An object of mass 2kg is placed in a frame `(s_1)` moving with velocity `10 veci +5 vecjm//s` and having acceleration `5 veci +10vec jm//s^(-2)`. This object is also seen an observer standing in frame `(s_2)` moving with velocity `5 i +10 hatjm//s` . The net force acting on object with respect to `s_2` frame is

Options

A. `-10 veci - 20vecj N `
B. `10 veci +20vecj N `
C. ` -5 veci+10 vecj N `
D. Zero

Answer

To solve the problem, we need to find the net force acting on an object of mass 2 kg from the perspective of an observer in frame \( S_2 \). Here’s a step-by-step solution:

### Step 1: Identify Given Information
- Mass of the object, \( m = 2 \, \text{kg} \)
- Velocity of frame \( S_1 \) (where the object is placed), \( \vec{v_1} = 10 \hat{i} + 5 \hat{j} \, \text{m/s} \)
- Acceleration of frame \( S_1 \), \( \vec{a_1} = 5 \hat{i} + 10 \hat{j} \, \text{m/s}^2 \)
- Velocity of frame \( S_2 \), \( \vec{v_2} = 5 \hat{i} + 10 \hat{j} \, \text{m/s} \)

### Step 2: Determine the Relative Acceleration
The observer in frame \( S_2 \) is moving with a constant velocity, which means it has no acceleration. Therefore, the relative acceleration of the object with respect to frame \( S_2 \) is simply the acceleration of the object in frame \( S_1 \).

\[ \vec{a_{r}} = \vec{a_1} - \vec{a_2} \]
Since \( \vec{a_2} = 0 \) (no acceleration in frame \( S_2 \)):
\[ \vec{a_{r}} = \vec{a_1} = 5 \hat{i} + 10 \hat{j} \, \text{m/s}^2 \]

### Step 3: Calculate the Net Force
Using Newton's second law, the net force acting on the object in frame \( S_2 \) can be calculated using the formula:
\[ \vec{F} = m \cdot \vec{a_{r}} \]
Substituting the values:
\[ \vec{F} = 2 \, \text{kg} \cdot (5 \hat{i} + 10 \hat{j}) \, \text{m/s}^2 \]
\[ \vec{F} = 10 \hat{i} + 20 \hat{j} \, \text{N} \]

### Step 4: Final Result
The net force acting on the object with respect to frame \( S_2 \) is:
\[ \vec{F} = 10 \hat{i} + 20 \hat{j} \, \text{N} \]

### Summary
The net force acting on the object with respect to frame \( S_2 \) is \( 10 \hat{i} + 20 \hat{j} \, \text{N} \).

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