Periodic Motion - 1

Oscillatory/Periodic MotionRepetitive Motion

Simple Harmonic Motion SHMspecial case of oscillatory.

• Object w equilibrium position has restoring force pushing it back to equilibrium.

Simple Harmonic MotionA spring exerts a restoring force directily proportional to the displacement fr. eq:

Remember Hooke’s Law

Negative sign Fnet opposite to s.

Pendulum is not SHM

• Fnet not directly opposite s

• for small displacement angles it approximates SHM.

unit is Hertz:

• Period T: time required for one cycle of periodic motion (sec).

• Frequency: number of oscillations per unit time.

Vocabulary

• Equilibrium (0): the spot the mass would come to rest when not disturbed – Fnet = 0.

• Displacement: (x) distance from equilibrium.

• Amplitude (xo) – max displacement from eq.

• Angular frequency , multiply f by 2. Reported as cy/sec or rad/sec. 2 = 1 full cycle.

• Angular frequency is a measure that comes from circular motion.

• It is measured in radians per second.

• f, Hz is cycles per second.

An object makes 1 revolution per second. How many radians does it complete in 1 second?

• 2 rad s-1 = 1 revolution per second.

• 2 rad s-1 = also describes 1 non-circular cycle or oscillation.

To find from frequency f, or period T, use:

= 2f or = 2 T

• T = 0.125 s

= 2/T = 50 rad/sec.

Ex 1. A pendulum completes 4 swings in 0.5 seconds. What is its angular frequency?

Free Body Diagram Mass on Spring

• Complete sheet.

•

Simple Harmonic Motion2 Conditions.

• 1. Acceleration/Fnet proportional to displacement.

• 2. Acceleration/Fnet directed toward equilibrium.

• Defining Equation for SHM• a = -2x

Sketch Graphs.

Simple Harmonic Motion On a bobbing mass:

• No friction

• Weight stays constant.

• Tension increases with stretch.

• Fnet / accl toward equilibrium position.

• Fnet increases with is proportional to displacement.

Graphical TreatmentEquations of SHM

Displacement, x, against time x = xo cos t

start point at max ampl.

** Set Calculator in Radians.

Displacement against time x = xosint

• Speed = d/t = 2R/T but

• 2so

• v = r but r the displacement x so.

• v = x.

Velocity against time v = vocost

Starting where? Midpoint = max velocity.

Equations of Graphs

• x = xo cos t x = xo sin t

• v = -vo sin t v = vo cos t

• a = -aocos t -ao sin t

• Released from Released equilibrium.

top.

• x = xo cos t xo = 3 cm

= 2f. = 0.4 Hz =1.26 rad/s.

• t = 10.66 s

• x = 0.03 cos (x 10.66) = 0.019 m

• You must use radians on calculator.

Ex 2. A mass on a spring is oscillating with f = 0.2 Hz and xo = 3 cm. What is the displacement of the mass 10.66 s after its release from the top?

SHM and Circular Motion

• Use the relationship to derive equations.

If an object moving with constant speed in a circular path is observed from a distant point (in the plane of the motion), it will appear to be oscillating with SHM.

The shadow of a pendulum bob moves with s.h.m. when the pendulum itself is either oscillating (through a small angle) or moving in a circle with constant speed, as shown in the diagram.

For any s.h.m. we can find a corresponding circular motion.

When a circular motion "corresponds to" a given s.h.m.,

i) the radius of the circle is equal to the amplitude of the s.h.m. 

ii) the time period of the circular motion is equal to the time period of the s.h.m.

From circular motion ac = v2/r and vc = 2r/TOscillating systems have acceleration too.

• But = 2/T

• vc = 2r/T

• vc = r

• But ac = v2/r

• So ac = (r)2/r

• but r is related to displacement x.

Derive Relationship between accl & for SHM.

For any displacement:

a = -²x

ao = -²xo

The negative sign shows Fnet & accl direction opposite displacement.

Derivation of accl in Hamper pg 76. Or use 2nd derivative of displacement.

Ex 3. A pendulum swings with f = 0.5 Hz. What is the size & direction of the acceleration when the bob has displacement of 2 cm right?

• a = -²x

• = 2f = • a = -()2 (0.02 m) = -0.197 m/s2. left.

Ex 4: A mass is bobbing on a spring with a period of 0.20 seconds. What is its angular acceleration at a point where its displacement is 1.5 cm?

• = 2/T

• . = 31 rad/s

• a = -²x

• a = (31rad/s)(1.5 cm) = 1480 cm/s2.

• 15 m/s2.

SHM, Hooke’s Law & k.

For a mass undergoing SHM on a spring, what is the relation between angular frequency , and k the spring constant?

• Use Hooke’s law and make substitutions to derive a relation in terms of angular frequency, k, and mass.

ma = - k x.

So a = -k x m

F= - kx.

a = -² x

So 2 = -k/m

To find the velocity of an oscillating mass or pendulum at any displacement:

When the mass is at equilibrium, x = 0, and velocity is maximum:

vo = ± xo.

Derivation on H pg 77.

At max velocity vo = xo.

f = 1) = rad/s

vo = (2 rad/s)(0.03)

vo = 0.188 m/s

Ex 4. A pendulum swings with f = 1 Hz and amplitude 3 cm. At what position will be its maximum velocity &what is the velocity?

vo = 0.2 m/s

Hwk Hamper pg 75- 77 Show equations and work, hand in virtual solar system lab.

http://www.learner.org/resources/series42.html?pop=yes&pid=565

Mechanical Universe w/questions

Units of Chapter 4

• The Pendulum

• Damped Oscillations

• Driven Oscillations and Resonance

The Period of a Mass on a Spring

Since the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that .

Make substitutions to find the relationship between T and k.

The Period of a Mass on a Spring

Therefore, the period is

How does T change as mass increases? Sketch it!

Period of pendulum

Energy Conservation in Oscillatory Motion

In an ideal system the total mechanical energy is conserved. A mass on a spring:

Horizontal mass no PEg.

Determining the max KE & PE:

• PE = ½ k x2 for a stretched spring.

• So PEmax = PE = ½ k xo

• KE = ½ mv2

• vmax = xo,

• KEmax = ½ m(2xo 2),

At any point:

• KE = ½ m2 (xo2 - x2 )

• How could you determine PE from Etot?

• Subtract KE from Etot.

Since the total E will always equal the max KE (or PE), we can calculate the number of Joules of total E from the KE equation:

ET = ½ m2xo2

Ex 5: A 200-g pendulum bob is oscillating with Amplitude = 3 cm, and f = 0.5 Hz. How much KE will it have as it passes through the origin?

• KEmax = ½ 2xo 2,

• xo = 0.03 m

= .

• KE = 8.9 x 10-4 J.

Energy Conservation in Oscillatory Motion

The total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.

Energy Conservation in Oscillatory Motion

The E transforms from potential to kinetic & back, the total energy remains the same.

The Pendulum

A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

The Pendulum

The restoring force is actually proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement x.

The Pendulum

However, for small angles, sin θ and θ are approximately equal.

The Pendulum

Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring.

Therefore, we find that the period of a pendulum depends only on the length of the string:

Damped Oscillations

In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed:

This causes the amplitude to decrease exponentially with time:

Damped Oscillations

This exponential decrease is shown in the figure:

The image shows a system that is underdamped – it goes through multiple

oscillations before coming to rest.

Damped Oscillations

A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time;

an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

Driven Oscillations and Resonance

An oscillation can be driven by an oscillating driving force; the f of the driving force may or may not be the same as the natural f of the system.

Driven Oscillations and Resonance

If the driving f is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

Summary

• Period: time required for a motion to go through a complete cycle

• Frequency: number of oscillations per unit time

• Angular frequency:

• Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.

Summary

• The amplitude is the maximum displacement from equilibrium.

• Position as a function of time:

• Velocity as a function of time:

Summary

• Acceleration as a function of time:

• Period of a mass on a spring:

• Total energy in simple harmonic motion:

Summary

• Potential energy as a function of time:

• Kinetic energy as a function of time:

• A simple pendulum with small amplitude exhibits simple harmonic motion

Summary

• Period of a simple pendulum:

• Period of a physical pendulum:

Summary

• Oscillations where there is a nonconservative force are called damped.

• Underdamped: the amplitude decreases exponentially with time:

• Critically damped: no oscillations; system relaxes back to equilibrium in minimum time

• Overdamped: also no oscillations, but slower than critical damping

Summary

• An oscillating system may be driven by an external force.

• This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance

• Resonance occurs when the driving frequency is equal to the natural frequency of the system