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Chapter 12: Problem 2

The fundamental premise of simple harmonic motion is that a force must be proportional to an object's displacement. Is anything else required?

Short Answer

Expert verified
For simple harmonic motion to take place, the force exerted on an object needs to be directly proportional to the displacement and directed towards the equilibrium position.

Step by step solution

01

Defining Simple Harmonic Motion

Simple harmonic motion, or SHM, is a type of periodic motion where the restoring force is directly proportional to the displacement. It occurs when the force that restores the object to its equilibrium position is linearly proportional to the displacement from equilibrium.
02

The relation between force and displacement in SHM

In simple harmonic motion, the force acting on an object should not just be proportional, it should be directly proportional to the displacement. This means that as the displacement increases, the force also increases.
03

Direction of force in SHM

Another key characteristic of SHM is that the force should always be directed towards the equilibrium position. This means that the force is always trying to restore the object back to its original position.
04

Concluding the requirements for SHM

To sum up, for an object to exhibit simple harmonic motion, the force exerted on it needs to be directly proportional to the displacement and must be directed towards the equilibrium position.

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Key Concepts

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

Periodic Motion
In the context of physics, periodic motion refers to the motion that repeats itself at regular time intervals. This concept is a cornerstone for understanding many natural phenomena. Examples of periodic motion include the swinging of a pendulum, the oscillation of a spring, and even the rotation of Earth around the Sun. What characterizes periodic motion is its consistency鈥攊t occurs over and over again, in the same sequence, at fixed durations.

For a motion to be periodic, it must have:
  • A regular time period: This is the time it takes for the cycle of motion to repeat itself.
  • A repetitive pattern: The motion returns to its starting point in every cycle.
Periodic motion is not exclusive to simple harmonic motion (SHM), but all SHM is periodic. Understanding this helps in identifying and analyzing different types of motion in various physical systems.
Restoring Force
A restoring force is a crucial concept when discussing simple harmonic motion. It is the force that acts to bring an object back to its equilibrium position. This force is directly responsible for the object's oscillatory behavior.

Key characteristics of a restoring force include:
  • It is proportional to displacement: The force grows stronger as the object moves further from equilibrium. Mathematically, this can be expressed as \( F = -kx \), where \( F \) is the restoring force, \( k \) is a constant, and \( x \) is the displacement.
  • It is directed towards equilibrium: This ensures the object accelerates back towards its original position, enabling the characteristic to-and-fro motion.
The negative sign in the formula \( F = -kx \) signifies that the force's direction is opposite to that of the displacement. Understanding how restoring forces work is essential for grasping why systems like springs and pendulums exhibit simple harmonic motion.
Equilibrium Position
The equilibrium position in simple harmonic motion is the point where the net force acting on the object is zero. It is the position where the object would remain at rest if not disturbed. This concept plays a central role in analyzing how and why objects oscillate.

Important aspects of the equilibrium position include:
  • Zero net force: At this position, there is no resultant force acting on the object, meaning it won't accelerate.
  • Central point of motion: All oscillations in simple harmonic motion are centered around this point. The object repeatedly moves away from and then returns to this position.
Recognizing the equilibrium position helps us understand the dynamics of simple harmonic motion. It explains why objects tend to return to a specific point after experiencing a displacement. Additionally, determining the equilibrium position allows for easier calculation of other parameters, such as the amplitude and potential energy in simple harmonic systems.

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Most popular questions from this chapter

Compare \(x(t)=A \cos \omega t\) to \(x(t)=A \cos (\omega t+\phi)\). What is the phase angle \(\phi\) and how does it change the solution to simple harmonic motion?

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 simple harmonic oscillator is observed to start its oscillations at the maximum amplitude when \(t=0\). Devise an appropriate solution that is consistent with this initial condition. Repeat when the oscillations start at the equilibrium position when \(t=0\).

Euler's formula states \(e^{i \omega t}=\cos (\omega t)+i \sin (\omega t)\) (where \(i=\sqrt{-1})\). (a) Show that \(x(t)=A e^{i \omega t}\) solves the force equation for simple harmonic motion. (b) What are the units of the constant \(A\) ?

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?
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