:

\[ |\alpha| \approx 10.10 \, \text{rad/s}^2 \]

### Final Answer
The magnitude of the angular acceleration of the motor is approximately \(10.10 \, \text{rad/s}^2\).

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