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Chapter 15: Problem 1

What conditions must be met to produce SHM?

Short Answer

Expert verified
For a motion to be classified as Simple Harmonic Motion, the following conditions must be met: 1. A restoring force must be present in the system (\(F = -kx\)). 2. The restoring force must be linear with respect to the displacement from the equilibrium position (\(F(x) = -kx\)). 3. The restoring force must always act in the opposite direction of the displacement. 4. The object's acceleration must be proportional to the negative of its displacement from the equilibrium position (\(a(x) = -c x\)).

Step by step solution

01

Condition 1: Restoring Force

For a motion to be classified as SHM, a restoring force must be present in the system. The restoring force is the force that acts on the object to bring it back to its equilibrium position. Mathematically, this force can be represented as \(F = -kx\), where F is the restoring force, k is a positive constant, and x is the displacement from the equilibrium position.
02

Condition 2: Linearity

The second condition for SHM is that the restoring force acting on the object must be proportional to its displacement from the equilibrium position. This can be represented as \(F(x) = -kx\), which tells us that the force applied on the object is linear with respect to its displacement.
03

Condition 3: Force Direction

The third condition for SHM is that the restoring force must always act in the opposite direction of the displacement. This means that if the object is displaced to the right from its equilibrium position, the restoring force will act towards the left, and vice versa.
04

Condition 4: Acceleration

The fourth condition for SHM is that the object must have an acceleration proportional to the negative of its displacement from the equilibrium position. Mathematically, this can be represented as \(a(x) = -c x\), where a(x) is the acceleration, c is a positive constant, and x is the displacement from the equilibrium position. In summary, for a motion to be classified as Simple Harmonic Motion, the following conditions must be met: 1. A restoring force must be present in the system. 2. The restoring force must be linear with respect to the displacement from the equilibrium position. 3. The restoring force must always act in the opposite direction of the displacement. 4. The object's acceleration must be proportional to the negative of its displacement from the equilibrium position.

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Key Concepts

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

Restoring Force
In Simple Harmonic Motion (SHM), the concept of restoring force is crucial. The restoring force is the driving factor that brings the object back to its equilibrium position. It acts like an invisible hand constantly guiding the object to its original state. Mathematically, this force is expressed as \( F = -kx \).
The negative sign indicates that the force always acts in the direction opposite to the displacement.
Here:
  • F is the restoring force.
  • k is a spring constant, a positive value which dictates the stiffness of the spring (or similar mechanism).
  • x is the displacement from the equilibrium position.
Whenever an object is displaced from its equilibrium position, the restoring force works diligently to return it to balance.
Equilibrium Position
The equilibrium position is the heart of Simple Harmonic Motion. It鈥檚 the central point to which the force and motion are always referenced. Imagine when you are on a swing, the position where you naturally come to rest without external forces is your equilibrium position.
In physics, this is the point where the net force acting on the object is zero. No push, no pull, just balance.
This concept is essential because:
  • It's the point where the energy of the system is minimized.
  • It serves as a reference point from which displacement (and thus forces and acceleration) are measured.
In SHM, every movement away from this position increases the restoring force trying to bring the object back to equilibrium.
Proportionality and Linearity
A key condition for SHM is that the restoring force should be directly proportional and linear with respect to displacement. This is crucial because it ensures the predictable, oscillatory nature of SHM.
When a force is linear, it implies a straight-line relationship, which is expressed as \( F(x) = -kx \).
This tells us:
  • The magnitude of the force changes consistently in response to any displacement.
  • The relationship between force and displacement is predictable and constant.
In simpler terms, the further you are from the equilibrium, the stronger the push or pull to return you, keeping the motion in harmony and rhythm.
Acceleration and Displacement
Understanding the link between acceleration and displacement in SHM is vital. The acceleration in SHM is always proportional to the negative of its displacement from equilibrium. It's what keeps the motion in perfect cycle.
Mathematically, it's expressed as \( a(x) = -c x \), where:
  • a(x) is the acceleration.
  • c is a constant determined by the system's characteristics.
  • x is the displacement.
The negative sign indicates this acceleration always resists the displacement, pulling the object back to the starting line once again. Without this proportionality, the uniform back-and-forth motion of SHM would be impossible.

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Most popular questions from this chapter

Reciprocating motion uses the rotation of a motor to produce linear motion up and down or back and forth. This is how a reciprocating saw operates, as shown below. If the motor rotates at \(60 \mathrm{Hz}\) and has a radius of \(3.0 \mathrm{cm}\) estimate the maximum speed of the saw blade as it moves up and down. This design is known as a scotch yoke.

A mass \(m_{0}\) is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and released. The mass oscillates with a frequency \(f_{0} .\) If the mass is replaced with a mass nine times as large, and the experiment was repeated, what would be the frequency of the oscillations in terms of \(f_{0} ?\)

Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction \(\mu_{\mathrm{s}}=0.100\) (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is \(\mu_{\mathrm{k}}=0.0850,\) what total distance does it travel before stopping? Assume it starts at the maximum amplitude.

The motion of a mass on a spring hung vertically, where the mass oscillates up and down, can also be modeled using the rotating disk. Instead of the lights being placed horizontally along the top and pointing down, place the lights vertically and have the lights shine on the side of the rotating disk. A shadow will be produced on a nearby wall, and will move up and down. Write the equations of motion for the shadow taking the position at \(t=0.0 \mathrm{s}\) to be \(y=0.0 \mathrm{m}\) with the mass moving in the positive y-direction.

(a) What is the effect on the period of a pendulum if you double its length? (b) What is the effect on the period of a pendulum if you decrease its length by \(5.00 \% ?\)
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