1. Simple Harmonic Motion
ISAT 241
Fall 2004
David J. Lawrence

2. Simple Harmonic Motion
Mass Attached to a Spring m
x = 0 “Equilibrium Position”
x > 0
x < 0 x

3. Simple Harmonic Motion
Mass Attached to a Spring
When the mass is stationary, it is in its equilibrium position (Net force = 0).
If we pull the mass down slightly and then release it, what happens?
Graph the displacement from equilibrium as a function of time.
x(t) = A cos (w t)
where A = amplitude = a constant, and
w = angular frequency = a constant.
We’ll learn more about these quantities soon.

5. Serway & Jewett, Principles of Physics
Figure 12.1
Brooks/Cole – Thomson Learning

6. Simple Harmonic Motion
Time
(s)
Displacement
x(t)
(m) -A wA
Velocity
v(t)
(m/s)
-wA
w2A
Acceleration
a(t)
(m/s2)
-w2A

7. Simple Harmonic Motion
Equations for this Special Case
Displacement from Equilibrium:
x(t) = A cos (w t)
Velocity: v(t) = =
Acceleration: a(t) = =
- w A sin (w t)
- w2 A cos (w t)

8. Simple Harmonic Motion
Definitions of Terms
Amplitude = A = the maximum displacement of the moving object from its equilibrium position.
(unit = m)
Period = T = the time it takes the object to complete one full cycle of motion.
(unit = s)
Frequency = f = the number of cycles or vibrations per unit of time.
(unit = cycles/s = 1/s = Hz = hertz)

9. Label This Graph !
Time
(s)

10. Simple Harmonic Motion
Definitions of Terms (continued)
Angular Frequency = w
(unit = radians/s = rad/s)
Phase Constant = Phase Angle = f
(unit = radians)
In general, simple harmonic motion cannot be described by a “pure” sine or cosine function, so a phase constant, f , or phase angle must be introduced.
E.g., x(t) = A cos (w t + f)
(wt + f) is called the phase of the motion

11. Summary Graphs for SHM – General Case
In the most general case, the displacement graph doesn’t begin at a peak.
This means that f ¹ 0.

12. Simple Harmonic Motion
General Equations
Displacement from Equilibrium:
x(t) =
Velocity: v(t) = =
Acceleration: a(t) = =
A cos (w t + f)
- wA sin (w t + f)
- w2A cos (w t + f)

13. Simple Harmonic Motion -- Example
An object oscillates with SHM along the x-axis. Its displacement varies with time according to the equation
where t is in seconds and the two angles in parentheses are in radians. (See figure on next slide.)
(a) Determine the amplitude, phase constant, angular frequency, frequency, and period of the motion.
(b) Calculate the velocity and acceleration of the object at any time t.
(c) Determine the position, velocity, and acceleration of the object at t = 1 s.

14. Serway & Jewett, Principles of Physics
Figure 12.1
Brooks/Cole – Thomson Learning

15. S. H. M. – Example (continued)
An object oscillates with SHM along the x-axis. Its displacement varies with time according to the equation
where t is in seconds and the two terms in parentheses are in radians.
(d) Determine the maximum displacement from the origin, maximum speed, and maximum acceleration of the object.
(e) Find the displacement of the object between t = 0 and t = 1 s.
(f ) What is the phase of the motion at t = 2.00 s?

16. Simple Harmonic Motion
Important properties of an object moving in simple harmonic motion:
The displacement, velocity, and acceleration all vary sinusoidally with time, but are not in phase.
The acceleration is proportional to the displacement, but in the opposite direction.
The frequency and period of the motion are independent of the amplitude.

17. Simple Harmonic Motion
Whenever the force acting on an object is linearly proportional to the displacement and in the opposite direction, the object exhibits simple harmonic motion.
We have been considering the simple example of a mass attached to a spring.

18. Mass Attached to a Spring
Hooke’s Law: Fspring = - kx
Newton’s Second Law: SF = ma = m (d2x/dt2)
Therefore, (d2x/dt2) = - (k/m) x
This is a differential equation, which can be solved for x(t).
The solution is the equation we have been using all along:
x(t) = A cos (w t + f)
Show that this is a solution to the differential equation by substitution.

19. Mass Attached to a Spring
For this case, the angular frequency is
so the frequency and period are

20. Mass-Spring System -- Example 1
Car hitting a pothole in the road.

21. Mass-Spring System -- Example 2
A 200 g mass is connected to a light spring with force constant 5 N/m, and is free to oscillate on a horizontal, frictionless surface. The mass is displaced 5 cm to the right from equilibrium and released from rest. (See figure on next slide.)
(a) Find the period of the mass’ motion.
(b) Find the displacement, speed, and acceleration as functions of time.
(c) Determine the max. speed of the mass.
(d) Determine the max. acceleration of the mass.

23. Summary Graphs for SHM – General Case
In the most general case, the displacement graph doesn’t begin at a peak.
This means that f ¹ 0.