Consider a simple pendulum. The period of oscillation of the simple pendulum depends on its length and acceleration due to gravity. Then the expression for its time period is

Options

A. `t=kgsqrtl`
B. `t=ksqrt(lg)`
C. `t=ksqrt((l)/(g))`
D. `t=k(l)/(sqrtg)`

Explanation

To derive the expression for the time period of a simple pendulum, we will follow these steps:

### Step 1: Understand the Variables
The time period \( T \) of a simple pendulum depends on:
- Length of the pendulum \( L \)
- Acceleration due to gravity \( g \)

### Step 2: Establish the Relationship
We can express the time period \( T \) as being proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity:
\[ T \propto \sqrt{\frac{L}{g}} \]

### Step 3: Introduce a Constant of Proportionality
To convert the proportionality into an equation, we introduce a constant \( k \):
\[ T = k \sqrt{\frac{L}{g}} \]

### Step 4: Determine the Dimensions
Next, we analyze the dimensions of the variables involved:
- The dimension of time \( [T] \) is \( T \).
- The dimension of length \( [L] \) is \( L \).
- The dimension of acceleration due to gravity \( [g] \) is \( LT^{-2} \).

### Step 5: Set Up the Dimensional Equation
We can express the time period in terms of its dimensions:
\[ [T] = k [L]^A [g]^B \]
Substituting the dimensions:
\[ T = k L^A (LT^{-2})^B \]
This simplifies to:
\[ T = k L^A L^B T^{-2B} = k L^{A+B} T^{-2B} \]

### Step 6: Compare the Powers
To find the values of \( A \) and \( B \), we compare the powers of \( L \) and \( T \):
1. For \( L \): \( A + B = 0 \) (Equation 1)
2. For \( T \): \( -2B = 1 \) (Equation 2)

### Step 7: Solve the Equations
From Equation 2, we find:
\[ B = -\frac{1}{2} \]
Substituting \( B \) into Equation 1:
\[ A - \frac{1}{2} = 0 \implies A = \frac{1}{2} \]

### Step 8: Substitute Back
Now we can substitute \( A \) and \( B \) back into our expression for \( T \):
\[ T = k L^{1/2} g^{-1/2} \]
This can be rewritten as:
\[ T = k \sqrt{\frac{L}{g}} \]

### Step 9: Final Expression
The final expression for the time period of a simple pendulum is:
\[ T = k \sqrt{\frac{L}{g}} \]

### Conclusion
Thus, the time period \( T \) of a simple pendulum is directly proportional to the square root of its length and inversely proportional to the square root of the acceleration due to gravity.