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Chapter 14: Problem 45

You pull a simple pendulum 0.240 m long to the side through an angle of \(3.50^{\circ}\) and release it. (a) How much time does it take the pendulum bob to reach its highest speed? (b) How much time does it take if the pendulum is released at an angle of \(1.75^{\circ}\) instead of \(3.50^{\circ} ?\)

Short Answer

Expert verified
(a) Time to highest speed: 0.245 s; (b) Time is the same: 0.245 s.

Step by step solution

01

Understand the Pendulum Motion

A simple pendulum exhibits simple harmonic motion when displaced at small angles. The time it takes to reach the highest speed from rest is a quarter of its period. The period (T) of a simple pendulum is given by:\[ T = 2\pi \sqrt{\frac{L}{g}} \]where L is the length of the pendulum and g is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)).
02

Calculate the Period of the Pendulum

Using the formula for the period, substitute the known values:\[ T = 2\pi \sqrt{\frac{0.240}{9.81}} \]Calculate the result to find the period of the pendulum.
03

Determine Time to Highest Speed for Initial Angle 3.50掳

The time to reach the highest speed is a quarter of the period because the pendulum swings from rest to maximum speed within a quarter cycle. Thus, the time is:\[ t = \frac{T}{4} \]
04

Calculate Time for Initial Angle 1.75掳

The period of a simple pendulum only depends on its length and gravity, not the initial angle. Therefore, for an initial angle of \(1.75^{\circ}\), repeat steps 2 and 3 with the same values for length and gravity.

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Key Concepts

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

Understanding Harmonic Motion
A simple pendulum moves in what is called simple harmonic motion, especially when it's released from small angles. This type of motion refers to the back-and-forth oscillation that happens in a repetitive and periodic manner. For a pendulum, this occurs between its highest points on either side and its lowest point in the middle.
  • At the highest points, the pendulum comes to a temporary stop before swinging back.
  • At the lowest point, the pendulum achieves its highest speed.
Simple harmonic motion is predictable. The pendulum's motion can be described using a sine or cosine function in physics. This is what makes pendulums helpful not only in clocks but also in various experiments where precise timing is required.
Influence of Pendulum Length
The length of a pendulum significantly affects how it swings and the period of its swing. The length is the distance from the pivot point to the center of mass of the pendulum bob. Longer pendulums have longer periods, meaning they take more time to complete a full swing.
  • A short pendulum swings faster and has a shorter period.
  • A long pendulum swings slower and has a longer period.
It's important to note that this length should remain constant for the pendulum to maintain uniform motion. In a classroom or lab, ensuring precise measurement of the pendulum's length is vital to achieving accurate results.
Role of Acceleration due to Gravity
The acceleration due to gravity, denoted as 'g', is crucial in determining the period of a pendulum. On Earth, this value is approximately 9.81 m/s虏, but it can vary slightly depending on location. This factor is part of the formula used to calculate the pendulum's period: \[ T = 2\pi \sqrt{\frac{L}{g}} \]Here, 'L' represents the pendulum's length, and 'g' is the local gravitational acceleration. On planets with higher gravity, a pendulum would swing faster, reducing the period. Conversely, on planets with lower gravity, the pendulum's oscillations would slow down, increasing the period. Thus, understanding 'g' is key in predicting the pendulum's behavior in different environments.
Calculating the Pendulum Period
The period of a pendulum is the time it takes to complete one full back-and-forth swing. It's crucial for determining how fast or slow a pendulum moves. The period is influenced by both the length of the pendulum and the acceleration due to gravity.
For a simple pendulum, the period can be calculated using the formula: \[ T = 2\pi \sqrt{\frac{L}{g}} \]From this formula, you can see that the period is independent of the initial angle of release, as long as the swings remain small. Understanding how to calculate the pendulum period is important for solving problems related to timekeeping and in physics experiments where timing accuracy is crucial.

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Most popular questions from this chapter

A uniform beam is suspended horizontally by two identical vertical springs that are attached between the ceiling and each end of the beam. The beam has mass 225 \(\mathrm{kg}\) , and a 175 -kg sack of gravel sits on the middle of it. The beam is oscillating in \(\mathrm{SHM}\) , with an amplitude of 40.0 \(\mathrm{cm}\) and a frequency of 0.600 cycle/s. (a) The sack of gravel falls off the beam when the beam has its maximum upward displacement. What are the frequency and amplitude of the subsequent SHM of the beam? (b) If the gravel instead falls off when the beam has its maximum speed, what are the frequency and amplitude of the subsequent SHM of the beam?

On a horizontal, frictionless table, an open-topped \(5.20-\mathrm{kg}\) box is attached to an ideal horizontal spring having force constant 375 \(\mathrm{N} / \mathrm{m}\) . Inside the box is a 3.44 -kg stone. The system is oscillating with an amplitude of 7.50 \(\mathrm{cm} .\) When the box has reached its maximum speed, the stone is suddenly plucked vertically out of the box without touching the box. Find (a) the period and (b) the amplitude of the resulting motion of the box. (c) Without doing any calculations, is the new period greater or smaller than the original period? How do you know?

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is 0.120 \(\mathrm{m} .\) The maximum speed of the block is 3.90 \(\mathrm{m} / \mathrm{s}\) . What is the maximum magnitude of the acceleration of the block?

A harmonic oscillator has angular frequency \(\omega\) and amplitude \(A .\) (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that \(U=0\) at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to \(A / 2,\) what fraction of the total energy of the system is kinetic and what fraction is potential?

CP SHM of a Butcher's Scale. A spring of negligible mass and force constant \(k=400 \mathrm{N} / \mathrm{m}\) is hung vertically, and a 0.200 -kg pan is suspended from its lower end. A butcher drops a 2.2 -kg steak onto the pan from a height of 0.40 \(\mathrm{m}\) . The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?
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