Centripetal force (F) on a body of mass (m) moving with uniform speed (v) in a circle of radius (r ) depends upon m, v and r. The formula for the centripetal force using theory of dimensons.

Options

A. `kmv^(2)`
B. `(kmv^(2))/(r )`
C. `(km^(2)v^(2))/(r )`
D. `(km^(2)v)/(r )`

Answer

To derive the formula for centripetal force (F) using dimensional analysis, we will follow these steps:

### Step 1: Identify the Variables
The centripetal force depends on three variables:
- Mass (m)
- Velocity (v)
- Radius (r)

### Step 2: Write the Dimensional Formulae
Next, we need to write the dimensional formulae for each of these variables:
- Mass (m) has the dimensional formula: \([M]\)
- Velocity (v) is distance per time, so its dimensional formula is: \([L][T]^{-1}\)
- Radius (r) is a length, so its dimensional formula is: \([L]\)

### Step 3: Formulate the Relationship
We assume that the centripetal force \(F\) can be expressed as:
\[ F = k \cdot m^a \cdot v^b \cdot r^c \]
where \(k\) is a dimensionless constant and \(a\), \(b\), and \(c\) are the powers we need to determine.

### Step 4: Write the Dimensional Formula for Force
The dimensional formula for force (F) is:
\[ [F] = [M][L][T]^{-2} \]

### Step 5: Substitute the Dimensional Formulae
Now we substitute the dimensional formulae into the equation:
\[ [M][L][T]^{-2} = [M]^a \cdot [L]^b \cdot [L]^c \cdot [T]^{-b} \]

This simplifies to:
\[ [M][L][T]^{-2} = [M]^a \cdot [L]^{b+c} \cdot [T]^{-b} \]

### Step 6: Equate the Powers
Now we equate the powers of each dimension on both sides:
1. For mass (M): \(1 = a\)
2. For length (L): \(1 = b + c\)
3. For time (T): \(-2 = -b\)

### Step 7: Solve the Equations
From the third equation, we get:
\[ b = 2 \]

Substituting \(b = 2\) into the second equation:
\[ 1 = 2 + c \]
This gives:
\[ c = -1 \]

Now substituting \(a = 1\), \(b = 2\), and \(c = -1\) into our original formula:
\[ F = k \cdot m^1 \cdot v^2 \cdot r^{-1} \]
This simplifies to:
\[ F = k \cdot \frac{mv^2}{r} \]

### Step 8: Conclusion
Thus, the formula for centripetal force is:
\[ F = \frac{mv^2}{r} \]