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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 block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of \(7.0 \mathrm{rad} / \mathrm{s} .\) The drawing indicates the position of the block when the spring is unstrained. This position is labeled " \(x=\) \(0 \mathrm{m} .\) "The drawing also shows a small bottle located \(0.080 \mathrm{m}\) to the right of this position. The block is pulled to the right, stretching the spring by \(0.050 \mathrm{m},\) and is then thrown to the left. In order for the block to knock over the bottle, it must be thrown with a speed exceeding \(v_{0} .\) Ignoring the width of the block, find \(v_{0}\).

A heavy-duty stapling gun uses a 0.140 -kg metal rod that rams against the staple to eject it. The rod is attached to and pushed by a stiff spring called a "ram spring" \((k=32000 \mathrm{N} / \mathrm{m})\). The mass of this spring may be ignored. The ram spring is compressed by \(3.0 \times 10^{-2} \mathrm{m}\) from its unstrained length and then released from rest. Assuming that the ram spring is oriented vertically and is still compressed by \(0.8 \times 10^{-2} \mathrm{m}\) when the downward-moving ram hits the staple, find the speed of the ram at the instant of contact.

A simple pendulum is made from a 0.65-m-long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

A spring lies on a horizontal table, and the left end of the spring is attached to a wall. The other end is connected to a box. The box is pulled to the right, stretching the spring. Static friction exists between the box and the table, so when the spring is stretched only by a small amount and the box is released, the box does not move. The mass of the box is \(0.80 \mathrm{kg}\), and the spring has a spring constant of \(59 \mathrm{N} / \mathrm{m}\). The coefficient of static friction between the box and the table on which it rests is \(\mu_{\mathrm{s}}=0.74 .\) How far can the spring be stretched from its unstrained position without the box moving when it is released?

A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is \(45.0 \mathrm{cm} .\) Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?
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