REPRESENTING SIMPLE HARMONIC MOTION

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/shm.gif

1

not simple

simple

y(t)

Simple Harmonic Motion

Watch as time evolves

2

-A

A

amplitude phase angle

determined by initial conditions

period

angular freq (cyclic) freq

determined by physical system3

Position (cm)

Velocity (cm/s)

Acceleration (cm/s2)

time (s)4

These representations of the position of a simple harmonic oscillator as a function of time are all equivalent - there are 2 arbitrary constants in each. Note that A, , Bp and Bq are REAL; C and D are COMPLEX. x(t) is real-valued variable in all cases.

Engrave these on your soul - and know how to derive the relationships among A & ; Bp & Bq; C; and D .

5

A:

B:

C:

D:

m = 0.01 kg; k = 36 Nm-1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms-1.

Express the motion in form A

form B

x

m

mk

k

Example: initial conditions

6

7

8

A particle executes simple harmonic motion.

When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle.

(a)potential energy is maximum and acceleration is maximum.(b)potential energy is maximum and acceleration is zero.(c)potential energy is minimum and acceleration is maximum.(d)potential energy is minimum and acceleration is zero.

9

A particle executes simple harmonic motion.

When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle.

(a)potential energy is maximum and acceleration is maximum.(b)potential energy is maximum and acceleration is zero.(c)potential energy is minimum and acceleration is maximum.(d)potential energy is minimum and acceleration is zero.

Answer (d). When velocity is maximum displacement is zero so potential energy and acceleration are both zero.

10

A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of

(a)1/4 (b) ½ (c) 2 (d) 4

11

A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of

(a)1/4 (b) 1/2 (c) 2 (d) 4

Answer (a). Since the frequency has increased the mass must have decreased. Frequency is inversely proportional to the square root of mass, so to double frequency the mass mustchange by a factor of 1/4.

12

A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass:

(a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled.

13

A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass:

(a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled.

Answer (d). Acceleration is proportional to frequency squared. If frequency is doubled than acceleration is quadrupled.

14

A particle oscillates on the end of a spring and its position as a function of time is shown below.

At the moment when the mass is at the point P it has(a) positive velocity and positive acceleration(b) positive velocity and negative acceleration(c) negative velocity and negative acceleration(d) negative velocity and positive acceleration

15

A particle oscillates on the end of a spring and its position as a function of time is shown below.

At the moment when the mass is at the point P it has(a) positive velocity and positive acceleration(b) positive velocity and negative acceleration(c) negative velocity and negative acceleration(d) negative velocity and positive acceleration

Answer (b). The slope is positive so velocity is positive. Since the slope is getting smaller with time the acceleration is negative.

Real

Imag

a

b

|z|

Complex numbers

Argand diagram

16

Euler’s relation

17

Consistency argument

If these represent the same thing, then the assumed Euler relationship says:

Equate real parts:

Equate imaginary parts:

18

Real

Imag

t = 0, T0, 2T0

t = T0/4

t = t

PHASOR

19

Adding complex numbers is easy in rectangular form

Real

Imag

ab

d

c

20

Multiplication and division of complex numbers is easy in polar form

Real

Imag

|z|

|w|

21

Real

Imag

ab

Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c., change i -> -i

The product of a complex number and its complex conjugate is REAL.We say “zz* equals mod z squared”

|z|

22

And finally, rationalizing complex numbers, or: what to do when there's an i in the denominator?

23

m = 0.01 kg; k = 36 Nm-1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms-1. Express the motion in

form C

form D

x

m

mk

k

Using complex numbers: initial conditions. Same example as before, but now use the "C" and "D" forms

24

mm)16.1425()16.1425()(11 6060 tsitsi eieitx

mm3.2850Re)(160 tsieitx

25