Chapter 13
Oscillations About Equilibrium
Dr. Jie Zou PHY 1151G
Department of Physics
1
Outline

Oscillations


Periodic motions


Basic cause for oscillations
Period and frequency
Simple harmonic motions



Classical example: A mass attached to a spring
Position as a function of time for a simple
harmonic motion
Period of a mass on a spring
Dr. Jie Zou PHY 1151G
Department of Physics
2
Oscillations


Oscillations: When systems are
displaced from equilibrium, it often
results in oscillations back and
forth from one side of the
equilibrium position to the other.
Basic cause of oscillations:
When an object is displaced from a
position of stable equilibrium it
experiences a restoring force that
is directed back toward the
equilibrium position.
Dr. Jie Zou PHY 1151G
Department of Physics
3
Periodic Motion


Periodic motion: A motion that
repeats itself over and over is
referred to as periodic motion.
Definition of period, T : time
required for one cycle (one
oscillation) of a periodic motion.
The trace of an
 SI unit: second or s.
electrocardiogram
 Definition of frequency, f : the
(ECG or EKG).
number of oscillations per unit of
time. f = 1/T.

SI unit: per second = 1/s = Hz.
Dr. Jie Zou PHY 1151G
Department of Physics
4
Example: Periodic motion

If the processing speed of a personal
computer is 1.80 GHz, how much time is
required for one processing cycle?

A tennis ball is hit back and forth between two
players warming up for a match. If it takes
2.31 s for the ball to go from one player to the
other, what are the period and frequency of
the tennis ball’s motion?
Dr. Jie Zou PHY 1151G
Department of Physics
5
Simple harmonic motion


A classical example of
simple harmonic motion:
The mass-spring system.
Key feature of a massspring system: A spring
exerts a restoring force that is
proportional to the
displacement from equilibrium,
F = - kx.
Dr. Jie Zou PHY 1151G
Department of Physics
6
Position versus time for simple
harmonic motion

Position versus time in simple
harmonic motion: x  Acos2 t
 T 



Here we suppose that the cart is
released at t = 0 from rest at x = A .
Amplitude, A : it represents the
maximum displacement of the cart
on either side of equilibrium.
Period, T : the cart’s motion
repeats with a period T.
Dr. Jie Zou PHY 1151G
Department of Physics
7
Example

An air-track cart attached to a spring
completes one oscillation every 2.4 s. At t = 0
the cart is released from rest at a distance of
0.10 m from its equilibrium position. What is
the position of the cart at (a) 0.30 s, (b) 0.60
s, (c) 2.7 s, and (d) 3.0 s?
Dr. Jie Zou PHY 1151G
Department of Physics
8
Period of a mass on a spring

m
A horizontal spring: T  2
k
 Unstretched at their



equilibrium position.
Here m is the mass (kg), and
k is the force constant of the
spring (N/m), and T is the
period (s).
The period, T, increases with
the mass and decreases with
the spring’s force constant.
The period, T, is independent
of the amplitude, A .
Dr. Jie Zou PHY 1151G
Department of Physics
9
A vertical spring


A vertical spring is in
equilibrium at y=-y0.
At this equilibrium position, the
spring stretches by an amount
y0 given by

ky0 = mg or y0 = mg/k
A mass on a vertical spring
oscillates about the equilibrium
point y = -y0.
m
 The period is also T  2
k
Dr. Jie Zou PHY 1151G

Department of Physics
10
Example


When a 0.420-kg mass is attached to a
spring, it oscillates with a period of 0.350
s. If, instead, a different mass, m2, is
attached to the same spring, it oscillates
with a period of 0.700 s. Find (a) the
force constant of the spring and (b) the
mass m2.
Real world physics: the relationship
between the mass and period is used by
NASA to measure the mass of
astronauts in orbit - The Body Mass
Measurement Device (BMMD)
Dr. Jie Zou PHY 1151G
Department of Physics
11
Homework

See online homework on
www.masteringphysics.com
Dr. Jie Zou PHY 1151G
Department of Physics
12