chapter 13 oscillations about equilibriumnsmn1.uh.edu/hpeng5/peng13_lectureoutline.pdfa glider with...

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Chapter 13

Oscillations about Equilibrium


Units of Chapter 13

• Periodic Motion

• Simple Harmonic Motion

• Connections between Uniform Circular Motion and Simple Harmonic Motion

• The Period of a Mass on a Spring

• Energy Conservation in Oscillatory Motion


Units of Chapter 13

• The Pendulum

• Damped Oscillations

• Driven Oscillations and Resonance


13-1 Periodic Motion

Period: time required for one cycle of periodic motion

Frequency: number of oscillations per unit time

This unit is called the Hertz:


13-2 Simple Harmonic MotionA spring exerts a restoring force that is proportional to the displacement from equilibrium:


13-2 Simple Harmonic Motion

A mass on a spring has a displacement as a function of time that is a sine or cosine curve:

Here, A is called the amplitude of the motion.


13-2 Simple Harmonic Motion

If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time:

It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.


13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion

An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:


Here, the object in circular motion has an angular speed of

where T is the period of motion of the object in simple harmonic motion.



The position as a function of time:

The angular frequency:

Question 13.1aQuestion 13.1a Harmonic Motion IHarmonic Motion Ia) 0a) 0

b) b) AA/2/2

c) c) AA

d) 2d) 2AA

e) 4e) 4AA

A mass on a spring in SHM has

amplitude A and period T. What

is the total distance traveled by

the mass after a time interval T?

Question 13.1aQuestion 13.1a Harmonic Motion IHarmonic Motion Ia) 0a) 0

b) b) AA/2/2

c) c) AA

d) 2d) 2AA

e) 4e) 4AA


amplitude A and period T. What

is the total distance traveled by

the mass after a time interval T?

In the time interval time interval TT (the period), the mass goes through one complete oscillationcomplete oscillation back to the starting point. The distance it covers is The distance it covers is A + A + A + AA + A + A + A (4(4AA).).

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T?

a) 0b) A/2c) Ad) 2Ae) 4A

Question 13.1bQuestion 13.1b Harmonic Motion IIHarmonic Motion II

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T?

a) 0b) A/2c) Ad) 2Ae) 4A

The displacement is Δx = x2 – x1. Because the initial and final positions of the mass are the same (it ends up back at its original position), then the displacement is zero.

Question 13.1bQuestion 13.1b Harmonic Motion IIHarmonic Motion II

FollowFollow--upup:: What is the net displacement after a half of a period?What is the net displacement after a half of a period?

A mass on a spring in SHM has amplitude A and period T. How long does it take for the mass to travel a total distance of 6A ?

a) ½Tb) ¾Tc) 1¼Td) 1½Te) 2T

Question 13.1cQuestion 13.1c Harmonic Motion IIIHarmonic Motion III

A mass on a spring in SHM has amplitude A and period T. How long does it take for the mass to travel a total distance of 6A ?

We have already seen that it takes one period T to travel a total distance of 4A. An additional 2A requires half a period, so the total time needed for a total distance of 6A is 1 T.

Question 13.1cQuestion 13.1c Harmonic Motion IIIHarmonic Motion III

FollowFollow--upup:: What is the net displacement at this particular time?What is the net displacement at this particular time?

a) ½Tb) ¾Tc) 1¼Td) 1½Te) 2T

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The velocity as a function of time:

Found by taking x components of the circular motion quantities.



The acceleration:

Found by taking x components of the circular motion quantities.

Question 13.2 Question 13.2 Speed and AccelerationSpeed and Acceleration

a) x = A

b) x > 0 but x < A

c) x = 0

d) x < 0

e) none of the above


amplitude A and period T. At

what point in the motion is v = 0

and a = 0 simultaneously?

Question 13.2 Question 13.2 Speed and AccelerationSpeed and Acceleration

a) x = A

b) x > 0 but x < A

c) x = 0

d) x < 0

e) none of the above


amplitude A and period T. At

what point in the motion is v = 0

and a = 0 simultaneously?

If bothIf both vv andand aa were zero at were zero at the same time, the mass the same time, the mass would be at rest and stay at would be at rest and stay at rest!rest! Thus, there is NONOpointpoint at which both vv and aaare both zero at the same time.

FollowFollow--upup:: Where is acceleration a maximum?Where is acceleration a maximum?


13-4 The Period of a Mass on a SpringSince the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that .

Substituting the time dependencies of a and xgives


13-4 The Period of a Mass on a Spring

Therefore, the period is

A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?

a) period will increaseb) period will not changec) period will decrease

Question 13.6aQuestion 13.6a Period of a Spring IPeriod of a Spring I

A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?

a) period will increaseb) period will not changec) period will decrease

The period is proportional to the square root of the mass. So an increase in mass will lead to an increase in the period of motion.

Question 13.6aQuestion 13.6a Period of a Spring IPeriod of a Spring I

FollowFollow--upup:: What happens if the amplitude is doubled?What happens if the amplitude is doubled?

km

T =2π

(a)

(b)


13-5 Energy Conservation in Oscillatory Motion

In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:

Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:



As a function of time,

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



This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

a) total energy will increaseb) total energy will not changec) total energy will decrease

Question 13.5aQuestion 13.5a Energy in SHM IEnergy in SHM I

A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

a) total energy will increaseb) total energy will not changec) total energy will decrease

The total energy is equal to the initial value of the elastic potential energy, which is PEs = kA2. This does not depend on mass, so a change in mass will not affect the energy of the system.

Question 13.5aQuestion 13.5a Energy in SHM IEnergy in SHM I

FollowFollow--upup:: What happens if you double the amplitude?What happens if you double the amplitude?

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13-6 The PendulumA 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.


13-6 The Pendulum

Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ,whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).


13-6 The Pendulum

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


13-6 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:


13-7 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:



This exponential decrease is shown in the figure:



The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. 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.


13-8 Driven Oscillations and ResonanceAn oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.


13-8 Driven Oscillations and Resonance

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


Summary of Chapter 13• 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 of Chapter 13

• The amplitude is the maximum displacement from equilibrium.

• Position as a function of time:

• Velocity as a function of time:



• Acceleration as a function of time:

• Period of a mass on a spring:

• Total energy in simple harmonic motion:



• Potential energy as a function of time:

• Kinetic energy as a function of time:

• A simple pendulum with small amplitude exhibits simple harmonic motion



• Period of a simple pendulum:



• 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



• 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

chapter 13 oscillations about equilibriumnsmn1.uh.edu/hpeng5/peng13_lectureoutline.pdfa glider with...

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