Springs

1
Springs

• And pendula, and energy

2
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.

3
Hookes Law Restoring Force

• 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).
• Hookes Law
• Felastic -kx
• Where
• k spring constant stiffness of spring
  (N/m)
• x displacement

4
Hookes 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.
• 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

5
Hookes 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

6
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

7
Simple Harmonic Motion Springs

• P / V / A graphs?

8
Simple Harmonic Motion Springs
9
The Pendulum

• Like a spring, pendula go through simple harmonic
  motion as follows.
• T 2pvl/g
• Where
• T period
• l length of pendulum string
• g acceleration of gravity
• Note
• This formula is true for only small angles of ?.
• The period of a pendulum is independent of its
  mass.

10
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
• P / V / A graphs?

11
Conservation of Energy The Pendulum

• (mechanical) Potential Energy is force acting
  through a distance
• If I lift an object, I increase its energy
• Gravitational potential energy
• We say potential because I dont have to drop
  the rock off the cliff
• Peg Fg h mgh

12
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?

13
Conservation of Energy The Pendulum

• Units
• Newton ?

14
Conservation of Energy The Pendulum

• Units
• Newton kg-m/sec2
• Energy
• Newton-m
• Kg-m2/sec2

15
Conservation of Energy

• Energy is conserved
• PE KE constant
• For springs,
• KE ½ kx2
• For objects in motion,
• KE ½ mv2

16
Conservation of Energy The Pendulum

• Conservation of Mechanical Energy
• PEi KEi PEf KEf
• mg?h ½ mv2
• g?h ½ v2
• If you solve for v
• v v 2g?h
• v v 2(9.81 m/s2)(0.45 m)
• v 2.97 m/s

17
Conservation of Energy The Pendulum

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