If n is the rigidity modulus, r is the radius , l is the length and C is the moment of the couple, then `(2lc)/(pi n r^4)` has the dimensions of

Options

A. Angle
B. Mass
C. Length
D. Frequency

Answer

To solve the problem, we need to find the dimensions of the expression \((2lc)/(πnr^4)\) where \(n\) is the rigidity modulus, \(r\) is the radius, \(l\) is the length, and \(C\) is the moment of the couple.

### Step 1: Determine the dimensions of each variable

1. **Rigidity Modulus (n)**:
- The rigidity modulus is defined as stress divided by strain.
- Stress = Force/Area = \(F/A\) = \(N/m^2\) = \(kg \cdot m^{-1} \cdot s^{-2}\).
- Strain = Change in length/original length (dimensionless).
- Therefore, the dimensions of rigidity modulus \(n\) are:
\[ [n] = [\text{Stress}] = [F/A] = [M L^{-1} T^{-2}] \]

2. **Length (l)**:
- The dimension of length is:
\[ [l] = [L] \]

3. **Moment of Couple (C)**:
- The moment of a couple is defined as force multiplied by the distance (lever arm).
- Dimensions of force \(F\) = \(kg \cdot m/s^2\) = \(M L T^{-2}\).
- Therefore, the dimensions of the moment of couple \(C\) are:
\[ [C] = [F] \cdot [\text{Distance}] = [M L T^{-2}] \cdot [L] = [M L^2 T^{-2}] \]

4. **Radius (r)**:
- The dimension of radius is:
\[ [r] = [L] \]

### Step 2: Substitute the dimensions into the expression

Now we substitute these dimensions into the expression \((2lc)/(πnr^4)\):

\[ \frac{2lc}{\pi n r^4} \]

- The dimensions of \(2lc\) are:
\[ [l] \cdot [C] = [L] \cdot [M L^2 T^{-2}] = [M L^3 T^{-2}] \]

- The dimensions of \(n\) and \(r^4\) are:
\[ [n] = [M L^{-1} T^{-2}], \quad [r^4] = [L^4] \]

### Step 3: Combine the dimensions

Now we can combine everything:

\[ \frac{[M L^3 T^{-2}]}{[M L^{-1} T^{-2}] \cdot [L^4]} = \frac{[M L^3 T^{-2}]}{[M L^{4-1}] \cdot [T^{-2}]} = \frac{[M L^3 T^{-2}]}{[M L^3 T^{-2}]} = 1 \]

### Step 4: Conclusion

The expression \((2lc)/(πnr^4)\) is dimensionless.

### Final Answer

Thus, the dimensions of \((2lc)/(πnr^4)\) are dimensionless, which corresponds to the option of "angle" since angles are often considered dimensionless.