Different lengths of pendulum are taken from Earth's surface to `h=R` . Time period remains same for

Options

A. Conical pendulum
B. Simple pendulum
C. Physical pendulum
D. Spring pendulum

Answer

To solve the question, we need to analyze how the time period of different types of pendulums changes when taken from the Earth's surface to a height equal to the radius of the Earth (h = R).

### Step-by-Step Solution:

1. **Understanding the Time Period of Different Pendulums**:
- The time period (T) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity.
- For a physical pendulum, the time period is also dependent on \(g\).
- The time period of a conical pendulum is similarly affected by \(g\).
- However, the time period of a spring pendulum is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \(m\) is the mass of the object attached to the spring and \(k\) is the spring constant. This formula does not include \(g\).

2. **Effect of Height on Gravity**:
- When we move to a height \(h = R\) (where \(R\) is the radius of the Earth), the acceleration due to gravity \(g\) changes. The new value of \(g\) at this height can be calculated using:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} = \frac{g}{(1 + 1)^2} = \frac{g}{4} \]
- This means that for simple, physical, and conical pendulums, the time period will increase because \(g\) decreases.

3. **Conclusion**:
- Since the time period of the simple pendulum, physical pendulum, and conical pendulum all depend on \(g\), their time periods will change when moving to a height \(h = R\).
- The spring pendulum, however, is independent of \(g\) and thus its time period remains the same.

4. **Final Answer**:
- The correct answer is **spring pendulum**.