Download Kinematics of simple harmonic motion (SHM)

Physics 12
Resource: Giancoli Chapter 11
Objectives
Describe examples of oscillations
 Define the terms displacement, amplitude,
frequency, period and phase difference.
 Define simple harmonic motion (SHM).
 Solve problems using the defining
equation for SHM.

Objectives
Apply equations for the kinematics of
SHM.
 Solve problems both graphically and by
calculation, for acceleration, velocity and
displacement during SHM.

Waves and oscillations
To oscillate means to move back and forth.
 Can you give examples of oscillation?

Waves and oscillations
All things that oscillate / vibrate are
ultimately linked.
 Their motion can be explained using the
concept of waves.


For simplicity, let us take the example of a
simple pendulum.
Kinematics of SHM

Consider the following pendulum:
A block of mass m attached to a spring
that oscillates horizontally on a
frictionless surface.

The equilibrium position is the point at
which the mass rests, without external
stretching or compression.
Kinematics of SHM

The force exerted by the spring is
represented by the following expression:
F = -kx
where F is the force exerted by the spring
k is the spring constant (dependent on the material of
the spring)
and x is the displacement of the mass m
What does the negative sign signify?
Kinematics of SHM

The negative sign connotes that the
restorative force of a spring is always in
the opposite direction of the
displacement.
 When the spring is stretched and
displacement x is to the right, the spring exerts
a force that restores it to the left (back to
equilibrium position x = 0)
 When the spring is compressed and
displacement is to the left, the spring exert a
force to the right.
Kinematics of SHM

Consider the following simple pendulum:
A mass m hanging vertically from a spring
with spring constant k.

Would the equilibrium position x0 be the
same as the pendulum which oscillates
horizontally?
Kinematics of SHM

The spring would be stretched an extra
amount related to the weight of the mass:
F = mg

The equilibrium point may be defined as
the point where
ΣF = 0
ΣF = mg – kx0
0 = mg – kx0
Kinematics of SHM
A family of 200 kg steps into a 1200kg car and the car lowers 3.0 cm.
(a) What is the spring constant k of
the car’s springs?
(b) How much further would the car
lower if the family was 300 kg?
Kinematics of SHM
(a)
(b)
6.5 x 104 Nm-1
4.5 cm
Definition of SHM

When the family’s mass is 200 kg, the
springs compress 3.0 cm
When half of the mass is added, i.e. the
family’s mass is 300 kg, the springs
compress 4.5 cm.

What do you notice?

Definition of SHM


An oscillator for which the force exerted
is proportional to its displacement is
called a simple harmonic oscillator.
In other words, simple harmonic motion
(SHM) is a type of motion for which
F = -kx holds true.
Definition of SHM
Are the following oscillators simple
harmonic in nature?
(a) F = 0.5x2
(b) F = -2.3y
(c) F = 8.6x
(d) F = -40t
Definition of SHM
no
(b) yes
(c) no
(d) yes
(a)
Why isn’t (c) an SHO?
Periodic nature of SHM
Imagine the motion of a simple pendulum
oscillating vertically.
 Consider the following characteristics:

maximum v
v=0
maximum a
a=0
maximum PE
minimum PE
maximum KE
minimum KE
Periodic nature of SHM

Consider the following characteristics:
maximum v
v=0
maximum a
maximum PE
minimum PE
maximum KE
a=0
minimum KE
Periodic nature of SHM

Computer simulation
Periodic nature of SHM
Recall the graph of an SHO’s motion.
 Is acceleration constant?
 How would you describe the shape of the
graph?

Periodic nature of SHM
Sinusoidal nature of SHM
Definition of terms
Term
Definition
displacement, x
the movement of particles above and below the
mean position
amplitude, A
the difference between the maximum
displacement and the mean position
period, T
the amount of time required to complete one
cycle (between two identical positions)
frequency, f
the number of complete cycles passing a given
point in one second (Hz)
wavelength, λ
the distance covered in a complete wave cycle.
Sinusoidal nature of SHM
Wave motion
Sinusoidal nature of SHM
Construct the following graphs:
displacement-time
velocity – time
acceleration – time
for a pendulum starting at maximum
displacement and one starting at
equilibrium position.
Sinusoidal nature of SHM

SHM is said to be sinusoidal in nature.

Depending on the starting point, the
relationship between certain variables
(displacement, velocity, acceleration) and
time can either be a sine or cosine
function.
Sinusoidal nature of SHM
Relationship between period T and
frequency f
f (in s-1 or Hz) =
T (in s) =
What is the relationship between period and
frequency?
Sinusoidal nature of SHM
Period T of SHM
T = 2π
Sinusoidal nature of SHM
A spider of mass 0.30 g waits in its web of
negligible mass. A slight movement causes
the web to vibrate with a frequency of
about 15 Hz.
(a) Estimate the value of the spring stiffness
constant k for the web
(b) At what frequency would you expect the
web to vibrate if an insect of mass 0.10 g
were trapped with the spider.
Sinusoidal nature of SHM
(a)
(b)
k = 2.7 N/m
f = 13 Hz
Energy of SHM
Is work done when a spring is stretched
or compressed?
 How is the energy stored?

Energy of SHM

When a spring is stretched or
compressed, work is done and converted
into the potential energy of the spring.

Elastic potential energy is given by the
expression:
PE = ½ kx2
Describe what happens to a spring in terms
of energy as it completes one cycle.
Energy of SHM
Total mechanical E = KE +PE
Total mechanical E = ½ mv2 + ½ kx2
Derive expressions for total mechanical
energy, E, at maximum displacement and
equilibrium position.
Energy of SHM
At x = A and x = -A, v = 0, therefore:
E = ½ m(02) + ½ kA2
E = ½ kA2
At equilibrium point, x = 0 and v = vmax,
therefore:
E = ½ mvmax2 + ½ k(0)2
E = ½ mvmax2
Energy of SHM
Use the conservation of mechanical energy
to deduce an expression for the
instantaneous velocity of an SHO (velocity
v at any time) in terms of vmax, x, and A:
v = ± vmax
Energy of SHM
Suppose a spring oscillator is stretched to
twice the amplitude (x = 2A). What
happens to the:
(a) energy of the system
(b) maximum velocity of the oscillator
(c) maximum acceleration of the mass
Energy of SHM
(a)
(b)
(c)
energy is quadrupled
maximum velocity is doubled
acceleration is doubled
Energy of SHM
A spring stretches 0.150 m when a 0.300-kg mass is
gently lowered on it. The spring is set up on a
frictionless table. The mass is pulled so that the
spring is stretched 0.100 m from the equilibrium
point then released from rest. Determine the:
(a) spring stiffness constant k
(b) amplitude of horizontal oscillation A
(c) magnitude of maximum velocity vmax
(d) magnitude of velocity v when the mass is 0.050 m
from equilibrium and
(e) magnitude of the maximum acceleration amax of the
mass
Energy of SHM
k = 19.6 N/m
(b) A = 0.100 m
(c) vmax = 0.808 m/s
(d) v = 0.70 m/s
(e) amax = 6.53 m/s2
(a)
Simple pendulum
Consider this simple pendulum.
Does it oscillate?
Does F = - kx still apply?
Simple pendulum
Derive expressions for:
(a) displacement along the arc
(b) the restoring force F tangent to the arc
Simple pendulum
Simple pendulum
Period, simple pendulum
T = 2π