Periodic Motion

"And your head is shaking
and your arms are shaking
and your feet are shaking,
cause the Earth is shaking"
- REM 

Lecture Outline: 

  1. Periodic Motion
  2. Simple Harmonic Oscillations
  3. Velocity and Acceleration
  4. Springs and Pendulums

 

Periodic Motion

A common type of motion is periodic motion. Any object that repeats its movements over a regular, well-defined cycle is said to be in periodic motion. The common back-and-forth motion of a pendulum or a repeated steps of a dancer are examples of this type of movement. We will use the block and spring system as our example.

Periodic motion is characterized by a period, T, that is the amount of time needed to make one complete cycle. The motion can also be described by frequency, f, which is the number of cycles that take place during a given unit of time. The MKS unit for frequency is the cycles/sec or Hertz, (Hz). The frequency and the period are related by

T = 1/f

 

Simple Harmonic Oscillations

Many objects undergo oscillations: they move back and forth about an equilibrium position. The most common example is a spring that obeys Hooke's Law.

This can be written as

where

The equation above is the differential equation for the Simple Harmonic Oscillator. Any system whose equation of motion can be written in this way will behave like a S.H.O. The quantity w is called the angular frequency. The identification above is for the spring; for different systems the physical quantities that give the angular frequency will be different.

The solution to the S.H.O. equation is

where A is the amplitude and f is called the phase angle.

The angular frequency can be related to the period (and frequency) of the oscillation.

 

If we image a pen attached to an oscillating mass on spring the plot it traces out on a steadily moving piece of paper is a sinusoidal function.

The phase angle is used to set the initial position of system. If time is measured from when the system is at a distance x = A from the equilibrium point then f = 0 and the graph of the position vs. time looks like

If time is measured from when the system is at x = 0 with a velocity in the positive direction then f = -90 degrees (or -p/2) and the graph looks like

Velocity and Acceleration

The velocity and acceleration can be found by taking derivatives of the S.H.O solution for x.

 

 

Spring and block system

position

velocity

acceleration

A

0

-w2A

0

-wA

0

-A

0

w2A

0

wA

0

 

 

 

 

B. Which of the following is necessary in order for an object to oscillate?

  1. A stable equilibrium point.
  2. A disturbance in its position.
  3. Little or no frictional force.
  4. Both 1 and 3.
  5. All of the above.

Answer.

 

 

C. A mass connected to a spring oscillates back and forth. Which of the following is true?

  1. At some point during the oscillation the mass has zero velocity but a non-zero acceleration.
  2. At some point during the oscillation the mass has zero velocity and zero acceleration.
  3. Both 1 and 2.
  4. Neither 1 nor 2.

Answer.

 

 

D. A mass attached to a spring oscillates back and forth with the position vs time graph shown below. At point P, the mass has

  1. positive velocity and positive acceleration.
  2. positive velocity and negative acceleration.
  3. positive velocity and zero acceleration.
  4. negative velocity and positive acceleration.
  5. negative velocity and negative acceleration.
  6. negative velocity and zero acceleration.

Answer.

 

E. A mass is attached to a spring on a frictionless horizontal surface. The mass is pulled a distance d from equilibrium position in the negative direction. At time t = 0 the mass is released. Which of the following correctly expresses its position, x, as a function of time?

  1. x = dcos(wt + p) .
  2. x = dcos(wt + p/2) .
  3. x = dcos(wt - p/2) .
  4. x = dcos(wt - p) .
  5. Both 1 and 4

Answer.

 

Springs and Pendulums

Returning to the spring and mass system we can relate the spring constant and mass to the period and frequency through the angular frequency.

Another system that exhibits simple harmonic motion is the pendulum. For small angular displacements the equation of motion for a pendulum can be written as a simple harmonic oscillator equation. For the simple pendulum, this yields an expression for the period of the oscillation of the pendulum:

Notice that the period does not depend on the mass of the pendulum.

Answers to ConcepTests:

A. The answer is 1. The widening of the blood vessel causes the fluid to slow. The slower fluid has a higher pressure.

Back to question.

 

 

B. The answer is 5. All of these are needed for harmonic oscillations.

Back to question. 

 

C. The answer is 1. At the endpoints of the motion the velocity of the mass is momentarily zero but the acceleration is -kA.

 Back to question.

 

 

D. The answer is 2. The slope of the curve is positive and decreasing.

Back to question.

 

 

E. The answer is 5. For t= 0 the SHO equation becomes x = dcos(f). For x = -d at t= 0 the possible solutions are f = p or -p.

Back to question.

 

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