A body of mass m is moving in a circle of radius angular velocity `omega`. Find the expression for centripetal force acting on it by the method of dimensions

Answer

To find the expression for the centripetal force acting on a body of mass \( m \) moving in a circle of radius \( r \) with angular velocity \( \omega \) using the method of dimensions, we can follow these steps:

### Step 1: Identify the quantities involved
We have three quantities that we will consider:
- Mass (\( m \))
- Radius (\( r \))
- Angular velocity (\( \omega \))

### Step 2: Assume a relationship
We assume that the centripetal force \( F \) depends on these quantities. We can express this relationship as:
\[ F \propto m^a r^b \omega^c \]
where \( a \), \( b \), and \( c \) are the powers to which the respective quantities are raised.

### Step 3: Introduce a proportionality constant
To remove the proportionality, we introduce a dimensionless constant \( k \):
\[ F = k \cdot m^a \cdot r^b \cdot \omega^c \]

### Step 4: Write the dimensions of each quantity
The dimensions of the quantities are:
- Force (\( F \)): \( [F] = [M L T^{-2}] \)
- Mass (\( m \)): \( [m] = [M] \)
- Radius (\( r \)): \( [r] = [L] \)
- Angular velocity (\( \omega \)): \( [\omega] = [T^{-1}] \)

### Step 5: Substitute dimensions into the equation
Substituting the dimensions into our equation gives:
\[ [M L T^{-2}] = k \cdot [M]^a \cdot [L]^b \cdot [T^{-1}]^c \]
Since \( k \) is dimensionless, it does not contribute to the dimensions.

### Step 6: Rearrange the equation
This can be rearranged as:
\[ [M L T^{-2}] = [M^a L^b T^{-c}] \]

### Step 7: Compare dimensions on both sides
Now, we can compare the dimensions on both sides:
1. For mass \( M \):
\[ 1 = a \quad \text{(1)} \]
2. For length \( L \):
\[ 1 = b \quad \text{(2)} \]
3. For time \( T \):
\[ -2 = -c \quad \Rightarrow \quad c = 2 \quad \text{(3)} \]

### Step 8: Solve for \( a \), \( b \), and \( c \)
From equations (1), (2), and (3), we find:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 2 \)

### Step 9: Substitute back into the expression
Now substituting these values back into our expression for \( F \):
\[ F = k \cdot m^1 \cdot r^1 \cdot \omega^2 = k \cdot m \cdot r \cdot \omega^2 \]

### Step 10: Conclusion
Thus, the expression for the centripetal force is:
\[ F = k \cdot m \cdot r \cdot \omega^2 \]
If we assume \( k = 1 \) for simplicity, we can write:
\[ F = m \cdot r \cdot \omega^2 \]