Physics 1B03summer - Lecture 7 Test 2 Tuesday June 2 at 9:30 am CNH-104 Energy, Work, Power, Simple...

Physics 1B03summer - Lecture 7

Test 2

Tuesday June 2 at 9:30 am

CNH-104

Energy, Work, Power, Simple Harmonic Motion

Class will start at 11am


Energy of a SHO


Energy of a SHO

Recall that for a spring:

ETot = K + U = 1/2mv2 + 1/2kx2

And we know that: x=A cos(ωt+φ)

v=-A ω sin(ωt+φ)


Energy in SHM

MLook again at the block & spring

energy) mechanical (totalconstant a

)(cos)(sin

)(cos)(sin

!

)(cos

)(sin

221

22221

222221

22212

21

222212

21

kA

ttkA

tktmAUK

k

tkAkxU

tAmmvK

We could also write E = K+U = ½ m(vmax )2

Hence: ETot = ½kA2


Suppose you double the amplitude of the motion, whathappens to the total energy?

a) Doublesb) 4 x Largerc) Doesn’t change

Quiz



Quiz

Suppose you double the amplitude of the motion, whathappens to the maximum speed ?


Suppose you double the mass and amplitude of the object, what happens to the maximum acceleration ?


Quiz


Energy

Since we know the total energy of a SHM, we can calculate the or displacement velocity at any point in time:

ETot = ½ kA2 = K+U = ½ mv2 + ½ kx2

So, if x=0, all E is in kinetic, and v is at max

if x=A, all E is in potential, and v is zero


A 100g block is 5cm from the equilibrium position moving at 1.5m/s. The angular frequency is 2 rad/s.

a) What is the total energy of the system ?

b) What is the amplitude of the oscillations ?

Example 1


Example 2

A 500g block on a spring is pulled 20cm and released.The motion has a period of 0.8s.

What is the velocity when the block is 15.4cm from the equilibrium ?


A 1.0kg block is attached to a spring with k=16N/m. While the block is at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40cm/s.

a) what is the amplitude of the subsequent oscillations ?

b) what is the block’s speed at the point where x=A/2 ?

Example 3


10 min rest


SHM and Circular Motion

)sin(sin

)cos(cos

0

0

tAAy

tAAx

A

Uniform circular motion about in the xy-plane, radius A, angular velocity :

(t) = 0 + t

(similar to, x=xo+vt)

and so

Hence, a particle moving in one dimension can be expressedas an ‘imaginary’ particle moving in 2D (circle), or vice versa -the ‘projection’ of circular motion can be viewed as 1D motion.


)sin(sin

)cos(cos

tAAy

tAAx

o

o

Compare with our expression for 1-D SHM.

Result:

SHM is the 1-D projection of uniform circular motion.

)cos( tAx


Phase Constant, θo

A

For circular motion,the phase constantis just the angle atwhich the motion started.


Example

An object is moving in circular motion with an angular frequency of 3π rad/s, and starts with an initial angle of π/6. If the amplitude is 2.0m, what is the objects angular position at t=3sec ?

What are the x and y values of the position at this time ?


Simple Pendulum

Gravity is the “restoring force” taking the place of the “spring” in our block/spring system.

Instead of x, measure the displacement as the arc length s along the circular path.

Write down the tangential component of F=ma:

sinmg

L

θ T

sin

But

)sin(

2

2

2

2

L

g

dt

d

Ls

mgmadt

sdm t

mg sin θ

Restoring force

s

mg


sin

2

2

L

g

dt

dSimple pendulum:

xdt

xd 22

2

SHM:

The pendulum is not a simple harmonic oscillator!

Lg

Lg

dtd sin 2

2

sinHowever, take small oscillations: (radians) if is small.

Then

22

2

dt

d

Compare:


Simple Pendulum

L

g

dt

d

2

2

Using sin(θ)~θ for small angles, we have thefollowing equation of motion:

With:Lg

Hence:2

2

2 4gTg

L

Application - measuring height - finding variations in g → underground resources

or: 2

22 4

T

LLg

-------------------------------------------------------------------------


This looks like

Lg

dtd 2

2

For small :

xdt

xd 22

2

, with angle instead of x.

So, the position is given by )cos()( o tt

amplitude

phase constant

(2/ period)


a) longer.b) same.c) shorter.

Quiz:

A geologist is camped on top of a large deposit of nickel ore, in a location where the gravitational field is 0.01% stronger than normal. The period of his pendulum will be

(by how much, in percent?)


Application

Pendulum clocks (“grandfather clocks”) often have a swinging arm with an adjustable weight. Suppose the arm oscillates with T=1.05sec and you want to adjust it to 1.00sec. Which way do you move the weight?

?


Quiz

A simple pendulum hangs from the ceiling of an elevator. If the elevator accelerates upwards, the period of the pendulum:

a) Gets shorterb) Gets largerc) Stays the same

Question: What happens to the period of a simple pendulum if the mass m is doubled?


A physicist wants to know the height of a building. He notes that a long pendulum that extends from the ceiling almost to the floor has a period of 12 s. How tall is the building?

Example


Compare Springs and Pendulum

xm

k

dt

xd

2

2

kxmaF

Newton’s 2nd Law:

mk

L

g

dt

d

2

2

Lg

singmmaF


t

x

x

t

SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant.

Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time.

SHM and Damping – EXTRA !!!


)cos()( 2

tAetxt

m

b

For weak damping (small b), the solution is:

f = bv where b is a constant damping coefficient

x

t

A damped oscillator has external nonconservativeforce(s) acting on the system. A common example is a force that is proportional to the velocity.

eg: green water

A e-(b/2m)t

2

2

dt

xdm

dt

dxbkx F=ma give:


10 min rest


Wave Motion Wave Motion – Chapter 20– Chapter 20

•Qualitative properties of wave motion•Mathematical description of waves in 1-D•Sinusoidal waves


A wave is a moving pattern. For example, a wave on a stretched string:

The wave speed v is the speed of the pattern. No particles move at this wave speed – it is the speed of the wave. However, the wave does carry energy and momentum.

v

Δx = v Δt

time t

t +Δt


The wave moves this way

The particles move up and down

If the particle motion is perpendicular to the direction the wave travels, the wave is called a “transverse wave”.

Transverse waves

Examples: Waves on a string; waves on water; light & other electromagnetic waves; some sound waves in solids (shear waves)


The wave moves long distances parallel to the particle motions.

Longitudinal waves

Example: sound waves in fluids (air, water)

Even in longitudinal waves, the particle velocities are quite different from the wave velocity. The speed of the wave can be orders of magnitude larger than the particle speeds.

The particles move back and forth.


QuizQuiz

wave motionA

B

C

Which particle is moving at the highest speed?

A) AB) BC) CD) all move with the same speed


For these waves, the wave speed is determined entirely by the medium, and is the same for all sizes & shapes of waves.

T

v lengthmass/unit

tension

eg. stretched string:

(A familar example of a dispersive wave is an ordinary water wave in deep water. We will discuss only non-dispersive waves.)

Non-dispersive waves: the wave always keeps the same shape as it moves.


Principle of SuperpositionPrinciple of Superposition

When two waves meet, the displacements add:

),(),(),( 21 txytxytxy observed

So, waves can pass through each other:

v

v

(for waves in a “linear medium”)


Quiz: Quiz: “equal and opposite” waves“equal and opposite” waves

v

v

Sketch the particle velocities at the instant the string is completely straight.

v

v


ReflectionsReflections

Waves (partially) reflect from any boundaryin the medium:

1) “Soft” boundary:

Reflection is upright

light string, or free end


Reflection is inverted

ReflectionsReflections

2) “Hard” boundary:

heavy rope, or fixed end.


QuizQuiz

A) y= f(xvt) B) y= f(x+vt) C) y= f( xvt) D) y= f( x+vt)

A wave show can be described by the equation y=f(xvt), as it travels along the x axis. It reflects from a fixed end at the origin. The reflected wave is described by:


The math: Suppose the shape of the wave at t = 0 is given by some function y = f(x).

y y = f (x)v

at time t :y = f (x - vt)

y

vt

at time t = 0:

Note: y = y(x,t), a function of two variables; f is a function of one variable


Non-dispersive waves:

y (x,t) = f (x ± vt)

+ sign: wave travels towards –x sign: wave travels towards +x

f is any (smooth) function of one variable.

eg. f(x) = A sin (kx)

Physics 1B03summer - Lecture 7 Test 2 Tuesday June 2 at 9:30 am CNH-104 Energy, Work, Power, Simple...

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