To solve this problem, we’ll use dimensional analysis to determine the formula for the period $t$ of a simple pendulum, involving the mass $m$, string length $l$, and gravitational acceleration $g$.

Step-by-Step Solution

1. Identify the Dimensions of Each Quantity:

   • Period $t$: The time for one complete swing, so its dimension is $[T]$.
   • Mass $m$: Mass, with dimension $[M]$.
   • Length $l$: The length of the pendulum, with dimension $[L]$.
   • Gravitational Acceleration $g$: This is acceleration due to gravity, with dimension $[L][T]^{-2}$.

2. Formulate the Dimensional Equation:

   We assume the period $t$ depends on $m$, $l$, and $g$ in some way. Thus, we can express $t$ as: $t = k \cdot m^a \cdot l^b \cdot g^c$ where $k$ is a dimensionless constant, and $a$, $b$, and $c$ are exponents to be determined.

3. Substitute Dimensions and Set Up Equations:

   Replacing each quantity with its dimensional form, we get: [ [T] = [M]^a \cdot [L]^b \cdot [L][T]^{-2}^c ] Expanding this: [ [T] = [M]^a \cdot [L]^{b+c} \cdot [T]^{-2c} ]

4. Equate Dimensions:

   Now, we match the powers of each fundamental dimension on both sides of the equation:

   • For $M$: $a = 0$
   • For $L$: $b + c = 0$
   • For $T$: $-2c = 1$

5. Solve the System of Equations:

   From $a = 0$, we see that mass $m$ does not affect the period, so $m$ can be disregarded.

   Solving $-2c = 1$ gives $c = -\frac{1}{2}$.

   Substitute $c = -\frac{1}{2}$ into $b + c = 0$: $b - \frac{1}{2} = 0 \Rightarrow b = \frac{1}{2}$

6. Construct the Formula:

   Substituting the values of $a$, $b$, and $c$ back, we get: $t = k \cdot l^{1/2} \cdot g^{-1/2}$ Simplifying further: $t = k \cdot \frac{\sqrt{l}}{\sqrt{g}} = k \cdot \sqrt{\frac{l}{g}}$

7. Conclusion:

   The formula for the period $t$ of a simple pendulum, based on dimensional analysis, is: $t = k \cdot \sqrt{\frac{l}{g}}$ where $k$ is a dimensionless constant, often determined experimentally to be close to $2\pi$ for a simple pendulum.

Would you like more details on any part of this solution, or have any questions about how we approached the dimensional analysis?

Additional Questions to Explore:

1. How does the mass $m$ affect the period of a simple pendulum if at all?
2. Why does gravitational acceleration $g$ appear in the denominator in the formula?
3. What is the significance of the constant $k$, and how is it determined?
4. How would the formula change if this were a different type of pendulum (e.g., a physical pendulum)?
5. Can we derive the period for other oscillatory systems using similar dimensional analysis?

Tip:

Dimensional analysis is a powerful tool to predict relationships between physical quantities without knowing the exact equations.