Derivation of Time Period of Simple Pendulum using Dimensional Analysis:
To derive an expression for the time period of a simple pendulum using dimensional analysis, we can follow these steps:
1. **Identify the variables:**
- Length of the pendulum (L)
- Acceleration due to gravity (g)
- Time period of the pendulum (T)
2. **Write the formula for the time period of a simple pendulum:**
- The time period (T) of a simple pendulum is given by the formula:
T = 2π√(L/g)
3. **Determine the dimensions of each variable:**
- [L] = L^1 (length)
- [g] = LT^(-2) (acceleration due to gravity)
- [T] = T^1 (time)
4. **Express the formula in terms of dimensions:**
- T = kL^a g^b
- T = kL^1 g^(-1/2)
- T = k(L/g)^(1/2)
5. **Equating dimensions on both sides:**
- [T] = [k(L/g)^(1/2)]
- T = k(L/g)^(1/2)
6. **Determine the dimension of k:**
- [LHS] = T^1
- [RHS] = k(L/g)^(1/2)
- Equating the dimensions, k = 1
7. **Final expression for the time period of a simple pendulum:**
- T = √(L/g)
By using dimensional analysis, we have derived the expression for the time period of a simple pendulum as T = √(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.