Chapter 13

Simple Harmonic Motion

Lecture 1OutlinePeriodic Motion and WavesHookes LawSimple Harmonic Motion (SHM)SHM Motion as a Function of TimeAngular Frequency, Period and Frequency of SHMElastic Potential EnergyMotion of PendulumDamped Harmonic MotionResonance

spring constant

Units: N/m

Applied force (external force) on a springHookes Law

Exercise 1

The spring constant of the spring is 320 N/m and the bar indicator extends 2.0 cm. What force does the air in the tire apply to the spring?

SolutionSimple Harmonic Motion Horizontal

If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.

Hookes Law Applied to a Spring Mass System

When applied to Spring-mass systemHookes Law becomes FS =-kxRestoring ForceFormula:Fs = - k x

Fs is the spring forcek is the spring constantx is the displacement of the object from its equilibrium positionThe negative sign indicates that the force is always directed towards the equilibrium position.Restoring ForceThe force acts toward the equilibrium positionIt is called the restoring forceThe direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position

Restoring ForceThe force is called a restoring force because at both situations, the spring exerts a force on the mass that acts in the direction of returning the mass to the equilibrium position.

This is the force by the spring itself. When to use F applied or F restoring?When external force is used to cause the extension or compression, use

When the mass is oscillating, use

Acceleration of mass

Example 2:

A 0.350-kg object attached to a spring of force constant 1.30102 N/m is free to move on a frictionless horizontal surface. If the object is released from rest at x = 0.100 m, find the force on it and its acceleration at x = 0.100 m, x = 0.0500 m, x = 0 m, x = -0.0500 m, and x = -0.100 m.

Solution:Simple Harmonic Motion (SHM)Motion that occurs when the net force along the direction of motion obeys Hookes LawThe force is proportional to the displacement and always directed toward the equilibrium positionThe motion of a spring-mass system is an example of Simple Harmonic Motion

Simple Harmonic MotionA mass spring system is a example of Simple Harmonic Motion SHM for spring-mass system

Therefore it is a SHMAmplitude, Period & FrequencyAmplitude, A is maximum position of the object from its equilibrium position.Oscillate between the positions x = A.The period, T, is the time that it takes for the object to complete one complete cycle of motion From x = A to x = - A and back to x = AThe frequency, , is the number of complete cycles or vibrations per unit timeFrequency is the reciprocal of the period = 1 / T

Simple Harmonic Motion and the Reference Circle

Simple Harmonic Motion and the Reference Circle

Motion as a Function of Time

Use of a reference circle allows a description of the motion

x is the position at time tx varies between +A and -A

Simple Harmonic Motion and the Reference Circle

period T: the time required to complete one cyclefrequency f: the number of cycles per second (measured in Hz)

amplitude A: the maximum displacementMotion as a Function of TimeUse of a reference circle allows a description of the motionx = A cos (t + )x is the position at time tx varies between +A and A is called the angular frequencyUnits are rad/s is the phase constant or the initial phase angle

Position, Velocity, Acceleration as a Function of Time

The graphs show:displacement as a function of time

velocity as a function of time

acceleration as a function of time

Graphical Representation of MotionWhen x is a maximum or minimum, velocity is zeroWhen x is zero, the velocity is a maximumWhen x is a maximum in the positive direction, a is a maximum in the negative direction

Derive the angular frequency

Period and Frequency from Circular MotionPeriod

This gives the time required for an object of mass m attached to a spring of constant k to complete one cycle of its motion

Frequency

Units are cycles/second or Hertz, Hz

Equations of SHMWhen t = 0, object is at the maximum displacement

Example 3:A mass of 0.250 kg hanging at a lower end of spring causes the spring to extend by 7.0 cm. Calculatethe spring constant the period of the vertical oscillation of the 0.250 kg mass.Solution:Example 4:A 1.30103-kg car is constructed on a frame supported by four springs. Each spring has a spring constant of 2.00104 N/m. If two people riding in the car have a combined mass of 1.60102 kg, find the frequency of vibration of the car when it is driven over a pothole in the road. Find also the period and the angular frequency. Assume the weight is evenly distributed. (Hint: period, frequency and angular frequency stand for singular, this means all the four springs have the same values for all the 3 quantities above!)

Solution:Exercise 5An object is moving in a simple harmonic motion which can be described by the equation x(t) = (15 mm) sin (10t).What is the amplitude and the period of the motion?Calculate the displacement of the object when t = 0.040 s.Calculate the velocity thenCalculate the maximum acceleration of the objectSolutionExample 6:

The device consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of the astronaut.

Solution:Elastic Potential EnergyA compressed spring has potential energyThe compressed spring, when allowed to expand, can apply a force to an objectThe potential energy of the spring can be transformed into kinetic energy of the objectThe energy stored in a stretched or compressed spring or other elastic material is called elastic potential energyPEs = kx2Energy in the SHM

If the mass is at the limits of its motion, the energy is all potential.If the mass is at the equilibrium point, the energy is all kinetic.We know what the potential energy is at the turning points:

Energy of a spring-mass system(horizontal)

Variation of K and U with t and x

More on spring mass systemExplanation

Example 7A particle performs simple harmonic motion with amplitude A. What fraction of the total energy is kinetic energy when the displacement is A ?SolutionExample 8A 0.6 kg mass attached to a spring vibrates three times per second with an amplitude of 120 cm. What is the velocity when it is 20 cm from the equilibrium position ?(Find k first from )

Solution

Example 9:

A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring? (Note: Momentary stop -> lowest point -> the kinetic energy = 0.)Solution:A 0.500-kg cart connected to a light spring for which the force constant is 20.0 N/m oscillates on a frictionless, horizontal air track.Calculate the maximum speed of the cart if the amplitude of the motion is 3.00 cm.What is the velocity of the cart when the position is 2.00 cm?Compute the kinetic and potential energies of the system when the position of the cart is 2.00 cm.Example 10:Solution:Simple Pendulum

Hookes law is F = - kxFor small angles, sin

Period of Simple PendulumThis shows that the period is independent of the amplitude and the massThe period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum

Simple Pendulum Compared to a Spring-Mass System

Example 11:Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.Solution:Damped OscillationsOnly ideal systems oscillate indefinitelyIn real systems, friction retards the motionFriction reduces the total energy of the system and the oscillation is said to be damped

More Types of DampingWith a higher viscosity, the object returns rapidly to equilibrium after it is released and does not oscillateThe system is said to be critically dampedWith an even higher viscosity, the piston returns to equilibrium without passing through the equilibrium position, but the time required is longerThis is said to be overdampedGraphs of Damped OscillatorsCurve a shows an underdamped oscillatorCurve b shows a critically damped oscillatorCurve c shows an overdamped oscillator

Resonance

Two resonance curves for an oscillator with natural frequency f0. The amplitude of the driving force is constant. In the red graph, the oscillator has one fourth as much damping as in the blue graph.When the driving frequency is equal to the natural frequency of the system, the amplitude of the motion is a maximum. This condition is called resonance.Example of Resonance

Turbulent winds set up torsional vibrations in the Tacoma Narrow Bridge, causing it at a frequency near one of the natural frequencies of the bridge structure. The resonance condition led to the bridges collapse.Summary