Problem 26 - Master Resource Book in JEE Main Physics

Question

A clock \(S\) is based on oscillation of a spring and a clock \(P\) is based on pendulum motion. Both clock run at the same rate on earth. On a planet having the same density as earth but twice the radius, (a) \(S\) will run faster than \(P\) (b) \(P\) will run faster than \(S\) (c) both will run at the same rate as on the earth (d) both will run at the same rate which will be different from that on the earth

Step-by-Step Solution

Answer

(b) \( P \) will run faster than \( S \).

Step 1: Understanding Earth's Gravitational Influence on Clocks

On Earth, a spring clock's timekeeping depends on the spring constant and mass, both of which are independent of gravity. The pendulum clock, however, relies on gravity; its time period is given by \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the pendulum's length and \( g \) is the gravitational acceleration. Thus, changes in gravity affect the pendulum clock more.

Step 2: Gravitational Formula for the Planet

This planet has twice the radius of Earth and the same density. Gravitational acceleration \( g' \) on this planet can be found using the formula for gravity \( g = \frac{G M}{R^2}\), where \( M \) is the mass and \( R \) the radius. Given the density (\( \rho \)), and \( M = \rho \times \frac{4}{3}\pi R^3\), mass becomes proportional to the cube of the radius.

Step 3: Determining the Relation of \( g' \) to \( g \)

Since the density is constant and the radius \( R' = 2R \), mass \( M' = 8M \). Substituting in gravitational formula: \( g' = \frac{G \times 8M}{(2R)^2} = \frac{8G M}{4R^2} = 2g \). The gravity on the planet is twice that of Earth's gravity.

Step 4: Effect of Gravity on Clocks

The time period of a pendulum clock \( P \) on the planet is \( T' = 2\pi \sqrt{\frac{L}{2g}} = \frac{1}{\sqrt{2}} T \). It runs faster than on Earth because of higher \( g' \). The spring clock \( S \) remains unaffected by gravitational changes as it does not rely on \( g \).

Step 5: Comparison on Planet

On the planet, the pendulum clock \( P \) runs faster due to the increased gravity, while the spring clock \( S \) runs at the same rate as it does on Earth. Thus, \( P \) is faster than \( S \) on this planet.

Key Concepts

Pendulum Clock

A pendulum clock is an incredible device that relies on the motion of a swinging pendulum to keep time. The key component in the pendulum clock's operation is the pendulum itself, which moves back and forth under the influence of gravity. The time it takes for the pendulum to complete one full swing is called the period, denoted by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \). In this expression, \( L \) is the length of the pendulum, and \( g \) is the gravitational acceleration.

• The pendulum's motion is a simple harmonic motion, meaning it's predictable and regular.
• The period depends on the length of the pendulum and the gravitational pull, not the mass of the pendulum bob.

In a scenario where gravity increases, the pendulum will complete its swings more quickly, resulting in a faster-running clock. Conversely, a decrease in gravitational force will slow it down. This dependency on gravity means that a pendulum clock's accuracy can be significantly affected by the location's gravity field.

Spring Clock

Unlike the pendulum clock, a spring clock operates without relying on gravity's influence. Its timekeeping mechanism is based on the oscillation of a spring, which consists of a coiled wire that can be wound tight and then unwinds at a consistent rate. The period of oscillation of the spring is determined by

• the spring constant (which represents the stiffness of the spring) and
• the mass of the object attached to the spring.

These factors are independent of gravity, which means the characteristics of a spring clock remain unchanged whether it's on Earth, another planet, or even in space.
The operation of a spring clock is described by Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement, leading to a simple harmonic motion. Due to its independence from gravity, a spring clock is a reliable timekeeper across different environments, making it particularly useful in situations where gravitational variations occur.

Gravitational Acceleration

Gravitational acceleration, denoted by \( g \), is a crucial concept in understanding how objects fall under the influence of gravity. On Earth, the standard average gravitational acceleration is approximately \( 9.81 \ m/s^2 \), but it varies slightly depending on altitude and geographical position.
When considering other planets, such as one with twice Earth's radius but the same density, the gravitational acceleration changes significantly. For such a planet, using the relationship \[ g = \frac{G M}{R^2} \] where \( G \) is the gravitational constant, \( M \) is mass, and \( R \) is the radius, the acceleration increases due to the increased mass and radius. This results in a gravitational acceleration \( g' = 2g \) because the mass scales with the cube of the radius, and the force depends inversely on the square of the radius.
This change in gravitational acceleration will affect objects depending on their reliance on gravity, such as pendulum clocks, which will operate differently under different gravitational forces.

Density and Radius Relationship

Understanding the relationship between density and radius is essential when assessing the gravitational forces on different celestial bodies. Density is defined as mass per unit volume, and when it remains constant while the radius changes, the mass of an object is influenced proportionally to the cube of the radius. This can be expressed as \[ M \propto \rho \times R^3 \] where \( \rho \) is the density and \( R \) is the radius.
For a planet with the same density as Earth but twice the radius, the mass becomes eight times greater. This increase in mass has a direct impact on the gravitational attraction, effectively doubling the gravitational acceleration since \[ g' = \frac{8M}{4R^2} = 2g \] This transformed gravitational environment will influence timekeeping devices that depend on gravity, explaining why clocks can run at different rates on different planets.
This understanding of density and radius is crucial in fields like astrophysics and planetary science, where it helps explain and predict the behavior of objects in various planetary environments.