AP Physics – Unit 6 Simple Harmonic Motion Notes 1

Simple Harmonic Motion Learning Objectives

3.B.3 Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion.

Examples should include gravitational force exerted by the Earth on a simple pendulum, mass-spring oscillator.

a. For a spring that exerts a linear restoring force the period of a mass-spring oscillator increases with mass and decreases with spring stiffness.

b. For a simple pendulum oscillating the period increases with the length of the pendulum.

c. Minima, maxima, and zeros of position, velocity, and acceleration are features of harmonic motion. Students should be able to calculate force and acceleration for any given displacement for an object oscillating on a spring.

3.B.3.1 The student is able to predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties.

3.B.3.2 The student is able to design a plan and collect data in order to ascertain the characteristics of the motion of a system undergoing oscillatory motion caused by a restoring force.

3.B.3.3 The student can analyze data to identify qualitative or quantitative relationships between given values and variables (i.e., force, displacement, acceleration, velocity, period of motion, frequency, spring constant, string length, mass) associated with objects in oscillatory motion to use that data to determine the value of an unknown.

3.B.3.4 The student is able to construct a qualitative and/or a quantitative explanation of oscillatory behavior given evidence of a restoring force.

5.B.2 A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy.

5.B.2.1 The student is able to calculate the expected behavior of a system using the object model (i.e., by ignoring changes in internal structure) to analyze a situation. Then, when the model fails, the student can justify the use of conservation of energy principles to calculate the change in internal energy due to changes in internal structure because the object is actually a system.

AP Physics – Unit 6 Simple Harmonic Motion Notes 2

Simple Harmonic Motion

When an object is displaced, it may be subject to a restoring force, resulting in a periodic oscillating motion. If the displacement is directly proportional to the linear restoring force, the object undergoes simple harmonic motion (SHM). Examples of linear restoring forces causing simple harmonic motion include a mass on a spring, the pendulum on a grandfather clock, a tree limb oscillating after you brush it walking through the woods, a child on a swing, even the vibrations of atoms in a solid can be modeled as simple harmonic motion.

Horizontal Spring-Block Oscillators

A popular demonstration vehicle for simple harmonic motion is the spring-block oscillator. The horizontal spring-block oscillator consists of a block of mass m sitting on a frictionless surface, attached to a vertical wall by a spring of spring constant k, as shown in the diagram below.

The block is then displaced an amount A from its equilibrium position and allowed to oscillate back and forth. As the block sits on a frictionless surface, in the ideal scenario the block would continue its periodic motion indefinitely.

Definitions:

Equilibrium Position – when a spring is at a length that it does not exert a force on a mass

Displacement – the distance of a mass from the equilibrium position

Amplitude – the maximum displacement (greatest distance from the equilibrium position)

Cycle – the complete to and fro motion from some initial point back to that same point

Period – (abbreviated T, or Ts for period of a spring) the amount of time required to complete one cycle

Frequency – (abbreviated f, or fs for frequency of a spring) the number of complete cycles per second

Note that the period of oscillation for a spring depends only on the mass of the block and the spring constant. There is no dependency on the magnitude of the displacement of the block.

AP Physics – Unit 6 Simple Harmonic Motion Notes 3

Example #1 :A 5-kg block is attached to a 2000 N/m spring as shown and displaced a distance of 8 cm from its equilibrium position before being released.

Determine the period of oscillation and the frequency for the block.

Example #2

It’s also interesting to look at the energy of the spring-block oscillator while it’s undergoing simple harmonic motion. Because the surface is frictionless, the total energy of the system remains constant. However, there is a continual transfer of kinetic energy into elastic potential energy and back.

When the block is at its equilibrium position, there is no elastic potential energy stored in the spring, therefore all of the energy of the block is kinetic. The block has achieved its maximum speed. At this position, there is also no net force on the block, therefore the block’s acceleration is zero.

AP Physics – Unit 6 Simple Harmonic Motion Notes 4

When the block is at its maximum amplitude position, all of its energy is stored in the spring as elastic potential energy. For an instant its kinetic energy is zero, therefore its velocity is zero. Further, at this position, the spring ex-hibits a maximum force on the block, providing the maximum acceleration. Let’s take a look at this graphically by examining a spring-block oscillator at various points in its periodic path.

AP Physics – Unit 6 Simple Harmonic Motion Notes 5

Of course, through the entire time interval, the total mechanical energy of the spring-block oscillator remains constant.

Vertical Spring-Block Oscillators

Spring-block oscillators can also be set up vertically as shown in the diagram.

Start your analysis by drawing a Free Body Diagram for the block, noting that gravity pulls the mass down, while the force of the spring provides the upward force. Call down the positive y-direction. At its equilibrium position, y=yeq. You can then write a Newton’s 2nd Law Equation for the block and solve for yeq.

Once the system has settled at equilibrium, you can displace the mass by pulling it some amount to either +A or lifting it an amount -A. The new system can be analyzed as follows:

This is the same analysis you would do for a horizontal spring system with spring constant k displaced an amount A from its equilibrium position. This means, in short, that to analyze a vertical spring system, all you do is find the new equilibrium position of the system, taking into account the effect of gravity, then treat it as a system with only the spring force to deal with, oscillating around the new equilibrium point. No need to continue to deal with the force of gravity!

AP Physics – Unit 6 Simple Harmonic Motion Notes 6

Example #3A 2-kg block attached to an unstretched spring of spring constant k=200 N/m as

shown in the diagram below is released from rest. I) Determine the period of the block’s oscillation.

II) What is the maximum displacement of the block from its equilibrium while undergoing simple harmonic motion?

Example #4A 5-kg block is attached to a vertical spring (k=500 N/m). After the block comes to rest, it is

pulled down 3 cm and released. I) What is the period of oscillation?

II) What is the maximum displacement of the spring from its initial unstrained position?

Ideal Pendulums

Ideal Pendulums provide another demonstration vehicle for simple harmonic motion. Consider a mass m attached to a light string that swings without friction about the vertical equilibrium position. As the mass travels along its path, energy is continuously transferred between gravitational potential energy and kinetic energy. The restoring force in the case of the ideal pendulum is provided by gravity.

AP Physics – Unit 6 Simple Harmonic Motion Notes 7

Notice that the period of the pendulum is dependent only upon the length of the pendulum and the gravitational field strength... there is no mass dependence!

Example #5A grandfather clock is designed such that each swing (or half-period) of the pendulum takes one second. How long is the pendulum in a grandfather clock?

Example #6What is the period of a grandfather clock from the previous example if it were on the moon, where the acceleration due to gravity on the surface is roughly one-sixth that of Earth?

Example #7Rank the following pendulums of uniform mass density from highest to lowest frequency.

AP Physics – Unit 6 Simple Harmonic Motion Notes 8

Example #8The period of an ideal pendulum is T. If the mass of the pendulum is tripled while its length is quadrupled,

what is the new period of the pendulum? A) 0.5 T B) T C) 2T

D) 4T