13. Using dimensional analysis, derive the relationship between the period $$ T $$ of a simple pendulum, its length $$ l $$, and the acceleration due to gravity $$ g $$.

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We assume the period $$ T $$ depends on the length $$ l $$ and the acceleration due to gravity $$ g $$ as

$$
T = k \, l^a g^b,
$$

where $$ k $$ is a dimensionless constant and $$ a $$ and $$ b $$ are exponents to be determined.

**Step 1. Determine the dimensions of each quantity**

- The dimension of the period $$ T $$ is time: $$[T] = T$$.
- The dimension of the length $$ l $$ is length: $$[l] = L$$.
- The dimension of the acceleration due to gravity $$ g $$ is length per time squared: $$[g] = LT^{-2}$$.

**Step 2. Write the dimensional equation**

The dimensions of the right-hand side are

$$
[l^a g^b] = L^a \left( LT^{-2} \right)^b = L^{a+b} T^{-2b}.
$$

For dimensional consistency with the left-hand side $$[T]$$, we require:

$$
L^{a+b} T^{-2b} = T^1.
$$

This gives us two equations by equating the exponents of $$ L $$ and $$ T $$ on both sides.

**Step 3. Equate the exponents**

- For $$ L $$:

$$
a + b = 0.
$$

- For $$ T $$:

$$
-2b = 1.
$$

**Step 4. Solve for the exponents**

From the time exponent equation:

$$
-2b = 1 \quad \Longrightarrow \quad b = -\frac{1}{2}.
$$

Using $$ a + b = 0 $$:

$$
a = -b = \frac{1}{2}.
$$

**Step 5. Write the relationship**

Substitute the values of $$ a $$ and $$ b $$ back into the original equation:

$$
T = k \, l^{\frac{1}{2}} g^{-\frac{1}{2}} = k \, \sqrt{\frac{l}{g}}.
$$

Thus, by dimensional analysis, the relationship between the period $$ T $$, the length $$ l $$, and the acceleration due to gravity $$ g $$ is

$$
T \propto \sqrt{\frac{l}{g}},
$$

where the proportionality constant $$ k $$ can be determined by solving the full equations of motion (in the case of a simple pendulum, $$ k = 2\pi $$).
The period $$ T $$ of a simple pendulum is proportional to the square root of its length $$ l $$ divided by the acceleration due to gravity $$ g $$. So, $$ T \propto \sqrt{\frac{l}{g}} $$.

Using dimensional analysis, derive the relationship between the period of a simple pendulum, its length , and the acceleration due to gravity .

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