Spring Forces and Simple Harmonic Motion

1
Spring Forces and Simple Harmonic Motion

• Chapter 10

2
Expectations

• At the end of this chapter, students will be able
  to
• Apply Hookes Law to the calculation of spring
  forces.
• Conceptually understand how simple harmonic
  motion is caused by forces and torques obeying
  Hookes Law.
• Calculate displacements, velocities,
  accelerations, and frequencies for objects
  undergoing simple harmonic motion.

3
Expectations

• At the end of this chapter, students will be able
  to
• Calculate the elastic potential energy resulting
  from work done by spring forces.
• Understand how pendulums approximate simple
  harmonic motion.
• Calculate the natural frequencies of both simple
  and physical pendulums.
• Conceptually understand the ideas of damped,
  driven, and resonant simple harmonic motion.

4
Expectations

• At the end of this chapter, students will be able
  to
• Analyze the elastic deformation of objects in
  terms of stress, strain, and the elastic moduli
  of materials.
• Express Hookes Law in terms of stress and
  strain.

5
Spring Forces

• A spring resists being stretched or compressed.

6
Spring Forces

• The force with which the spring resists is
  proportional to the distance through which it is
  compressed or stretched

spring constant SI units N/m
7
Spring Forces Hookes Law

• This relationship is called Hookes Law.

8
Hookes Law

• Robert Hooke
• 1635 1703
• English mathematician and natural philosopher
  contemporary of Isaac Newton
• Early builder of microscopes and telescopes

9
Hookes Law

• The direction of the spring force is always
  opposite the direction of the stretching or
  compression of the spring hence, the minus sign.

10
A Consequence of Hookes Law

• Consider Newtons second law and Hookes Law
  simultaneously
• Now, a small and sneaky bit of calculus

differential equation
its solution
11
Simple Harmonic Motion

• Motion described by this equation
• is called simple harmonic motion (SHM). It is
• periodic (repeats itself in time)
• oscillatory (takes place over a limited spatial
  range)

angular frequency (rad/s)
displacement (m)
time (s)
amplitude (m)
12
Simple Harmonic Motion
A 1.00 m
13
SHM Reference Circle Representation

• A vector of magnitude
• A rotates about the
• origin with an angular
• velocity w.
• The x component of
• the vector represents
• the displacement.

14
SHM Frequency

• Since there are 2p radians in each trip (cycle)
  around the reference circle, the cycle
  frequency is related to the angular frequency by
• SI units of cycle frequency, f
• cycles / s Hertz (Hz)

15
SHM Velocity

• We can calculate the
• velocity from the
• reference circle
• representation

16
SHM Acceleration
17
SHM Velocity and Acceleration

• w must be expressed in rad/s.
• Like displacement, velocity and acceleration are
  periodic in time.
• Maximum velocity
• Maximum acceleration
• Acceleration has maximum magnitude at extremes of
  displacement.
• Velocity has maximum magnitude when displacement
  is zero.

18
Mass on a Spring System

• Natural frequency for a mass m on a spring with
  spring constant k

19
Work Done in Straining a Spring

• Stretch or compress a spring by a displacement x
  from its unstrained length.
• Initial force F0 0 Final force Fmax kx
• Average force
• Work over a displacement x

20
Elastic Potential Energy

• The spring force is a conservative force
• Like all conservative forces, its work is
  path-independent.
• Like all conservative forces, it is associated
  with a form of stored or potential energy.
• Elastic potential energy

21
Total Mechanical Energy

• We add another term

22
The Simple Pendulum

• A simple pendulum is a particle attached to one
  end of a massless cord of length L. It is able
  to swing freely and without friction from the
  other end of the cord.
• Its frequency

23
The Physical Pendulum

• A physical pendulum is any real object (mass m)
  suspended a distance L from its center of
  gravity, able to swing freely and without
  friction from the suspension point.
• Its frequency

24
Physical-Simple Correspondence

• Notice that a simple pendulum would have a moment
  of inertia
• Substitute
• As a physical pendulum becomes a simple one, its
  frequency collapses to that of a simple
  pendulum.

25
Small-Angle Approximation

• The restoring torque on a pendulum does not
  actually have the Hookes Law form
• Result the restoring torque increases with
  angle, but at less than a linear rate.

26
Small-Angle Approximation

• How much less?

q, under linear
0.10 0.0002
1.0 0.02
2.0 0.08
5.0 0.51
10 2.0
27
Damped Oscillations

• If the only force doing work on an object is the
  spring force (conservative), its mechanical
  energy is conserved. If frictional forces also
  do work, the objects mechanical energy
  decreases, and the SHM is called damped.
• If the frictional force is just large enough to
  prevent oscillation as the object reaches its
  equilibrium position, it is called critically
  damped.

28
Driven Oscillations

• If a driving force acts on an object in addition
  to a Hookes Law restoring force, the harmonic
  motion of the object is called driven.
• Example a tree in a gusty wind.

29
Driven Oscillations Resonance

• If the driving force is periodic, and is applied
  at the natural frequency of the oscillating
  object, the work done on the object adds up over
  multiple cycles of motion, and large-amplitude
  motion results. This is called resonance. The
  natural frequency is sometimes called the
  resonant frequency.
• Example a person on a swing, being pushed by
  another person.

30
Material Deformation Everything is a Spring

• Solid materials are interconnected,
  microscopically, by powerful intermolecular
  bonding forces.
• These forces behave like springs with really
  large spring constants.
• Because of them, material objects resist
  deformations, such as compression, elongation, or
  shearing.

31
Tension and Compression

• A force acts to increase the length of an object

fractional change in length
applied force
cross- sectional area
Youngs modulus SI units N/m2
32
Thomas Young

• 1773 - 1829
• English physicist,
• physician, and
• Egyptologist
• Famous mostly for his
• work in optics

33
Shear

• A pair of forces act to shear an object (deform
  it slantwise)

applied force
cross-sectional area
Shear modulus SI units N/m2
34
Shear

• 1945 - ?
• Has only one name
• English physicist and pop
• musician
• Inventor of the shear
• modulus
• Rumored to have appeared in the 1983 version of
  Dune

35
Volume Deformation

• In order to discuss volume deformation, it is
  necessary to define a new force-related quantity
  pressure.
• Pressure is the ratio of the magnitude of a force
  applied perpendicular to a surface to the area of
  that surface
• SI units N/m2 Pascals (Pa)

36
Blaise Pascal

• 1623 1662
• French mathematician
• Invented the first digital
• calculator (the Pascaline)

37
Volume Deformation

• A change in pressure changes the volume of an
  object

pressure change
fractional change in volume
bulk modulus SI units N/m2
38
Stress and Strain

• Stress is the deforming force applied to an
  object, divided by its cross-sectional area
• Stress has SI units of N/m2 (just as pressure and
  the elastic moduli have).

39
Stress and Strain

• Strain is the change in a dimensional quantity
  expressed as a fraction of its un-deformed value
• Strain is a dimensionless, unitless ratio.
• Strain is the result of stress on a material
  object.

40
Stress and Strain

• Consider the defining equation for Youngs
  modulus
• Rearrange
• To restate
• This is the stress-strain formulation of Hookes
  Law.

41
Summary Elastic Moduli

• deformation elastic modulus
  equation
• length Youngs modulus (Y)
• shear shear modulus (S)
• volume bulk modulus (B)

all moduli have the same SI units N/m2 Pa