Problem 686 - JEE Physics

Question

The time period of a simple pendulum on a freely moving artificial satellite is .......... sec (A) 0 (B) 2 (C) 3 (D) Infinite

Step-by-Step Solution

Answer

The time period of a simple pendulum on a freely moving artificial satellite is infinite (D), as the pendulum cannot oscillate in the absence of gravity.

Step 1: Understand the setup

In this problem, we have a simple pendulum on an artificial satellite moving freely in space. The satellite is not affected by any external force apart from gravity, which is acting as the centripetal force for its motion.

Step 2: Recall the formula for the time period of a simple pendulum

The time period (T) of a simple pendulum can be found using 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.

Step 3: Recognize the effect of gravity on the satellite

A satellite in space is in free-fall (experiencing weightlessness) as it is constantly falling towards Earth due to gravity. In the case of our simple pendulum, the effect of gravity is the only force acting on it. In satellite's reference frame, the apparent gravity is zero, as it is also in free-fall.

Step 4: Calculate the time period of the simple pendulum in the satellite's reference frame

Since the apparent acceleration due to gravity (g) is zero in the satellite's reference frame, we can plug the value of 0 for g into the time period formula: \( T = 2\pi \sqrt{\frac{l}{0}} \). However, there is an issue as we are dividing by zero. This means that the pendulum cannot oscillate in the absence of gravity.

Step 5: Choose the correct answer

Given the situation, the pendulum will not oscillate back and forth, and its time period is effectively infinite. So the correct answer option is: (D) Infinite

Key Concepts

Simple Pendulum

A simple pendulum consists of a weight suspended from a pivot point, allowed to swing freely. The movement of the pendulum is driven by the force of gravity acting on it, attempting to restore it to its equilibrium position. In normal conditions, the time period of a simple pendulum, which is the time it takes to complete one full swing, can be calculated using the formula:\( T = 2\pi \sqrt{\frac{l}{g}} \)where:

• \( T \) is the time period,
• \( l \) is the length of the pendulum,
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).

In these calculations, gravity plays a crucial role as it influences how fast the pendulum swings back and forth. The longer the pendulum or the weaker the gravity, the longer the time period. However, if gravity is significantly altered, such as in a satellite, these conditions change dramatically.

Artificial Satellite

An artificial satellite is a human-made object placed into orbit around Earth or another celestial body. In space, these satellites follow an orbital path due to the gravitational pull of the body they circle. Usually, artificial satellites are used for a variety of purposes, such as:

• Communications
• Weather monitoring
• Space exploration
• Scientific research

Inside these satellites, the effects of gravity are much different than on the surface of the Earth. As satellites move in their orbits, they are in a state of constant free-fall towards the Earth, balanced by their tangential velocity, which creates an environment of apparent weightlessness.

Weightlessness

Weightlessness, or microgravity, occurs when objects or people experience little to no gravitational force. This phenomenon is commonly experienced by astronauts aboard spacecraft or satellites orbiting Earth. In a weightless environment, all objects fall at the same rate, so they experience no net force acting on them. If you were holding a simple pendulum on a satellite, the pendulum would not swing back and forth as there is no gravitational force to pull it downwards. Instead, both the pendulum and yourself would float freely, leading to a unique environment where conventional dynamics of gravity do not apply. This is why the time period of a pendulum becomes undefined or infinite in such conditions.

Gravity in Space

Gravity is a fundamental force that attracts two bodies towards each other. It is responsible for the orbits of planets around the sun and satellites around Earth. In space, gravity still exists, but its effects are perceived differently. In the reference frame of an orbiting satellite:

• The satellite is in free-fall, constantly being pulled by Earth's gravity, but its horizontal velocity keeps it orbiting.
• This creates a sensation of weightlessness onboard the satellite.
• The apparent gravity inside the satellite is essentially zero, making conditions vastly different from those on Earth.

For a simple pendulum in such a setting, the absence of apparent gravity means that the pendulum doesn’t have a restoring force to enable oscillation. This results in the pendulum's time period being infinite, as it cannot complete a single swing in the absence of a gravitational pull.