Harmonic Motion (IV)

1
Harmonic Motion (IV)

• Energy of a simple harmonic oscillator

2
Energy of a SHO

• Recall ETot K U
  1/2mv2 1/2kx2
• for a spring. But we know that
  xA cos(?tf) v-A ? sin(?tf)

3
Energy in SHM
Look again at the block spring
Hence ETot ½kA2
We could also write E KU ½ m(vmax )2
4
Concept Quiz
Suppose you double the amplitude of the motion,
what happens to the maximum speed ?

• Doubles
• 4 x Larger
• Doesnt change

5
Concept Quiz

• Suppose you double the amplitude of the motion,
  what
• happens to the maximum acceleration?
• Doubles
• 4 x Larger
• Doesnt change

6
Concept Quiz

• Suppose you double the amplitude of the motion,
  what
• happens to the the total energy?
• Doubles
• 4 x Larger
• Doesnt change

7
Energy

• Since we know the total energy of a SHM, we can
  calculate the or displacement velocity at any
  point in time ETot1/2kA2 KU 1/2mv2
  1/2kx2
• So, if x0, all E is in kinetic, and v is at max
  if xA, all E is in potential, and v is zero

8
Example

• A 100g block is 5cm from the equilibrium position
  moving at 1.5m/s.
• a) What is the total energy of the system ?
• b) What is the amplitude of the oscillations ?

9
Example
A 500g block on a spring is pulled 20cm and
released. The motion has a period of 0.8s.
What is the velocity when the block is 15.4cm
from the equilibrium ?
10
Example

• A 1.0kg block is attached to a spring with
  k16N/m. While the block is at rest, a student
  hits it with a hammer and almost instantaneously
  gives it a speed of 40cm/s.
• what is the amplitude of the subsequent
  oscillations ?
• what is the blocks speed at the point where
  xA/2 ?