1) A spring is an elastic object that stores mechanical energy and exerts a restoring force proportional to its displacement from equilibrium. 2) The motion of an object attached to a spring is called simple harmonic motion (SHM), where the restoring force causes the object to oscillate back and forth periodically over time. 3) In an undamped spring system without friction or energy losses, the object will oscillate indefinitely; a damped system includes forces proportional to velocity that cause the oscillations to decay over time until the object reaches equilibrium.

1. PGSWE-511

2. A spring is an elastic object that stores mechanical
energy. Springs are typically made of spring steel. There
are many spring designs. In everyday use, the term often
refers to coil springs.
What is a spring ?

3. Simple Harmonic Motion—Spring
Oscillations
© 2014 Pearson Education, Inc.
If an object vibrates or
oscillates back and forth
over the same path, each
cycle taking the same
amount of time, the motion
is called periodic. The mass
and spring system is a
useful model for a periodic
system.

4. We assume that the surface is frictionless. There
is a point where the spring is neither stretched nor
compressed; this is the equilibrium position. We
measure displacement from that point (x = 0 on the
previous figure).
The force exerted by the spring depends on the
displacement:
Simple Harmonic Motion—Spring
Oscillations
(11-1)

5. Simple Harmonic Motion—Spring
Oscillations
• The minus sign on the force indicates that it is a
restoring force—it is directed to restore the mass to
its equilibrium position.
• k is the spring constant
• The force is not constant, so the acceleration is not
constant either

6. Simple Harmonic Motion—Spring
Oscillations
• Displacement is measured from the
equilibrium point
• Amplitude is the maximum
displacement
• A cycle is a full to-and-fro motion;
this figure shows half a cycle
• Period is the time required to
complete one cycle
• Frequency is the number of cycles
completed per second

7. Simple Harmonic Motion—Spring
Oscillations
If the spring is hung
vertically, the only
change is in the
equilibrium position,
which is at the point
where the spring force
equals the
gravitational
force.

8. Simple Harmonic Motion—Spring
Oscillations
Any vibrating system where the restoring force is
proportional to the negative of the displacement is in
simple harmonic motion (SHM), and is often called a
simple harmonic oscillator.

9. Describing Simple Harmonic Motion
1. The mass starts at its maximum
positive displacement, y = A. The
velocity is zero, but the
acceleration is negative because
there is a net downward force.
2. The mass is now moving
downward, so the velocity is
negative. As the mass nears
equilibrium, the restoring force—
and thus the magnitude of the
acceleration—decreases.
3. At this time the mass is moving
downward with its maximum
speed. It’s at the equilibrium
position, so the net force—and
thus the acceleration—is zero.

10. Describing Simple Harmonic Motion
4. The velocity is still negative but
its magnitude is decreasing, so
the acceleration is positive.
5. The mass has reached the
lowest point of its motion, a
turning point. The spring is at
its maximum extension, so
there is a net upward force and
the acceleration is positive.
6. The mass has begun moving
upward; the velocity and
acceleration are positive.

11. Describing Simple Harmonic Motion
7. The mass is passing through the
equilibrium position again, in the
opposite direction, so it has a
positive velocity. There is no net
force, so the acceleration is zero.
8. The mass continues moving
upward. The velocity is positive
but its magnitude is decreasing, so
the acceleration is negative.
9. The mass is now back at its
starting position. This is another
turning point. The mass is at rest
but will soon begin moving
downward, and the cycle will
repeat.

12. Describing Simple Harmonic Motion
• The position-versus-time graph for oscillatory motion is a
cosine curve:
• x(t) indicates that the position is a function of time.
• The cosine function can be written in terms of frequency:

13. Describing Simple Harmonic Motion
• The velocity graph is an upside-down sine function with
the same period T:
• The restoring force causes an acceleration:
• The acceleration-versus-time graph is inverted from the
position-versus-time graph and can also be written

14. Describing Simple Harmonic Motion

15. Energy in Simple Harmonic Motion
We already know that the potential energy of a spring is
given by:
PE = ½ kx2
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are
assuming the system is frictionless.
(11-3)

16. Energy in Simple Harmonic Motion
If the mass is at the limits of its
motion, the energy is all potential.
If the mass is at the equilibrium
point, the energy is all kinetic.
We know what the potential energy
is at the turning points:
(11-4a)

17. The total energy is, therefore ½ kA2
And we can write:
This can be solved for the velocity as a function of
position:
where
Energy in Simple Harmonic Motion
(11-4c)
(11-5b)
(11-5a)

23. What is an undamped system?
"Undamped" means that there are no energy losses with
movement (whether intentional, by adding dampers, or
unintentional, through drag or friction). An undamped
system will vibrate forever without any additional
applied forces.

27. What is an damped system?
A damped spring is just like our simple spring with an
additional force. In the simple spring case, we had a
force proportional to our distance from the equilibrium
position. ... In order to settle at equilibrium, we add a
force proportional to our velocity in the case of damped
springs.