Problem 9 - Applied Physics

Question

If you double the length of a pendulum, what happens to its period?

Step-by-Step Solution

Answer

When the pendulum length is doubled, its period increases by a factor of \( \sqrt{2} \) or about 1.414.

Step 1: Understand the Pendulum Period Formula

The period of a pendulum, which is the time it takes to complete one oscillation, is given by the formula: \[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity. This equation shows that the period depends on the square root of the pendulum's length.

Step 2: Analyze the Effect of Doubling the Length

When the length \( L \) of the pendulum is doubled, it becomes \( 2L \). Substitute \( 2L \) into the period formula:\[ T' = 2\pi \sqrt{\frac{2L}{g}} \]This is the new period of the pendulum after doubling the length.

Step 3: Simplify the New Period Formula

Simplify the equation for the new period:\[ T' = 2\pi \sqrt{\frac{2L}{g}} = 2\pi \sqrt{2} \cdot \sqrt{\frac{L}{g}} \]Comparing this to the original period formula \( T = 2\pi \sqrt{\frac{L}{g}} \), we see that:\[ T' = \sqrt{2} \cdot T \]This shows that the new period is \( \sqrt{2} \) times the original period.

Step 4: Calculate the Numerical Value

The square root of 2 is approximately 1.414. Therefore, doubling the length of the pendulum causes the period to increase by a factor of approximately 1.414.

Key Concepts

Oscillation

In the context of a pendulum, an oscillation refers to one complete movement of the pendulum bob from its starting point, to the opposite side, and back again to the starting point. It's a repeating, predictable motion.
This regular back-and-forth movement is what defines the stability and usefulness of pendulums in timekeeping, such as in grandfather clocks. The time it takes to complete one oscillation is known as the period.

• Each oscillation is consistent if initial conditions remain unchanged.
• Understanding oscillation is fundamental when exploring pendulum mechanics, including the effect of altering parameters like pendulum length.

The consistency in oscillations makes pendulums ideal for studying physics principles, as they offer predictable motion that can be objectively analyzed for various conditions.

Pendulum Length

Pendulum length plays a critical role in determining the period of the pendulum's oscillation. When referring to pendulum length, it means the distance from the pivot point to the center of mass of the pendulum bob.
According to the formula \[ T = 2\pi \sqrt{\frac{L}{g}} \] the period of the pendulum is directly related to the square root of its length. This means:

• If the length increases, the period also increases.
• The relationship isn't linear; instead, it's a square root relationship, meaning doubling the length doesn't double the period—rather, it multiplies the period by \( \sqrt{2} \).

Therefore, manipulating the pendulum length allows us to control how long each oscillation takes, which was historically used to regulate the precision of clocks.

Acceleration Due to Gravity

The term "acceleration due to gravity" (symbolized as \( g \)) manifests as a crucial aspect of a pendulum's movement. This value, approximately 9.81 m/s² on Earth, affects how swiftly a pendulum accelerates towards the ground during its swing.
In the pendulum period formula, \[ T = 2\pi \sqrt{\frac{L}{g}} \], \( g \) is in the denominator, indicating:

• A greater gravitational force decreases the pendulum's period, making oscillations quicker.
• Conversely, in environments with lesser gravity, a pendulum would oscillate slower.

Thus, while changing Earth's gravitational force isn't feasible, understanding \( g \)'s role helps explain pendulum behavior under different gravitational conditions, such as on other planets or sizeable altitude changes on Earth.