Boing

1
Boing!

• Things that vibrate

2
Objectives

• Analyze force, acceleration, velocity, and
  position at any point in a vibration cycle.
• Trace the evolution of kinetic and potential
  energy, velocity, and displacement in an
  oscillation.
• Identify the factors determining the period of a
  spring and a pendulum.

3
Springs

• Hookes law F kx
• F force exerted by the spring
• k spring constant (characteristic of the
  particular spring)
• x distance spring is displaced from equilibrium

4
How does a spring mass move?

• Newtons second law F ma
• Force F depends on position by Hookes law F
  kx

5
Group Work

• Predict the motion of a mass acted on only by a
  Hookes law spring.
• Express your prediction as a distance-time graph.
  
• Explain why you believe that the mass will move
  in the manner you predicted.

6
CPS Question

• The net force on an oscillating object is
• zero at the top and bottom
• maximum at the top and bottom
• minimum but not zero at the top and bottom

7
CPS Question

• The acceleration of an oscillating object is
• zero at the top and bottom
• maximum at the top and bottom
• minimum but not zero at the top and bottom

8
CPS Question

• The velocity of an oscillating object is
• maximum at the equilibrium position
• maximum at the top and bottom
• maximum midway between equilibrium and top or
  bottom

9
Uniform Circular Motion

• Centripetal force F mv2/r inwards
• Constant magnitude F0 direction depends on
  position

• Force in y-direction is proportional to y

10
Uniform Circular Motion

• Angle changes at a steady rate.
• Projection on y-axis has Hookes law force.
• So, projection on y-axis must have Hookes law
  motion too!
• What is the projection of an angle on the y-axis?

11
Hookes Law Motion Position
y
time
period T
12
Hookes Law Motion
13
Group Work

• What is velocity (min, max, gt 0, lt 0, 0) of the
  oscillating mass at the top? At the equilibrium
  point? At the bottom?
• What is acceleration (min, max, gt 0, lt 0, 0) of
  the oscillating mass at the top? At the
  equilibrium point? At the bottom?

14
Group Work

• What is the objects position when it is not
  accelerating?
• What is the objects velocity when it is not
  accelerating?
• What is the objects position when it is not
  moving?
• What is the objects acceleration when it is not
  moving?

15
CPS Question

• The potential energy of an oscillating mass is
  greatest

• at its extreme positions.
• at its equilibrium (middle) position.
• between the middle and an extreme position.

16
CPS Question

• The kinetic energy of an oscillating mass is
  greatest

• at its extreme positions.
• at its equilibrium (middle) position.
• between the middle and an extreme position.

17
Group Work

• What is the objects kinetic energy when it is
  not accelerating?
• What is the objects potential energy when it is
  not accelerating?

18
Energy

• Potential energy of a stretched spring

• Conservation of energy

PE KE constant
(This of course ignores the nasty reality of
energy dispersal by friction and drag.)
19
Group Work

• Show that the total energy E of an oscillating
  spring is always positive.

20
Energy
y
time
21
Period and Frequency

• Period T
• time of one cycle (units s)
• Frequency f
• cycles per unit time (units 1/s Hz)
• f 1/T

22
CPS Question

• Increasing the spring constant k makes the period

• shorter.
• longer.
• k has no effect on the period.

23
CPS Question

• Increasing the mass m makes the period

• shorter.
• longer.
• m has no effect on the period.

24
Period

• Period

• So
• increased mass m ? longer period
• increased k ? shorter period
• period does not depend on amplitude

25
CPS Question

• When a spring mass oscillates under the influence
  of gravity,

• the spring constant k changes.
• the vibration period T changes.
• Both of these.
• None of these.

26
Effect of Gravity

• Less than you might expect
• Changes equilibrium position x 0
• Does not change k

27
Spring Gravity
spring alone
force
0
0
position
28
Spring Gravity
force
0
position
29
Spring Gravity
different equilibrium position same k
net force
0
0
position