Due to some unknown interaction, force F experienced by a particle is given by the following equation.
`vecF=-(A)/(r^(3))vecr`
Where A is positive constant and r distance of the particle from origin of a coordinate system. Dimensions of constant A are :-

Options

A. `ML^(2)T^(-2)`
B. `ML^(3)T^(-2)`
C. `ML^(4)T^(-2)`
D. `ML^(0)T^(0)`

Answer

To find the dimensions of the constant \( A \) in the equation

\[ \vec{F} = -\frac{A}{r^3} \vec{r} \]

we will follow these steps:

### Step 1: Understand the given equation
The force \( \vec{F} \) is expressed in terms of the constant \( A \) and the distance \( r \) from the origin. The vector \( \vec{r} \) indicates the direction of the force.

### Step 2: Identify the dimensions of force
We know that force \( F \) can be expressed in terms of mass \( m \) and acceleration \( a \):

\[ F = m \cdot a \]

The dimensions of mass \( m \) are \( [M] \), and the dimensions of acceleration \( a \) (which is \( \text{length}/\text{time}^2 \)) are \( [L][T^{-2}] \). Therefore, the dimensions of force \( F \) are:

\[ [F] = [M][L][T^{-2}] = [M L T^{-2}] \]

### Step 3: Write down the dimensions of \( r \)
The distance \( r \) has dimensions of length:

\[ [r] = [L] \]

### Step 4: Substitute dimensions into the equation
From the equation \( \vec{F} = -\frac{A}{r^3} \vec{r} \), we can express the dimensions as follows:

\[ [F] = \frac{[A]}{[r^3]} \cdot [r] \]

Substituting the dimensions we have:

\[ [M L T^{-2}] = \frac{[A]}{[L^3]} \cdot [L] \]

### Step 5: Simplify the equation
This simplifies to:

\[ [M L T^{-2}] = \frac{[A]}{[L^3]} \cdot [L] = \frac{[A]}{[L^2]} \]

### Step 6: Rearrange to find dimensions of \( A \)
To isolate \( [A] \), we multiply both sides by \( [L^2] \):

\[ [M L T^{-2}] \cdot [L^2] = [A] \]

This gives:

\[ [A] = [M L^3 T^{-2}] \]

### Final Answer
Thus, the dimensions of the constant \( A \) are:

\[ [A] = [M L^3 T^{-2}] \]

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