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Chapter 10: Problem 46

A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the center of the staircase. However, nothing is available with which to measure the length of the rope. Therefore, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be 9.2 s. What is the height of the tower?

Short Answer

Expert verified
The height of the tower is approximately 21.03 m.

Step by step solution

01

Understanding the Relationship between Period and Length

The period of a simple pendulum is related to its length through 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, approximately 9.81 m/s虏.
02

Solving for Length of the Pendulum

To find the length \( L \), rearrange the pendulum formula to solve for \( L \): \( L = \frac{gT^2}{4\pi^2} \). Substitute \( T = 9.2 \, s \) and \( g = 9.81 \, m/s^2 \).
03

Calculating Length with Given Values

Substitute the given period into the formula: \( L = \frac{9.81 \times (9.2)^2}{4\pi^2} \). Calculating this expression, we find \( L \approx 21.03 \, m \).
04

Interpreting the Result

The calculated length \( L \) represents the length of the pendulum, which in this scenario is also the height of the tower as the rope is hanging directly down the middle of the staircase.

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Key Concepts

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

Simple Pendulum
A simple pendulum is a basic yet fascinating physics concept. Imagine a weight, called a pendulum bob, suspended from a fixed point so that it can swing freely. This setup creates a simple pendulum. When you pull the bob slightly to one side and release it, it swings back and forth due to gravity. This motion is regular and repetitive, known as periodic motion. The beauty of a simple pendulum lies in its simplicity, which also makes it a great tool for studying fundamental physics principles. Its behavior is predictable and can be used to measure characteristics such as time and height.
Period of a Pendulum
The period of a pendulum is the time it takes to complete one full swing, starting from one side, swinging to the opposite side, and returning back. This time duration is influenced by the length of the pendulum and the strength of gravity, but interestingly, not by the mass of the bob or the arc of the swing. In simple terms, the longer the pendulum, the longer its period. This is described by the formula:
  • Period (\( T \)) = 2蟺 \( \sqrt{\frac{L}{g}} \)
Where \( T \) is the period, \( L \) is the length, and \( g \) is the acceleration due to gravity. From this formula, you can see that the period is proportional to the square root of the length. Hence, a pendulum's swing time can tell us much about its length.
Length Calculation
Calculating the length of a pendulum can be achieved by rearranging the period formula. By knowing the period and the acceleration due to gravity, you can solve for the length. Let's look at how this works:
  • Length (\( L \)) = \( \frac{gT^2}{4蟺^2} \)
Using this formula, if you measure the period (\( T \)) to be 9.2 seconds, and you know that gravity (\( g \)) is approximately 9.81 m/s虏, you can calculate the length of the pendulum. This calculation is vital in scenarios where direct measurement is difficult, such as determining the height of a tower using a pendulum as described in your problem.
Gravity Acceleration
Gravity plays an essential role in the motion of a simple pendulum. It acts as the restoring force that causes the pendulum to swing back to its equilibrium position. The standard acceleration due to gravity on Earth's surface is approximately 9.81 m/s虏, although this can vary slightly depending on your location on the planet. This constant value is crucial when calculating the period and length of a pendulum, as it influences how fast the pendulum swings.
  • Larger gravity = shorter period (faster swings)
  • Smaller gravity = longer period (slower swings)
Understanding how gravity impacts a pendulum helps in making accurate predictions and calculations, enhancing our ability to use pendulums in various practical applications like measuring heights or synchronizing clocks.

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Most popular questions from this chapter

A spring is hung from the ceiling. A \(0.450-\mathrm{kg}\) block is then attached to the free end of the spring. When released from rest, the block drops \(0.150 \mathrm{m}\) before momentarily coming to rest, after which it moves back upward. (a) What is the spring constant of the spring? (b) Find the angular frequency of the block's vibrations.

An object attached to a horizontal spring is oscillating back and forth along a frictionless surface. The maximum speed of the object is \(1.25 \mathrm{m} / \mathrm{s},\) and its maximum acceleration is \(6.89 \mathrm{m} / \mathrm{s}^{2} .\) How much time elapses between an instant when the object's speed is at a maximum and the next instant when its acceleration is at a maximum?

A loudspeaker diaphragm is producing a sound for 2.5 s by moving back and forth in simple harmonic motion. The angular frequency of the motion is \(7.54 \times 10^{4} \mathrm{rad} / \mathrm{s} .\) How many times does the diaphragm move back and forth?

To measure the static friction coefficient between a \(1.6-\mathrm{kg}\) block and a vertical wall, the setup shown in the drawing is used. A spring (spring constant \(=510 \mathrm{N} / \mathrm{m}\) ) is attached to the block. Someone pushes on the end of the spring in a direction perpendicular to the wall until the block does not slip downward. The spring is compressed by \(0.039 \mathrm{m} .\) What is the coefficient of static friction?

An 86.0-kg climber is scaling the vertical wall of a mountain. His safety rope is made of nylon that, when stretched, behaves like a spring with a spring constant of \(1.20 \times 10^{3} \mathrm{N} / \mathrm{m} .\) He accidentally slips and falls freely for \(0.750 \mathrm{m}\) before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?
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