Problem 77 - College Physics

Question

To study the effects of acceleration on the period of oscillation, a student puts a grandfather clock with a \(0.9929-\mathrm{m}-\) long pendulum inside an elevator. Find the period of the grandfather clock (a) when the elevator is stationary, (b) when the elevator is accelerating upward at \(1.50 \mathrm{~m} / \mathrm{s}^{2}\) (c) when the elevator is accelerating downward at \(1.50 \mathrm{~m} / \mathrm{s}^{2},\) (d) when the cable on the elevator breaks and the elevator simply falls, and (e) when the elevator is moving upward at a constant speed of \(5.00 \mathrm{~m} / \mathrm{s}\).

Step-by-Step Solution

Answer

(a) T = 2.00 s, (b) T = 1.88 s, (c) T = 2.11 s, (d) Pendulum won't oscillate, (e) T = 2.00 s.

Step 1: Understanding the Period in Simple Pendulum

The 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 length of the pendulum, and \( g \) is the acceleration due to gravity. This formula assumes the pendulum is in a stationary frame where gravity is the only acceleration acting upon it.

Step 2: Calculate the Period When the Elevator is Stationary

When the elevator is stationary, the effective acceleration \( g = 9.81 \, \mathrm{m/s^2} \). Substitute \( L = 0.9929 \, \mathrm{m} \) and \( g = 9.81 \, \mathrm{m/s^2} \) into the formula. \[T = 2\pi \sqrt{\frac{0.9929}{9.81}}\]This calculation gives you the period of the pendulum in a stationary elevator.

Step 3: Calculate the Period When the Elevator is Accelerating Upward

When the elevator accelerates upward at \( 1.50 \, \mathrm{m/s^2} \), the effective gravity is \( g = 9.81 + 1.50 = 11.31 \, \mathrm{m/s^2} \). Substitute \( g = 11.31 \, \mathrm{m/s^2} \) into the formula to compute the period:\[T = 2\pi \sqrt{\frac{0.9929}{11.31}}\]

Step 4: Calculate the Period When the Elevator is Accelerating Downward

When the elevator accelerates downward at \( 1.50 \, \mathrm{m/s^2} \), the effective gravity is \( g = 9.81 - 1.50 = 8.31 \, \mathrm{m/s^2} \). Substitute \( g = 8.31 \, \mathrm{m/s^2} \) into the formula to compute the period:\[T = 2\pi \sqrt{\frac{0.9929}{8.31}}\]

Step 5: Calculate the Period When the Cable Breaks

If the cable breaks, the effective gravity becomes zero because the elevator is in free fall. Hence, \( g = 0 \). Since the formula \( T = 2\pi \sqrt{\frac{L}{g}} \) involves division by \( g \), and \( g = 0 \), the expression becomes undefined, indicating that the pendulum would not oscillate.

Step 6: Calculate the Period When the Elevator is Moving at Constant Speed

If the elevator moves upward at a constant speed of \( 5.00 \, \mathrm{m/s} \), the acceleration is zero, implying the effective gravity remains \( g = 9.81 \, \mathrm{m/s^2} \). Thus, the period is the same as when it is stationary:\[T = 2\pi \sqrt{\frac{0.9929}{9.81}}\]

Key Concepts

Simple Harmonic Motion

Pendulums, like those in a grandfather clock, often exhibit a type of movement known as Simple Harmonic Motion (SHM). This motion is typified by the pendulum swinging back and forth in a regular and repeating cycle. What makes SHM special is its predictability—over time, it follows a sinusoidal pattern. Simple Harmonic Motion is centered around a stable equilibrium position, which is the point the pendulum tries to return to after being displaced.

The key characteristics of SHM extend to a few important factors:

• Amplitude: This is the maximum angle or distance the pendulum swings away from its equilibrium position.
• Period: Defined as the time it takes to complete one full swing back and forth.
• Frequency: The number of complete cycles the pendulum makes per unit of time. This is often closely linked to the period as frequency is inversely proportional to the period.

The simplicity of SHM makes it a fundamental concept in physics, applicable not just to pendulums but to a variety of other systems like springs and even atoms in a lattice.

Acceleration Effects

The acceleration of an elevator can significantly affect the motion of a pendulum inside it. When an elevator accelerates, it changes the effective gravitational force acting on the pendulum.

To understand how acceleration affects the pendulum, consider the different scenarios:

• Accelerating Upward: Effective gravitational force increases because gravity and acceleration in the same direction add up. This results in a shorter period of oscillation since the pendulum feels stronger gravity.
• Accelerating Downward: Effective gravitational force decreases because gravity and acceleration act in opposite directions. The pendulum has a longer period of oscillation as it experiences a weaker effective gravity.

In summary, the pendulum's period changes with varying levels of acceleration, illustrating the intricate relationship between acceleration and oscillation in a moving reference frame.

Gravitational Acceleration

Gravitational acceleration, denoted by the symbol \( g \), is a crucial factor in pendulum motion. It represents the natural pull that the Earth's gravity exerts on objects. The standard value of gravitational acceleration near the Earth's surface is \( 9.81 \, \mathrm{m/s^2} \).

In the context of a pendulum, gravitational acceleration determines how quickly and repetitively the pendulum swings back and forth. It plays a vital role in calculating the pendulum's oscillation period using the formula:\[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( L \) is the pendulum length.The interplay between length and gravitational acceleration means that a pendulum of a given length will oscillate more rapidly where gravity feels stronger and slower where it is weaker. Understanding this concept helps demonstrate why pendulum clocks keep different time under varying gravitational conditions.

Free Fall

Free fall occurs when the only force acting on an object is gravity—no other forces, like air resistance, interfere. In our pendulum scenario, if the elevator cable breaks, the elevator, along with everything inside, would be in a state of free fall. The effective gravitational acceleration would become zero as both the elevator and the pendulum fall at the same rate.

In this condition:

• The pendulum no longer experiences a restoring force to bring it back to its equilibrium position.
• It stops oscillating because the formula for period, \( T = 2\pi \sqrt{\frac{L}{g}} \), becomes invalid when \( g = 0 \).

This dramatizes a fundamental principle in physics where in free fall, all objects accelerate at the same rate irrespective of their mass, leading to a state where typical oscillations cease.