Simple Harmonic Motion

Page 1
This text is meant to accompany class discussions. It is not everything there is to know about simple harmonic motion. It is meant as a  prep for class.

Click for the questions that go with this reading
Springs

When you pull on a spring it pulls back. But hows does the force behave?

Hang various masses from the same spring. Below is what is may look like. (All the springs are identical.)

Hanging Weights from Springs

A graph of hanging weight versus displacement from starting point may look like the one below.

This chapter will examine the behavior of ideal springs. Ideal springs don't loose energy and are 100% elastic. This means that we are only going to examine the elastic region of the graph.

The elastic region is linear. It's equation is "w = mx + 0." This simplifies to "w = mx." The slope in the elastic region is special. It is called the "spring constant." Every spring has it own spring constant. The spring constant is represented with the variable "k." It has the units of slope from this graph, N/m. This makes the equation "w = kx."

Example 1: Check your understanding

 

 

Hooke's Law

 

Look closer at the forces acting on the hanging mass.

free body diagram

Because the hanging mass is not moving the weight equals the force of the spring pulling up. Since the spring's force is in the opposite direction of the weight, w = —F. Therefore the equation above becomes.

Fspring = —kx

This equation is Hooke's Law. Fspring is the spring's force. The spring force always opposes any force that is applied to the spring. This makes Fspring always negative. k is the spring constant. The spring constant describes the spring's stiffness. x is the displacement from a spring's equilibrium position.

 

The spring constant describes a spring's stiffness. They have a wide range. For example:

Jeep springs pen and spring
The red coils between the wheel and the engine are the springs. They could have spring constant from 30,000 to 300,000 N/m 4-the-love-of-jeeps.com (Citation:When are Jeep lifts needed? 4-the-love-of-jeeps.com,. 'When Are Jeep Lifts Needed?'. N. p., 2015. Web. 5 Mar. 2015.)

This pen has a spring constant of around 10 N/m.

A slinky has a spring constant approximately equal to 2 N/m.

 


The equilibrium position is the rest position of the spring before it is stretched or compressed. This is where x = 0.

Eqilibrium position

The spring constant can be determined by calculating the ratios of the force to displacement, k = Force/displacement or from the slope of a spring force vs. displacement graph.

Note: Technically the rest position of the hanging spring is a "quasi-equilibrium." For this course consider this position to be the equilibrium position.

 

Example 2: Check your understanding
  • Question
  • Answer
A 50.0 N force stretches a spring 10.0 cm. What is this spring's spring constant?
asnwer

by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)

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