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Chapter 10: Problem 43

A simple pendulum is made from a \(0.65-\mathrm{m}\) -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?

Short Answer

Expert verified
The pendulum takes approximately 0.81 seconds to attain its greatest speed.

Step by step solution

01

Understand the Problem

We are given a simple pendulum with a length of \(0.65\, \text{m}\). The ball attached to it is released from rest after being pulled to one side. We need to find the time it takes for the pendulum to reach its greatest speed, which occurs at the lowest point of the swing.
02

Recall the Pendulum Period Formula

For a simple pendulum, the period \(T\) is given by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity (approximately \(9.81\, \text{m/s}^2\)).
03

Calculate the Period \(T\)

Substitute the given length \(L = 0.65\, \text{m}\) into the period formula:\[ T = 2\pi \sqrt{\frac{0.65}{9.81}} \]Calculate this expression to find the period \(T\).
04

Compute Half the Period

The pendulum reaches its greatest speed at the lowest point of its swing, which is half the period. Compute this by dividing \(T\) by 2.
05

Solve for the Exact Values

Carry out the arithmetic to find the numerical value of \(T\) and then \(\frac{T}{2}\):\[ T = 2\pi \sqrt{\frac{0.65}{9.81}} \approx 1.62 \, \text{s} \]Therefore, \( \frac{T}{2} \approx 0.81 \, \text{s} \).

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Key Concepts

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

Pendulum Period Formula
The period of a simple pendulum is an important concept in understanding its motion. The formula to find the period, which is the time it takes for one complete back-and-forth swing, is given by:\[ T = 2\pi \sqrt{\frac{L}{g}} \]Here:
  • \(T\) represents the period.
  • \(L\) is the length of the pendulum from the point of suspension to the center of the mass.
  • \(g\) is the acceleration due to gravity, which is approximately \(9.81 \, \text{m/s}^2\) on Earth's surface.
This formula shows that the period is independent of the mass of the bob and depends only on the pendulum's length and the gravitational force. By substituting specific values into this equation, you can determine the period for any simple pendulum using its length.
Greatest Speed of Pendulum
A pendulum achieves its greatest speed at the lowest point of its swing. This is because all the potential energy stored at the highest point of the swing is converted into kinetic energy at the lowest point. The speed is maximized when the gravitational potential energy is minimum. The time it takes for a pendulum to reach this point after being released from a certain height is half the period of the pendulum's motion. This is because the pendulum first descends to its lowest point before ascending again to complete a full cycle. Thus, knowing the period allows us to calculate the time to reach maximum speed by dividing by two.
Length of Pendulum
The length of the pendulum, denoted as \(L\), is crucial for calculating the period. It is measured from the point of attachment to the center of mass of the pendulum bob. The longer the pendulum, the greater the period will be. This is because a longer pendulum arc means it takes more time to complete a swing.For example, if a pendulum has a length of \(0.65 \, \text{m}\), as in the exercise provided, you can substitute this length into the period formula to find out the time for one complete cycle. Thus, understanding and measuring the pendulum length correctly is key to predicting its motion accurately.
Acceleration Due to Gravity
The acceleration due to gravity, denoted by \(g\), is a force that pulls objects toward the center of the Earth. In pendulum calculations, it affects the speed and period of the pendulum's motion. On Earth, this value is approximately \(9.81 \, \text{m/s}^2\).This constant is crucial in the pendulum period formula and determines how fast the pendulum swings. Gravity assures that a longer pendulum with a higher arc will also have a longer period due to the gravitational pull needing more time to influence the motion.In other planetary conditions where gravity might differ, the pendulum's period would also change, showing how the acceleration due to gravity is a pivotal factor in pendulum dynamics.

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Most popular questions from this chapter

A hand exerciser utilizes a coiled spring. A force of 89.0 \(\mathrm{N}\) is required to compress the spring by 0.0191 \(\mathrm{m} .\) Determine the force needed to compress the spring by 0.0508 \(\mathrm{m} .\)

A 1.1\(\cdot \mathrm{kg}\) object is suspended from a vertical spring whose spring constant is 120 \(\mathrm{N} / \mathrm{m}\) . (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of 0.20 \(\mathrm{m}\) and released from rest. Find the speed with which the object passes through its original position on the way up.

A uniform \(1.4-\mathrm{kg}\) rod that is 0.75 \(\mathrm{m}\) long is suspended at rest from the ceiling by two springs, one at each end of the rod. Both springs hang straight down from the ceiling. The springs have identical lengths when they are unstretched. Their spring constants are 59 \(\mathrm{N} / \mathrm{m}\) and 33 \(\mathrm{N} / \mathrm{m}\) . Find the angle that the rod makes with the horizontal.

When an object of mass \(m_{1}\) is hung on a vertical spring and set into vertical simple harmonic motion, it oscillates at a frequency of 12.0 \(\mathrm{Hz}\) . When another object of mass \(m_{2}\) is hung on the spring along with the first object, the frequency of the motion is 4.00 \(\mathrm{Hz}\) . Find the ratio \(m_{2} / m_{1}\) of the mases.

Two physical pendulums (not simple pendulums) are made from meter sticks that are suspended from the ceiling at one end. The sticks are uniform and are identical in all respects, except that one is made of wood (mass \(=0.17 \mathrm{kg}\) and the other of metal (mass \(=0.85 \mathrm{kg}\) . They are set into oscillation and execute simple harmonic motion. Determine the period of \((\mathrm{a})\) the wood pendulum and \((\mathrm{b})\) the metal pendulum.
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