A body is rotating around a fixed axis with angular velocity 3 rad/s with constant angular acceleration of `1rad//s^(2)` at some time. Find the magnitude of acceleration of a particle 5 m away from the axis after the body has turned by `90^(@)`.

Options

A. `5sqrt(1+(9+pi)^(2))m//s^(2)`
B. `5(9+pi)m//s^(2)`
C. `5m//s^(2)`
D. `5sqrt(82)ms^(-2)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Given Data
- Initial angular velocity, \( \omega_0 = 3 \, \text{rad/s} \)
- Angular acceleration, \( \alpha = 1 \, \text{rad/s}^2 \)
- Distance from the axis, \( r = 5 \, \text{m} \)
- Angle turned, \( \theta = 90^\circ = \frac{\pi}{2} \, \text{rad} \)

### Step 2: Calculate the Final Angular Velocity
We can use the equation of motion for angular displacement to find the final angular velocity \( \omega \) after the body has turned by \( \theta \):

\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]

Substituting the values:

\[ \omega^2 = (3)^2 + 2 \cdot (1) \cdot \left(\frac{\pi}{2}\right) \]

Calculating this gives:

\[ \omega^2 = 9 + \pi \]

### Step 3: Calculate the Centripetal Acceleration
The centripetal acceleration \( a_c \) is given by the formula:

\[ a_c = \omega^2 r \]

Substituting the values we found:

\[ a_c = (9 + \pi) \cdot 5 \]

### Step 4: Calculate the Tangential Acceleration
The tangential acceleration \( a_t \) is given by:

\[ a_t = r \alpha \]

Substituting the values:

\[ a_t = 5 \cdot 1 = 5 \, \text{m/s}^2 \]

### Step 5: Calculate the Magnitude of Total Acceleration
The total acceleration \( a \) is the vector sum of the tangential and centripetal accelerations. Since they are perpendicular to each other, we can use the Pythagorean theorem:

\[ a = \sqrt{a_t^2 + a_c^2} \]

Substituting the values:

\[ a = \sqrt{(5)^2 + [(9 + \pi) \cdot 5]^2} \]

Calculating \( a_c \):

\[ a_c = 5(9 + \pi) \]

Now substituting this back into the equation for total acceleration:

\[ a = \sqrt{25 + [5(9 + \pi)]^2} \]

### Step 6: Final Calculation
Now we can compute the final value of the total acceleration.

1. Calculate \( 5(9 + \pi) \).
2. Square that result.
3. Add 25 to that squared result.
4. Take the square root to find \( a \).