Problem 18 - Master Resource Book in JEE Main Physics

Question

As the temperature is increased, the time period of a pendulum (a) increases as its effective length increases even though its centre of mass still remains at the centre of the bob (b) decreases as its effective length increases even though its centre of mass still remains at the centre of the bob (c) increases as its effective length increases due to shifting of centre of mass below the centre of the bob (d) decreases as its effective length increases remains same but the centre of mass shifts above the centre of the bob

Step-by-Step Solution

Answer

The period increases as the effective length increases due to thermal expansion, making option (a) correct.

Step 1: Understand the Pendulum Formula

The time period of a simple pendulum is given by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( T \) is the period, \( L \) is the effective length of the pendulum, and \( g \) is the acceleration due to gravity. This formula implies that the time period is proportional to the square root of the length of the pendulum.

Step 2: Analyze the Effect of Temperature on Length

As the temperature increases, the material of the pendulum expands, and the effective length \( L \) of the pendulum increases. According to the formula, an increase in \( L \) causes an increase in the period \( T \).

Step 3: Consider the Centre of Mass

The problem statement indicates that the center of mass remains at the center of the bob even with the temperature increase. This means that the shift purely affects the length and not the placement of the center of mass relative to the pendulum's position.

Step 4: Identify the Correct Option

Based on the analysis, the time period of the pendulum increases as the effective length increases due to thermal expansion. The center of mass remaining at the bob implies any changes in period are solely due to length change. Thus, option (a) is correct: "increases as its effective length increases even though its centre of mass still remains at the centre of the bob."

Key Concepts

Time Period

The time period of a simple pendulum is a critical concept in understanding pendulum motion. It represents the time it takes for the pendulum to complete one full oscillation or swing. The formula for the time period is given by:\[ T = 2\pi \sqrt{\frac{L}{g}} \]where:

• \( T \) is the time period of the pendulum.
• \( L \) represents the effective length of the pendulum, measured from the point of suspension to the center of mass of the bob.
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \) on the surface of Earth.

The time period increases with the square root of the length. This relationship explains why pendulums with longer strings swing slower, taking a longer time to complete an oscillation. Conversely, a shorter pendulum swings more quickly. This characteristic makes pendulums ideal timekeepers, like in grandfather clocks, where the length can be finely tuned to regulate time.

Thermal Expansion

Thermal expansion is an important factor to consider when dealing with physical structures and materials. It occurs because most substances expand when heated due to increased molecular activity.For a pendulum, as temperature rises, the rod or wire that the pendulum bob is attached to expands. This thermal expansion causes an increase in the effective length, \( L \), of the pendulum.

• The coefficient of thermal expansion (\( \alpha \)) quantifies how much a substance expands per degree of temperature increase.
• For metals, which are commonly used in pendulum rods, \( \alpha \) is a small, positive value.
• An increase in the effective length \( L \) leads to a longer time period, \( T \), due to the pendulum formula.

Even though the center of mass of the pendulum remains at the center of the bob, the change in length alone is enough to affect the pendulum's rate of swing. In practical applications, materials might be chosen that minimize expansion or other mechanisms might be incorporated to compensate for these changes in precise devices.

Center of Mass

The center of mass is a crucial point for an object, especially in physics, as it helps in predicting the motion of the object. For a pendulum, the bob's center of mass plays a significant role in defining the effective length \( L \) of the pendulum.

• The center of mass is the point where the mass of the body is concentrated and its position determines the balance of the body.
• In a simple pendulum, the center of mass typically coincides with the geometric center of the spherical bob.
• When the pendulum's length changes due to factors like thermal expansion, the position of the center of mass relative to the point of suspension is what affects the oscillation period.

In the exercise situation, the center of mass does not shift despite the length increase due to temperature changes. Instead, the expansion occurs in the supporting material (typically the wire or rod), thus increasing the effective length. This illustrates why it's crucial to understand these principles, as they enable us to predict behavior in pendulum systems accurately without needing to account for complex shifts in the bob's position.