When the angular velocity of a uniformly rotating body has increased thrice, the resultant of forces applied to it increases by 60 N. Find the accelerations of the body in the two cases. The mass of the body,m = 3 kg.

Options

A. `2.5ms^(-2),7.5ms^(-2)`
B. `7.5ms^(-2),22.5ms^(-2)`
C. `5 ms^(-2),45ms^(-2)`
D. `2.5ms^(-2),22.5ms^(-2)`

Answer

To solve the problem, we need to find the centripetal accelerations of a uniformly rotating body before and after its angular velocity is increased. Let's break it down step by step.

### Given Data:
- Mass of the body, \( m = 3 \, \text{kg} \)
- Increase in resultant forces, \( \Delta F = 60 \, \text{N} \)
- Final angular velocity is three times the initial angular velocity, \( \omega_f = 3 \omega_i \)

### Step 1: Understanding the Change in Forces
The change in the resultant force can be expressed in terms of the centripetal acceleration. The centripetal force required for circular motion is given by:
\[ F = m \cdot a_c \]
where \( a_c \) is the centripetal acceleration.

### Step 2: Setting Up the Equation
The change in force due to the change in angular velocity can be expressed as:
\[ F_f - F_i = \Delta F \]
Substituting the expressions for centripetal force:
\[ m \cdot a_{c_f} - m \cdot a_{c_i} = 60 \, \text{N} \]
This simplifies to:
\[ m (a_{c_f} - a_{c_i}) = 60 \]

### Step 3: Expressing the Accelerations
The centripetal accelerations can be expressed in terms of angular velocities:
\[ a_{c_i} = \omega_i^2 r \]
\[ a_{c_f} = \omega_f^2 r = (3 \omega_i)^2 r = 9 \omega_i^2 r \]

### Step 4: Substituting into the Equation
Substituting these into the force equation gives:
\[ m (9 \omega_i^2 r - \omega_i^2 r) = 60 \]
This simplifies to:
\[ m (8 \omega_i^2 r) = 60 \]

### Step 5: Solving for \( \omega_i^2 r \)
Now, substituting the mass \( m = 3 \, \text{kg} \):
\[ 3 (8 \omega_i^2 r) = 60 \]
\[ 8 \omega_i^2 r = \frac{60}{3} = 20 \]
\[ \omega_i^2 r = \frac{20}{8} = 2.5 \]

### Step 6: Finding the Initial Centripetal Acceleration
Now that we have \( \omega_i^2 r = 2.5 \):
\[ a_{c_i} = \omega_i^2 r = 2.5 \, \text{m/s}^2 \]

### Step 7: Finding the Final Centripetal Acceleration
Now, we can find the final centripetal acceleration:
\[ a_{c_f} = 9 \omega_i^2 r = 9 \times 2.5 = 22.5 \, \text{m/s}^2 \]

### Final Results
- Initial centripetal acceleration, \( a_{c_i} = 2.5 \, \text{m/s}^2 \)
- Final centripetal acceleration, \( a_{c_f} = 22.5 \, \text{m/s}^2 \)

### Summary
Thus, the accelerations of the body in the two cases are:
- Initial acceleration: \( 2.5 \, \text{m/s}^2 \)
- Final acceleration: \( 22.5 \, \text{m/s}^2 \)