#### Transcript Springs

```Springs
And pendula, and energy
Spring Constants
Spring
Small Spring
Long Spring
Medium spring
2 in series
2 in parallel
3 in series
3 in parallel
k
Units
Do these results make
of spring “stiffness”?
Hooke’s Law
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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 – Energy
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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.
Add graph, show work on graph as area under
triangle
Hooke’s Law – Energy
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F = kx
W = Fd
W = (average F)d
W = [F(final) – F(initial)]/2*d
W = [kx - 0 ]/2*x
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W = ½ kx^2 = D PE + D KE
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Hooke’s Law – Energy
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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
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The other form of energy of immediate interest is
gravitational potential energy
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PEg = mgh
And, for completeness, we have
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Kinetic Energy KE = 1/2mv2
Restoring Forces and Simple
Harmonic Motion
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Simple Harmonic Motion
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A motion in which the system repeats itself driven by a
restoring force
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Springs
Gravity
Pressure
Harmonic Motion
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Pendula and springs are examples of
things that go through simple harmonic
motion.
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Simple harmonic motion always contains a
“restoring” force that is directed towards the
center.
Simple Harmonic Motion &
Springs
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At maximum displacement (+ x):
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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):
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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
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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
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Note:
1.
2.
This formula is true for only small angles of θ.
The period of a pendulum is independent of its mass.
Simple Harmonic Motion &
Pendula
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At maximum displacement (+ y):
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The Gravitational Potential Energy will be at a
maximum.
The acceleration will be at a maximum.
At equilibrium (y = 0):
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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
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(mechanical) Potential Energy is stored
force acting through a distance
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If I lift an object, I increase its energy
Gravitational potential energy
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We say “potential” because I don’t have to drop
the rock off the cliff
Peg = Fg * h = mgh
Conservation of Energy
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Consider a system where a ball attached
to a spring is let go. How does the KE and
PE change as it moves?
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Let the ball have a 2Kg mass
Let the spring constant be 5N/m
Conservation of Energy
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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
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(mechanical) Potential Energy is stored
force acting through a distance
Work is force acting through a distance
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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?
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Peg = Fg * h = mgh
What are the units?
Conservation of Energy & The
Pendulum
Units
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Newton = ?
Conservation of Energy & The
Pendulum
Units
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Newton = kg-m/sec^2
Energy
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Newton-m
Kg-m^2/sec^2
Conservation of Energy
Energy is conserved
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PE + KE = constant
For springs,
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PE = ½ kx2
For objects in motion,
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KE = ½ mv2
Conservation of Energy & The
Pendulum
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Conservation of Mechanical Energy
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PEi + KEi = PEf + KEf
mgΔh = ½ mv2
gΔh = ½ v2
If you solve for v:
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v = √ 2gΔh
v = √ 2(9.81 m/s2)(0.45 m)
v = 2.97 m/s
Conservation of Energy & The
Pendulum
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