RECALL:Angular Displacement & Angular Velocity

Think about spinning a ball on a string or a ball on a turn-table in terms of SHM and UCM:

If you look at the ball from the side, its motion could be plotted as a sine wave.

is the angle between position 1 and 2.

is the amount of rotation—measured in degrees or radians (we’ll use radians since they are the SI unit for ).

1

2

r

For an object rotating about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise.

Angular Displacement,

For an object rotating, angular displacement is found by:

(in radians) = Arc length = s

= f – i where i is sometimes zero.

radius r

If determining for a complete circle, or one complete cycle:

Arc length = 2r (for one revolution)

= (arc length) / r

= (2r) / r = 2 radians

1 revolution = 2 radians = 360o

So, 1 radian = 360o / 2 = 57.3o

Once around the circular path is 360-degrees or 2-radians!

Angular Velocity Angular Velocity = the “turning rate”—expressed in

units of revolutions per second, or metric: radians per second.

Angular velocity = angular displacement / elapsed time

= /t units: radians/second or rad/s For one cycle: t = T = period = time for one cycle And, = (2r)/r = 2 radians = angular

displacement for one cycle = /t = 2 radians / T = 2 / T

= 2 / T And since T=1/f , where f = frequency = 2 / (1/f) = 2f

= 2f Angular velocity is sometimes called angular frequency!

A person bounces up and down on a trampoline, while always staying in contact with it.

The motion is simple harmonic motion, and it takes 1.90 seconds to complete one cycle.

The height of each bounce above the equilibrium position is 45.0 cm.

Determine the amplitude, and

determine the angular frequency of the motion.

The solution… The amplitude of simple harmonic motion

is the distance from the equilibrium position to the point of maximum height.

Since the distance from the equilibrium position to the point of maximum height is the amplitude of the motion…

A = 45.0 cm = 0.45 m

The angular frequency is found with = 2 / T

= 2 / T = 2 / 1.90 seconds

= 3.31 rad/second

Check this out!

www.acoustics.salford.ac.uk/feschools/waves/shm.php

The frequency of oscillation depends on physical properties of the oscillator; it does notdepend on the amplitude of the oscillation.

= 2f = (k/m)for a spring w/mass m on itwhere is in radians/second

Rearranged:

The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

= 2f = (k/m)for a spring w/mass m on itwhere is in radians/second

Rearranged:

The diaphragm of a loudspeaker moves back and forth in simple harmonic motion to create sound.

A) Where in the motion does this maximum speed occur?

B) Where in the motion does this maximum acceleration occur?

A) The speed of the diaphragm is momentarily zero at the two ends of its motion, x = +A and x = -A… its maximum speed occurs midway between these two points, or at x=0m.

B) The maximum acceleration occurs when the force acting on the diaphragm is a maximum. The maximum force acts when the diaphragm is at the ends of the path, where the displacement is the greatest… the maximum acceleration is at x=+A and x=-A.

Solution:

Auto Suspensions

The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports 320 kg. A person pushes down on the middle of the front end of the car and notices that it vibrates through five cycles in 3.0 seconds. Find the spring constant of either spring.

The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports 320 kg. A person pushes down on the middle of the front end of the car and notices that it vibrates through five cycles in 3.0 seconds. Find the spring constant of either spring.

=2f = (k/m)

2f = (k/m)

(2f)2 = k/m

m42 f2 = k

(320 kg)(42)(5cycles/3sec)2 = k

k = 35091.9 N/m

1. At which of the above times is the displacement zero? (Where is the spring in equilibrium?)

2. At which of the above times is the velocity zero?3. At which of the above times is the acceleration zero?4. At which of the above times is the kinetic energy a maximum?5. At which of the above times is the potential energy a maximum?6. At which of the above times is kinetic energy being transformed to

potential energy?7. At which of the above times is potential energy being transformed to

kinetic energy?

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows:

1. At which of the above times is the displacement zero? A, E, I2. At which of the above times is the velocity zero? C, G3. At which of the above times is the acceleration zero? A, E, I (Fnet = 0)4. At which of the above times is the kinetic energy a maximum? A, E, I5. At which of the above times is the potential energy a maximum? C, G6. At which of the above times is kinetic energy being transformed to

potential energy? B, F7. At which of the above times is potential energy being transformed to

kinetic energy? D, H

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows:

1. At which of the above times is the displacement zero?2. At which of the above times is the velocity zero?3. At which of the above times is the acceleration zero?4. At which of the above times is the kinetic energy a maximum?5. At which of the above times is the potential energy a maximum?6. At which of the above times is kinetic energy being transformed

to potential energy?7. At which of the above times is potential energy being

transformed to kinetic energy?

A pendulum is pulled to the side and released. Its subsequent motion appears as follows:

1. At which of the above times is the displacement zero? C, G2. At which of the above times is the velocity zero? A, E, I3. At which of the above times is the acceleration zero? C, G4. At which of the above times is the kinetic energy a maximum? C,

G5. At which of the above times is the potential energy a maximum?

A, E, I6. At which of the above times is kinetic energy being transformed to

potential energy? D, H7. At which of the above times is potential energy being transformed

to kinetic energy? B, F

A pendulum is pulled to the side and released. Its subsequent motion appears as follows:

Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram:

Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

A. a b c dB. d c b aC. a b c d

Additional Question:

Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram:

Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

A. a b c dB. d c b aC. a b c d

Answer:

Slide 14-35

Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram:

Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

A. a b c dB. d c b aC. a b c d

Additional Question:

Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram:

Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

A. a b c dB. d c b aC. a b c d

ANSWER:

Slide 14-36

Stiffer springs have greater k (spring constant) values

A pendulum is pulled to the side and released. Rank the following positions in terms of the speed, from highest to lowest. There may be ties.

Additional Question:

A pendulum is pulled to the side and released. Rank the following positions in terms of the speed, from highest to lowest. There may be ties.

Answer:

Fastest: C & G, B & D & F & H, A & E & I

ANSWER: