Simple Harmonic Motion

ISAT 241Fall 2004

David J. Lawrence

Simple Harmonic Motion Mass Attached to a Spring

x

m x = 0 “Equilibrium Position”

x > 0

x < 0

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 (t)• where A = amplitude = a constant, and

• = angular frequency = a constant.

• We’ll learn more about these quantities soon.

Serway & Jewett, Principles of Physics

Figure 12.1

Brooks/Cole – Thomson Learning

A

-A

Simple Harmonic Motion

Displacement x(t) (m)

Velocity v(t) (m/s)

Acceleration a(t) (m/s2)

0

0

0

Time (s)

Time (s)

Time (s)

A

2A

2A

A

Equations for this Special Case

– Displacement from Equilibrium:

x(t) = A cos (t)

– Velocity: v(t) = =

– Acceleration: a(t) = =

dt

dx

Simple Harmonic Motion

dt

dv

A sin (t)

A cos (t)

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)

0Time (s)

Label This Graph !

Simple Harmonic Motion Definitions of Terms (continued)

– Angular Frequency =

(unit = radians/s = rad/s)

– Phase Constant = Phase Angle = (unit = radians)

In general, simple harmonic motion cannot be described by a “pure” sine or cosine function, so a phase constant, , or phase angle must be introduced.

E.g., x(t) = A cos (t ) (t + ) is called the phase of the motion

T

2πf2πω

Summary Graphs for SHM – General Case

In the most general case, the displacement graph doesn’t begin at a peak.

This means that

General Equations– Displacement from Equilibrium:

x(t) =

– Velocity: v(t) = =

– Acceleration: a(t) = =

Simple Harmonic Motion

dt

dx

dt

dv

A cos (t + )

A sin (t + )

A cos (t + )

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.

)4

tcos()m0.4()t(xx

Serway & Jewett, Principles of Physics

Figure 12.1

Brooks/Cole – Thomson Learning

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?

x x t m t ( ) ( . )cos( )4 04

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.

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.

Mass Attached to a Spring Hooke’s Law: Fspring = kx

Newton’s Second Law: F = 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 (t + ) Show that this is a solution to the differential

equation by substitution.

Mass Attached to a Spring For this case, the angular frequency is

so the frequency and period are

m

k

T

f1

f

T1

m

k

2

1

2

k

m

22

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

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.

Summary Graphs for SHM – General Case

In the most general case, the displacement graph doesn’t begin at a peak.

This means that