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Chapter 13: Problem 40

A simple pendulum is \(5.00 \mathrm{~m}\) long. (a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at \(5.00 \mathrm{~m} / \mathrm{s}^{2} ?\) (b) What is its period if the elevator is accelerating downward at \(5.00 \mathrm{~m} / \mathrm{s}^{2} ?(\mathrm{c})\) What is the period of simple harmonic motion for the pendulum if it is placed in a truck that is accelerating horizontally at \(5.00 \mathrm{~m} / \mathrm{s}^{2} ?\)

Short Answer

Expert verified
(a) The period of the pendulum when the elevator is accelerating upward is \( T_{upward} = 2\pi\sqrt{\frac{5.00}{14.81}} \). (b) The period of the pendulum when the elevator is accelerating downward is \( T_{downward} = 2\pi\sqrt{\frac{5.00}{4.81}} \). (c) The period of the pendulum inside a truck accelerating horizontally is \( T_{horizontal} = 2\pi\sqrt{\frac{5.00}{9.81}} \).

Step by step solution

01

Defining the Variables

First, let's define the variables. The length of the pendulum \( l \) is 5.00m, and the acceleration due to gravity \( g \) is approximately 9.81 m/s虏. We also have the accelerations upward, downward, and horizontally, which are all 5.00 m/s虏.
02

The Period of a Pendulum

The period \( T \) of a simple pendulum is given by the formula \( T = 2\pi\sqrt{\frac{l}{g}} \), where \( g' \) is the effective acceleration due to gravity, it can change depending on the specific situation, like if the elevator is moving upward \( g' = g + a \), downward \( g' = g - a \), or horizontally \( g' = g \).
03

Calculate the Period in Different Conditions

(a) For the elevator accelerating upward, the acceleration due to gravity \( g' = g + a = 9.81 + 5.00 = 14.81 \, m/s虏 \). Plug it into the formula we get \( T = 2\pi\sqrt{\frac{5.00}{14.81}} \). (b) For the elevator accelerating downward, the acceleration due to gravity \( g' = g - a = 9.81 - 5.00 = 4.81 \, m/s虏 \). Plug it into the formula we get \( T = 2\pi\sqrt{\frac{5.00}{4.81}} \). (c) As for the horizontal acceleration, it does not affect the pendulum, so \( g' = g \), which gives \( T = 2\pi\sqrt{\frac{5.00}{9.81}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Harmonic Motion
Simple harmonic motion (SHM) is seen in systems where the force working on an object is directly proportional to the displacement from its equilibrium position.
In the context of a simple pendulum, SHM means the pendulum will swing back and forth in a regular, repeated manner.
This regular motion is akin to the pendulum trying to return to its center position with a force proportional to how far it has swung away from it. Understanding SHM:
  • The oscillation is periodic, meaning it repeats at regular time intervals.
  • The restoring force is gravity, pulling the pendulum back to its resting position.
  • Small-angle approximations help simplify the motion to SHM.
By considering these principles, analyzing the pendulum's behavior in different accelerating environments can be straightforward.
Period of a Pendulum
The period of a pendulum refers to the time it takes to complete one full oscillation, from one extreme side to the other and back. For a simple pendulum, the period can be calculated using:\[T = 2\pi\sqrt{\frac{l}{g'}}\]where:
  • \( T \) is the period.
  • \( l \) is the length of the pendulum.
  • \( g' \) is the effective acceleration due to gravity, adjusted for additional forces acting, like elevator motion or horizontal elements.
Understanding Different Situations:- **Elevator Moving Upward:** The effective gravity \( g' \) increases as the elevator's upward acceleration adds to gravity.- **Elevator Moving Downward:** Here, \( g' \) decreases because the elevator removes some effect of gravity.- **Horizontal Acceleration:** Pendulum motion stays unaffected as horizontal forces do not enter the equation for vertical motion.
Acceleration due to Gravity
Acceleration due to gravity \( g \) is a crucial concept affecting a pendulum's period. At the Earth's surface, \( g \) averages about 9.81 m/s虏.
This force constantly acts on the pendulum, helping it return to its central position. In real-world situations:- **Increasing \( g \):** When additional forces (like an upward elevator movement) add to \( g \), it feels stronger. The pendulum swings faster, reducing its period.- **Decreasing \( g \):** Opposite forces (downward elevator movement) make gravity feel weaker. The pendulum slows, lengthening its period.

Practical Implications

Gravity directly influences pendulum behavior in various environments:
  • Understanding effective \( g \) enables better design and analysis of time-keeping devices using pendulums.
  • Adjusting pendulum length or recognizing gravity changes helps maintain accurate oscillation timings.
These principles show how external accelerations modify a pendulum鈥檚 characteristics fundamentally by affecting \( g \).

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Most popular questions from this chapter

When a \(4.25-\mathrm{kg}\) object is placed on top of a vertical spring, the spring compresses a distance of \(2.62 \mathrm{~cm} .\) What is the force constant of the spring?

A simple pendulum has mass \(1.20 \mathrm{~kg}\) and length \(0.700 \mathrm{~m}\). (a) What is the period of the pendulum near the surface of Earth? (b) If the same mass is attached to a spring, what spring constant would result in the period of motion found in part (a)?

Ocean waves are traveling to the east at \(4.0 \mathrm{~m} / \mathrm{s}\) with a distance of \(20 \mathrm{~m}\) between crests. With what frequency do the waves hit the front of a boat (a) when the boat is at anchor and (b) when the boat is moving westward at \(1.0 \mathrm{~m} / \mathrm{s} ?\)

ecp A spring \(1.50 \mathrm{~m}\) long with force constant \(475 \mathrm{~N} / \mathrm{m}\) is hung from the ceiling of an elevator, and a block of mass \(10.0 \mathrm{~kg}\) is attached to the bottom of the spring. (a) By how much is the spring stretched when the block is slowly lowered to its equilibrium point? (b) If the elevator subsequently accelerates upward at \(2.00 \mathrm{~m} / \mathrm{s}^{2}\), what is the position of the block, taking the equilibrium position found in part (a) as \(y=0\) and upwards as the positive \(y\) -direction. (c) If the elevator cable snaps during the acceleration, describe the subsequent motion of the block relative to the freely falling elevator. What is the amplitude of its motion?

A pendulum clock that works perfectly on the Earth is taken to the Moon. (a) Does it run fast or slow there? (b) If the clock is started at 12:00 midnight, what will it read after one Earth day \((24.0 \mathrm{~h})\) ? Assume the free-fall acceleration on the Moon is \(1.63 \mathrm{~m} / \mathrm{s}^{2}\).
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