Transcript The Gravitational Potential Energy will be at a maximum. The

Springs
And pendula, and energy
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
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
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.
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.
Hooke’s Law – Energy

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




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
Simple Harmonic Motion & Springs

Simple Harmonic Motion:


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.
Restoring Forces and Simple
Harmonic Motion

Simple Harmonic Motion

A motion in which the system repeats itself
driven by a restoring force



Springs
Gravity
Pressure
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.
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
Simple Harmonic Motion &
Springs
1.5
1
0.5
Position
0
Velocity
0
5
10
15
20
25
Acceleration
-0.5
-1
-1.5
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:
1.
2.
This formula is true for only small angles of θ.
The period of a pendulum is independent of its mass.
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?
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
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
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
Conservation of Energy


What is the equilibrium position of the
ball?
How far will it fall before being pulled
Back up by the spring?
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
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?
Conservation of Energy & The
Pendulum
Units

Newton = ?
Conservation of Energy & The
Pendulum
Units


Newton = kg-m/sec^2
Energy


Newton-m
Kg-m^2/sec^2
Conservation of Energy
Energy is conserved

PE + KE = constant
For springs,

PE = ½ kx2
For objects in motion,

KE = ½ mv2
Conservation of Energy & The
Pendulum

http://zonalandeducation.com/mstm/physics/
mechanics/energy/springPotentialEnergy/spri
ngPotentialEnergy.html