At time t second, a particle of mass 3 kg has position vector r metre, where `r = 3that(i) - 4 cos t hat(j)`. Find the impulse of the force during the time interval `0 le t le(pi)/(2)`

Options

A. (a)`12 hatJ N -s`
B. (b)`9 hatJ N -s`
C. (c)`4 hatJ N -s`
D. (d)`14 hatJ N -s`

Answer

To solve the problem, we need to find the impulse of the force acting on a particle with a given position vector over a specified time interval. Let's break down the solution step by step.

### Step 1: Identify the position vector
The position vector \( \mathbf{r} \) of the particle is given as:
\[ \mathbf{r} = 3t \hat{i} - 4 \cos t \hat{j} \]

### Step 2: Calculate the velocity vector
The velocity \( \mathbf{v} \) is the derivative of the position vector with respect to time \( t \):
\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} = \frac{d}{dt}(3t \hat{i} - 4 \cos t \hat{j}) \]
Differentiating each component:
\[ \mathbf{v} = 3 \hat{i} + 4 \sin t \hat{j} \]

### Step 3: Calculate the acceleration vector
The acceleration \( \mathbf{a} \) is the derivative of the velocity vector with respect to time \( t \):
\[ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d}{dt}(3 \hat{i} + 4 \sin t \hat{j}) \]
Differentiating each component:
\[ \mathbf{a} = 0 \hat{i} + 4 \cos t \hat{j} = 4 \cos t \hat{j} \]

### Step 4: Calculate the force vector
Using Newton's second law, the force \( \mathbf{F} \) is given by:
\[ \mathbf{F} = m \mathbf{a} \]
Given that the mass \( m = 3 \, \text{kg} \):
\[ \mathbf{F} = 3 \cdot (4 \cos t \hat{j}) = 12 \cos t \hat{j} \]

### Step 5: Calculate the impulse
Impulse \( \mathbf{I} \) is given by the integral of the force over the time interval:
\[ \mathbf{I} = \int_{0}^{\frac{\pi}{2}} \mathbf{F} \, dt = \int_{0}^{\frac{\pi}{2}} 12 \cos t \hat{j} \, dt \]
Calculating the integral:
\[ \mathbf{I} = 12 \hat{j} \int_{0}^{\frac{\pi}{2}} \cos t \, dt \]
The integral of \( \cos t \) is \( \sin t \):
\[ \mathbf{I} = 12 \hat{j} [\sin t]_{0}^{\frac{\pi}{2}} = 12 \hat{j} (\sin(\frac{\pi}{2}) - \sin(0)) = 12 \hat{j} (1 - 0) = 12 \hat{j} \]

### Final Answer
The impulse of the force during the time interval \( 0 \leq t \leq \frac{\pi}{2} \) is:
\[ \mathbf{I} = 12 \, \text{N s} \, \hat{j} \]