1. Springs
And pendula, and energy

2. Elastic Potential Energy
What is it?
Energy that is stored in elastic materials as a result of their stretching.
Where is it found?
Rubber bands
Bungee cords
Trampolines
Springs
Bow and Arrow
Guitar string
Tennis Racquet

3. Hooke’s Law A spring can be stretched or compressed with a force.
The force by which a spring is compressed or stretched is proportional to the magnitude of the displacement (F a x).
Hooke’s Law:
Felastic = -kx
Where:
k = spring constant = stiffness of spring (N/m)
x = displacement

4. Hooke’s Law Felastic = -kx k = spring constant = 10 (N/m)
x = displacement = 0.2m
F = - (0.2m)(10 N/m) = -2N
Why negative? Because the direction of the Force and the displacement are in opposite directions.

5. Hooke’s Law – Energy
When a spring is stretched or compressed, energy is stored.
The energy is related to the distance through which the force acts.
In a spring, the energy is stored in the bonds between the atoms of the metal.

6. Hooke’s Law – Energy W = ½ kx2 = D PE + D KE F = kx W = Fd
W = (average F)d = d(average F)
W = d*[F(final) – F(initial)]/2
W = x[kx - 0 ]/2
W = ½ kx2 = D PE + D KE

7. Hooke’s Law – Energy
This stored energy is called Potential Energy and can be calculated by PEelastic = ½ kx2
Where:
k = spring constant = stiffness of spring (N/m)
x = displacement
The other form of energy of immediate interest is gravitational potential energy
PEg = mgh
And, for completeness, we have
Kinetic Energy KE = 1/2mv2

8. Simple Harmonic Motion & Springs
An oscillation around an equilibrium position in which a restoring force is proportional the the displacement.
For a spring, the restoring force F = -kx.
The spring is at equilibrium when it is at its relaxed length.
Otherwise, when in tension or compression, a restoring force will exist.

9. Restoring Forces and Simple Harmonic Motion
A motion in which the system repeats itself driven by a restoring force
Springs
Gravity
Pressure

10. Harmonic Motion
Pendula and springs are examples of things that go through simple harmonic motion.
Simple harmonic motion always contains a “restoring” force that is directed towards the center.

11. Simple Harmonic Motion & Springs
At maximum displacement (+ x):
The Elastic Potential Energy will be at a maximum
The force will be at a maximum.
The acceleration will be at a maximum.
At equilibrium (x = 0):
The Elastic Potential Energy will be zero
Velocity will be at a maximum.
Kinetic Energy will be at a maximum

12. Simple Harmonic Motion & Springs

13. The Pendulum
Like a spring, pendula go through simple harmonic motion as follows.
T = 2π√l/g
Where:
T = period
l = length of pendulum string
g = acceleration of gravity
Note:
This formula is true for only small angles of θ.
The period of a pendulum is independent of its mass.

14. 10.3 Energy and Simple Harmonic Motion
Example 3 Changing the Mass of a Simple Harmonic Oscilator
A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring?
What about a 0.4 kg ball?

15. Simple Harmonic Motion & Pendula
At maximum displacement (+ y):
The Gravitational Potential Energy will be at a maximum.
The acceleration will be at a maximum.
At equilibrium (y = 0):
The Gravitational Potential Energy will be zero
Velocity will be at a maximum.
Kinetic Energy will be at a maximum

16. Conservation of Energy & The Pendulum
(mechanical) Potential Energy is stored force acting through a distance
If I lift an object, I increase its energy
Gravitational potential energy
We say “potential” because I don’t have to drop the rock off the cliff
Peg = Fg * h = mgh

17. Conservation of Energy
Consider a system where a ball attached to a spring is let go. How does the KE and PE change as it moves?
Let the ball have a 2Kg mass
Let the spring constant be 5N/m

18. Conservation of Energy
What is the equilibrium position of the ball?
How far will it fall before being pulled
Back up by the spring?

19. Conservation of Energy & The Pendulum
(mechanical) Potential Energy is stored force acting through a distance
Work is force acting through a distance
If work is done, there is a change in potential or kinetic energy
We perform work when we lift an object, or compress a spring, or accelerate a mass

20. Conservation of Energy & The Pendulum
Does this make sense? Would you expect energy to be made up of these elements?
Peg = Fg * h = mgh
What are the units?

21. Conservation of Energy & The Pendulum
Units
Newton = ?

22. Conservation of Energy & The Pendulum
Units
Newton = kg-m/sec^2
Energy
Newton-m
Kg-m^2/sec^2

23. Conservation of Energy
Energy is conserved
PE + KE = constant
For springs,
PE = ½ kx2
For objects in motion,
KE = ½ mv2

24. Conservation of Energy & The Pendulum