Oscillatory motion (non-constant acceleration)

6.3.1. An elevator supported by a single cable descends a shaft at a constant speed. The only forces acting on the elevator are the tension in the cable and the gravitational force. Which one of the following statements is true?

a) The work done by the tension force is zero joules.

b) The net work done by the two forces is zero joules.

c) The work done by the gravitational force is zero joules.

d) The magnitude of the work done by the gravitational force is larger than that done by the tension force.

e) The magnitude of the work done by the tension force is larger than that done by the gravitational force.

6.3.5. Consider the box in the drawing. We can slide the box up the frictionless incline from point A and to point C or we can slide it along the frictionless horizontal surface from point A to point B and then lift it to point C. How does the work done on the box along path A-C,WAC, compare to the work done on the box along the two step path A-B-C, WABC?

a) WABC is much greater than WAC.

b) WABC is slightly greater than WAC.

c) WABC is much less than WAC.

d) WABC is slight less than WAC.

e) The work done in both cases is the same.

6.5.1. Two balls of equal size are dropped from the same height from the roof of a building. One ball has twice the mass of the other. When the balls reach the ground, how do the kinetic energies of the two balls compare?

a) The lighter one has one fourth as much kinetic energy as the other does.

b) The lighter one has one half as much kinetic energy as the other does.

c) The lighter one has the same kinetic energy as the other does.

d) The lighter one has twice as much kinetic energy as the other does.

e) The lighter one has four times as much kinetic energy as the other does.

5.2.1. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two?

a) Keep the radius fixed and increase the period by a factor of two.

b) Keep the radius fixed and decrease the period by a factor of two.

c) Keep the speed fixed and increase the radius by a factor of two.

d) Keep the speed fixed and decrease the radius by a factor of two.

e) Keep the radius fixed and increase the speed by a factor of two.

5.3.3. A ball is attached to a string and whirled in a horizontal circle. The ball is moving in uniform circular motion when the string separates from the ball (the knot wasn’t very tight). Which one of the following statements best describes the subsequent motion of the ball?

a) The ball immediately flies in the direction radially outward from the center of the circular path the ball had been following.

b) The ball continues to follow the circular path for a short time, but then it gradually falls away.

c) The ball gradually curves away from the circular path it had been following.

d) The ball immediately follows a linear path away from, but not tangent to the circular path it had been following.

e) The ball immediately follows a line that is tangent to the circular path the ball had been following

5.3.5. Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground (a horizontal circle). What is the force, the physical force, holding the bucket in a circular path?

a) the centripetal force

b) the centrifugal force

c) your hand on the handle

d) gravitational force

e) None of the above are correct.

Oscillatory motion (non-constant acceleration)

Simple Harmonic Motion

10.1 The Ideal Spring and Simple Harmonic Motion

xkF Appliedx

spring constant

Units: N/m


HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING

The restoring force on an ideal spring is xkFx

xmakxF

10.2 Simple Harmonic Motion and the Reference Circle

period T: the time required to complete one cycle

frequency f: the number of cycles per second (measured in Hz)

Tf

1

Tf

22

amplitude A: the maximum displacement

€

x = Acosωt

file:///Users/silvinagatica/Desktop/simulations/applets/sim08.htm

m

k


ACCELERATION

€

ax = −Aω2 cosωt

VELOCITY

€

vx = −Aω sinωt

€

x = AcosωtDISPLACEMENT

€

max = −mAω2 sinωt = −mω2Asinωt = −kx

€

k

m

€

mω2 = k


FREQUENCY OF VIBRATION

m

k

€

m1 < m2

ω1 >ω2

T1 < T2€

T =2π

ωf =1/T


Example 6 A Body Mass Measurement Device

The device consists of a spring-mounted chair in which the astronautsits. The spring has a spring constant of 606 N/m and the mass ofthe chair is 12.0 kg. The measured period is 2.41 s. Find the mass of theastronaut.


€

=k

mtotal

2total km

Tf

22

astrochair2total2

mmT

km

kg 77.2kg 0.124

s 41.2mN606

2

2

2

chair2astro

m

T

km

10.3 Energy and Simple Harmonic Motion

2212

21

elastic fo kxkxW

F

x

F=-kx

x0 xf

-kxf

-kx0

Work=Area =

€

12 x f × (−kx f ) − 1

2 x0 × (−kx0)


DEFINITION OF ELASTIC POTENTIAL ENERGY

The elastic potential energy is the energy that a springhas by virtue of being stretched or compressed. For anideal spring, the elastic potential energy is

221

elasticPE kx

SI Unit of Elastic Potential Energy: joule (J)


of EE

€

12mv f

2 + mgh f + 12 ky f

2 = 12mvo

2 + mgho + 12 kyo

2

oo mghkh 221

m 14.0

mN28

sm8.9kg 20.02

2

2

k

mgho

simulation


m

k

€

4m

ω /2

2T

€

T =2π

ωf =1/T

simulation horizontal oscilator

SUMMARY:

€

x = Acosωt

Period T: time when ωt = 2 π

Frequency f: # cycles per second

ω: angular frequency (how fast it ocillates)

€

4k

2ω

T /2

10.4 The Pendulum

A simple pendulum consists of a particle attached to a frictionlesspivot by a cable of negligible mass.

simulation

Does T depend on the mass?

10.4 The Pendulum

A simple pendulum consists of a particle attached to a frictionlesspivot by a cable of negligible mass.

only) angles (small L

g

€

2π f =2π

T=

g

L

€

T = 2πL

g

How much would you change L to double T?

simulation

10.4 The Pendulum

Example 10 Keeping Time

Determine the length of a simple pendulum that willswing back and forth in simple harmonic motion with a period of 1.00 s.

2

2L

g

Tf

m 248.0

4

sm80.9s 00.1

4 2

22

2

2

gTL

2

2

4gT

L


Example 1 A Tire Pressure Gauge

The spring constant of the springis 320 N/m and the bar indicatorextends 2.0 cm. What force does theair in the tire apply to the spring?


N 4.6m 020.0mN320 xkF Applied

x

10.5 Damped Harmonic Motion

In simple harmonic motion, an object oscillated with a constant amplitude.

In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes.

This is referred to as damped harmonic motion.

10.5 Damped Harmonic Motion

1) simple harmonic motion

2&3) underdamped

4) critically damped

5) overdamped

10.6 Driven Harmonic Motion and Resonance

When a force is applied to an oscillating system at all times,the result is driven harmonic motion.

Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity.

10.6 Driven Harmonic Motion and Resonance

RESONANCE

Resonance is the condition in which a time-dependent force can transmitlarge amounts of energy to an oscillating object, leading to a large amplitudemotion.

Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.

Oscillatory motion (non-constant acceleration)

Documents

Transcript of Oscillatory motion (non-constant acceleration)