When forces `F_(1)` , `F_(2)` , `F_(3)` are acting on a particle of mass m such that `F_(2)` and `F_(3)` are mutually prependicular, then the particle remains stationary. If the force `F_(1)` is now rejmoved then the acceleration of the particle is

Options

A. `F_(1)//m`
B. `F_(2)F_(3)//mF_(1)`
C. `(F_(2)-F_(3))//m`
D. `F_(2)//m`

Explanation

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

### Step 1: Understand the Forces Acting on the Particle
We have three forces acting on a particle of mass \( m \): \( F_1 \), \( F_2 \), and \( F_3 \). The forces \( F_2 \) and \( F_3 \) are mutually perpendicular.

### Step 2: Condition for the Particle to be Stationary
Since the particle remains stationary, the net force acting on it must be zero. This means:
\[ F_1 + F_2 + F_3 = 0 \]
This implies that \( F_1 \) is balancing the resultant of \( F_2 \) and \( F_3 \).

### Step 3: Calculate the Resultant of \( F_2 \) and \( F_3 \)
Since \( F_2 \) and \( F_3 \) are perpendicular, we can find their resultant using the Pythagorean theorem:
\[ R = \sqrt{F_2^2 + F_3^2} \]
Thus, we can express \( F_1 \) in terms of \( F_2 \) and \( F_3 \):
\[ F_1 = R = \sqrt{F_2^2 + F_3^2} \]

### Step 4: Remove \( F_1 \) and Analyze the Situation
Now, if we remove \( F_1 \), the only forces acting on the particle will be \( F_2 \) and \( F_3 \). Since \( F_2 \) and \( F_3 \) are not balanced by \( F_1 \) anymore, the particle will experience a net force equal to the resultant of \( F_2 \) and \( F_3 \).

### Step 5: Calculate the Acceleration of the Particle
According to Newton's second law of motion, the acceleration \( a \) of the particle can be calculated using the formula:
\[ F = ma \]
Here, the net force \( F \) acting on the particle is the resultant of \( F_2 \) and \( F_3 \):
\[ F = R = \sqrt{F_2^2 + F_3^2} \]
Thus, we can write:
\[ \sqrt{F_2^2 + F_3^2} = ma \]
From this, we can derive the acceleration \( a \):
\[ a = \frac{\sqrt{F_2^2 + F_3^2}}{m} \]

### Step 6: Relate the Resultant to \( F_1 \)
Since we established earlier that \( F_1 = \sqrt{F_2^2 + F_3^2} \), we can substitute this back into our equation for acceleration:
\[ a = \frac{F_1}{m} \]

### Final Answer
Thus, the acceleration of the particle when \( F_1 \) is removed is:
\[ a = \frac{F_1}{m} \]

---