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Chapter 12: Problem 13

The application of an external force on a simple pendulum can create many different outcomes, depending on how frequently the force is applied. Explain what will happen to the amplitude of the motion if an external force is applied to a simple pendulum at the same frequency as the natural frequency of the pendulum.

Short Answer

Expert verified
In case of a simple pendulum, when an external force is applied at the same frequency as its natural frequency, resonance occurs which leads to an increase in the amplitude of the pendulum's motion.

Step by step solution

01

Understanding Resonance

Picturing a pendulum under the impact of an external force, is the first step in understanding the principle of resonance. Resonance is a physical phenomenon that occurs when an object is forced to vibrate at its natural frequency. Once there is a match of the frequency of the applied external force and the natural frequency of the system (in this case, our simple pendulum), resonance occurs.
02

Resonance and Amplitude

Once resonance occurs, the amplitude - the maximum displacement or distance moved by the pendulum from its equilibrium position - increases. This is because the energy of the system accumulates, causing the pendulum to swing more widely. When force is applied consistently at the pendulum's natural frequency, its amplitude continues to increase.
03

Final Observation

So, if an external force is applied to a simple pendulum at the same frequency as the natural frequency of the pendulum, the amplitude of the pendulum's motion will increase. This is a direct result of the phenomenon of resonance.

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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 mechanical system often used in physics to illustrate the principles of oscillatory motion. It consists of a mass (called the bob) attached to the end of a string or rod, which swings back and forth around a fixed point. When this pendulum is set in motion by pulling it to one side and letting go, it will move back and forth due to the force of gravity.
  • The path it takes is an arc of a circle, and the point where the pendulum hangs at rest is its equilibrium position.
  • The time it takes to complete one full swing back and forth is known as the period of the pendulum.
  • The period of a simple pendulum depends on the length of the string or rod and the acceleration due to gravity, but not on the mass of the bob.
Understanding the simple pendulum is essential in grasping concepts such as resonance, as it provides a clear model to observe oscillations and how they respond to external forces.
Amplitude
Amplitude in the context of a pendulum refers to the maximum extent of its swing from the equilibrium position. It essentially measures how far the pendulum travels from its central rest point at the peak of its motion.
  • Amplitude is a critical factor in determining the energy within the pendulum system. Higher amplitude indicates greater energy.
  • When the pendulum is disturbed and set in motion, the potential energy at the highest point of the swing is converted to kinetic energy at the lowest point.
  • An increase in amplitude, especially due to resonance, means more energy has been added to the system.
In situations such as when resonance occurs, if the external force matches the pendulum's natural frequency, the energy is added at just the right time each cycle, causing the amplitude to grow significantly.
Natural Frequency
The natural frequency of a simple pendulum is the frequency at which it tends to oscillate in the absence of any external force or interference. Each pendulum has its own specific natural frequency based on its characteristics:
  • It is determined by the length of the pendulum and the gravitational force acting on it.
  • Longer pendulums have lower natural frequencies, meaning they swing more slowly, while shorter pendulums swing faster.
  • Natural frequency is crucial when discussing resonance, as applying an external force at this frequency can significantly affect the system's behavior.
When an external force is applied at this exact frequency, the energy inputs align perfectly with the pendulum's natural oscillation. This alignment is what leads to the phenomenon of resonance, causing an increase in amplitude and a more vigorous swinging motion.

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Most popular questions from this chapter

A spinning golf ball of radius \(R\) can be suspended in a stream of high velocity air (Figure 12-30). The ball is in equilibrium at the vertical center of the air stream \((y=0)\), and not moving except for its rotation at velocity \(v_{\text {Rot }}\). Show that the ball undergoes simple harmonic motion when it is released a small vertical distance \(y\) below the equilibrium position. Assume that for small \(y\) there is no net horizontal force, and that for small values of \(y\) the speed of the air drops off linearly from the center of the air stream. (For small \(y\), the speed varies as \(v(y)=v_{0}-b|y|\), where \(v_{0}\) is the speed of the air in the center of the stream and \(b\) is a constant.) SSM

A \( 100-\mathrm{g}\) object is fixed to the end of a spring that has a spring constant of \(15 \mathrm{~N} / \mathrm{m}\). The object is displaced \(15 \mathrm{~cm}\) to the right and released from rest at \(t=0\) to slide on a horizontal, frictionless table. (a) Find the first three times when the object is at the equilibrium position. (b) Find the first three times when the object is \(10 \mathrm{~cm}\) to the left of equilibrium. (c) What is the first time that the object is \(5 \mathrm{~cm}\) to the right of equilibrium, moving toward the left?

Not all oscillatory motion is simple harmonic, but simple harmonic motion is always oscillatory. Explain this statement and give an example to support your explanation. SSM

The position as a function of time for an object that has a mass \(m\), is attached to a spring that has a force constant \(k\), and is sliding on a horizontal frictionless table is given by \(x(t)=A \cos (\omega t+\phi)\) where \(\omega=\sqrt{k / m}\). As a function of time, determine an expression for (a) the potential energy of the object-spring system and (b) the kinetic energy of the object-spring system. (c) Show that the total energy of the object-spring system is conserved. SSM

Starting with the force equation for a damped harmonic oscillator, show that a solution of the form \(x(t)=A e^{-(b / 2 m) t} \sin \omega_{1} t\) works. The differential equation and the lightly damped oscillation frequency are \(m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=0 \quad\) and \(\quad \omega_{1}=\sqrt{\omega_{0}^{2}-\frac{b^{2}}{4 m^{2}}}\)
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