ub, phy101: chapter 10, pg 1 physics 101: lecture 17 simple harmonic motion l today’s lecture will...

UB, Phy101: Chapter 10, Pg 1

Physics 101: Physics 101: Lecture 17Lecture 17Simple Harmonic MotionSimple Harmonic Motion

Today’s lecture will cover Textbook Sections 10.1 - 10.4


Simple Harmonic MotionSimple Harmonic Motion

Period = T (seconds per cycle)

Frequency = f = 1/T (cycles per second)

Angular frequency = = 2f = 2/T


Simple Harmonic Motion:Simple Harmonic Motion:

x(t) = [A]cos(t)

v(t) = -[A]sin(t)

a(t) = -[A2]cos(t)

x(t) = [A]sin(t)

v(t) = [A]cos(t)

a(t) = -[A2]sin(t)

xmax = A

vmax = A

amax = A2

Period = T (seconds per cycle)

Frequency = f = 1/T (cycles per second)

Angular frequency = = 2f = 2/T

For spring: 2 = k/m

OR


SpringsSprings

Hooke’s Law: Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position.

FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality.

(often called “spring constant”)

relaxed position

FX = 0

xx=0


SpringsSprings




relaxed position

FX = -kx > 0

xx 0

x=0


SpringsSprings




FX = - kx < 0

xx > 0

relaxed position

x=0


X=0

X=AX=-A

X=A; v=0; a=-amax

X=0; v=-vmax; a=0

X=-A; v=0; a=amax

X=0; v=vmax; a=0

X=A; v=0; a=-amax

Springs and Simple Harmonic MotionSprings and Simple Harmonic Motion


Chapter 10, PreflightChapter 10, Preflight

A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The speed of the mass is constant

+A

t-A

x

CORRECT


Chapter 10, Preflight Chapter 10, Preflight

A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The acceleration of the mass is constant

+A

t-A

x

CORRECT


Review: Period of a SpringReview: Period of a Spring

For simple harmonic oscillator

= 2f = 2/T

For mass M on spring with spring constant k

k

m2 T

m

k


What does moving in a circle have to do with moving back & forth in a straight line ??

y

x

-R

R

0

1 1

2 2

3 3

4 4

5 5

6 62

R

8

7

8

7

23

x

x = R cos = R cos (t)since = t


Potential Energy of a SpringPotential Energy of a Spring

0x

2S kx

21

PE

Where x is measured fromthe equilibrium position PES

m

xx=0

Analogy: marble in a bowl:

PE KE PE KE PE

“height” of bowl x2


Same thing for a vertical spring:Same thing for a vertical spring:

0y

PES

m

y

2S ky

21

PE

Where y is measured fromthe equilibrium position

y=0



In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. In which case is the maximum kinetic energy of the mass the biggest?

1. Case 12. Case 23. Same

CORRECT


Chapter 10, Preflight Chapter 10, Preflight

PE = 1/2kx2 KE = 0

x=0 x=+Ax=-A x=0 x=+Ax=-A

samefor both

PE = 0 KE = KEMAX

same for both


Pendulum SummaryPendulum Summary

Lg

gL

T 2

2

For “small oscillation”, period does not depend on

•mass

•amplitude

DEMO



Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should: 1. Make the pendulum shorter 2. Make the pendulum longer

Lg

gL

T 2

2

CORRECT



A pendulum is hanging vertically from the ceiling of an elevator. Initiallythe elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be1. greater than T 2. equal to T3. less than T

Lg

gL

T 2

2

CORRECT

“Effective g” is larger when accelerating upward

(you feel heavier)



Imagine you have been kidnapped by space invaders and are being held prisoner in a room with no windows. All you have is a cheap digital wristwatch and a pair of shoes (including shoelaces of known length). Explain how you might figure out whether this room is on the earth or on the moon

Lg

gL

T 2

2

2

22T

Lg


using L=T^2 * g / 4pi^2, solve for gravity. Use the shoelaces to find the T and if g equals 9.8, you are on earth otherwise goodluck

All you would have to do is jump up in the air. The gravity on the moon is 1/6 that of earth's. If you were on the moon, you would come down a lot slower.

ub, phy101: chapter 10, pg 1 physics 101: lecture 17 simple harmonic motion l today’s lecture will...

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