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Class PHYSICS110SPRING2007

Chapter 14

Assignment is due at 11:00am on Monday, May 28, 2007

Credit for problems submitted late will decrease to 0% after the deadline has passed.There is no penalty for wrong answers to free response questions. Multiple choice questions are penalized as described in the online help.The unopened hint bonus is 2% per part.You are allowed 4 attempts per answer.

 

Good Vibes: Introduction to Oscillations

Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion.

Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth.

The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force

must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the

restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them.

Consider a block of mass attached to a spring with force constant , as shown in the figure

. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is

relaxed, the block is located at . If the block is pulled to the right a distance and then released,

will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from , it will

ANSWER:

remain at rest.

move to the left until it reaches equilibrium and stop there.

move to the left until it reaches and stop there.

move to the left until it reaches and then begin to move to the right.

As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and

the block will slow down, temporarily coming to rest at .

After is reached, the block will begin its motion to the right, pushed by the spring. The

block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we've assumed, there is no friction, the motion will repeat indefinitely.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds.

The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI

units, is measured in inverse seconds, or hertz ( ).

Part B

If the period is doubled, the frequency is

ANSWER:

unchanged.

doubled.

halved.

Part C

An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its

frequency ?Express your answer in hertz.

ANSWER:

   =

 10   

Part D

If the frequency is 40 , what is the period ?Express your answer in seconds.

ANSWER:

   =

 0.025   

The following questions refer to the figure that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x axis are located a distance from the equilibrium position?

ANSWER:

R only

Q only

both R and Q

Part F

Suppose that the period is . Which of the following points on the t axis are separated by the time

interval ?

ANSWER:

K and L

K and M

K and P

L and N

M and P

Now assume that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 .Part G

What is the period ?Hint G.1

How to approach the problem

Hint not displayed

Express your answer in seconds.

ANSWER:

   =

 0.02   

Part H

How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?Express your answer in seconds.

ANSWER:

   =

 0.01   

Part I

What distance does the object cover during one period of oscillation?Express your answer in meters.

ANSWER:

   =

 0.48   

Part J

What distance does the object cover between the moments labeled K and N on the graph?Express your answer in meters.

ANSWER:

   =

 0.36   

 

Position, Velocity, and Acceleration of an Oscillator

Learning Goal: To learn to find kinematic variables from a graph of position vs. time.The graph of the position of an oscillating object as a function of time is shown.

Some of the questions ask you to determine ranges on the graph over which a statement is true. When answering these questions, choose the most complete answer. For example, if the answer "B to D" were correct, then "B to C" would technically also be correct--but you will only recieve credit for choosing the most complete answer.

Part A

Where on the graph is ?

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to E

Part B

Where on the graph is ?

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to

E

Part C

Where on the graph is ?

ANSWER:

A only

C only

E only

A and C

A and C and E

B and D

Part D

Where on the graph is the velocity ?Hint D.1 Finding instantaneous velocity

Hint not displayed

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to E

Part E

Where on the graph is the velocity ?

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to E

Part F

Where on the graph is the velocity ?Hint F.1

How to tell if

Hint not displayed

ANSWER:

A only

B only

C only

D only

E only

A and C

A and C and E

B and D

Part G

Where on the graph is the acceleration ?Hint G.1 Finding acceleration

Hint not displayed

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to E

Part H

Where on the graph is the acceleration ?

ANSWER:

A to B

A to C

C to D

C to E

B to D

A to B and D to E

Part I

Where on the graph is the acceleration ?Hint I.1

How to tell if

Hint not displayed

ANSWER:

A only

B only

C only

D only

E only

A and C

A and C and E

B and D

 

Mass Hitting a Spring

A block sliding with velocity along a frictionless floor hits a spring at time (configuration 1). The spring compresses until the block comes to a momentary stop (configuration 2).

Finally, the spring expands, pushing the block back in the direction from which it came.

In this problem you will be shown a series of plots related to the motion of the block and spring, and you will be asked to identify what the plots represent. In each plot, the point labeled "1" refers to configuration 1 (when the block first comes in contact with the spring). The point labeled "2" refers to configuration 2 (when the block comes to rest with the spring compressed).

In the questions that follow, "force" refers to the x component of the force that the spring exerts on the block and "position" and "velocity" refer to the x components of the position and velocity of the block.

For all graphs, treat the origin as ; that is, the x axis represents and the y axis represents

.

Part A

Consider graph A.

What might this graph represent?

Part Specify the initial condition

A.1

Part not displayed

Hint A.2

Determine what the slope means

Hint not displayed

ANSWER:

position vs. time

velocity vs. time

force vs. time

force vs. position

Part B

Consider graph B.

What might this graph represent?

Hint B.1 Starting point and slope

Hint not displayed

ANSWER:

position vs. time

velocity vs. time

force vs. time

force vs.

position

Part C

Consider graph C.

What might this graph represent?

Part C.1 Specify the initial and final conditions

Part not displayed

ANSWER:

position vs. time

velocity vs. time

force vs. time

force vs. position

 

Cosine Wave

The graph shows the position of an oscillating object as a function of time .

The equation of the graph is

,

where is the amplitude, is the angular frequency, and is a phase constant. The quantities , ,

and are measurements to be used in your answers. Part A

What is in the equation?Hint A.1

Maximum of

Hint not displayed

ANSWER:

Part B

What is in the equation?Hint B.1 Period

Hint not displayed

ANSWER:

Part C

What is in the equation?

Hint C.1

Using the graph and trigonometry

Hint not displayed

Hint C.2

Using the graph and Part B

Hint not displayed

ANSWER:

 

Vertical Mass-and-Spring Oscillator

A block of mass is attached to the end of an ideal spring. Due to the weight of the block, the block

remains at rest when the spring is stretched a distance from its equilibrium length.

The spring has an unknown spring

constant . Part A

What is the spring constant ?Part A.1

Sum of forces acting on the block

Part not displayed

Express the spring constant in terms of given quantities and , the magnitude of the acceleration due to gravity.

ANSWE        

R: =

Part B

Suppose that the block gets bumped and undergoes a small vertical displacement. Find the resulting angular frequency of the block's oscillation about its equilibrium position.Hint B.1

Formula for angular frequency

Hint not displayed

Express the frequency in terms of given quantities and , the magnitude of the acceleration due to gravity.

ANSWER:

   =    

It may seem that this result for the frequency does not depend on either the mass of the block or the spring constant, which might make little sense. However, these parameters are what would determine

the extension of the spring when the block is hanging: .

One way of thinking about this problem is to consider both and as unknowns. By measuring and (both fairly simple measurements), and knowing the mass, you can determine the value of the

spring constant and the acceleration due to gravity experimentally.

 

Analyzing Simple Harmonic Motion

This applet shows two masses on springs, each accompanied by a graph of its position versus time. Part A

What is an expression for , the position of mass I as a function of time? Assume that position is measured in meters and time is measured in seconds.Part A.1

How to approach the problem

Part not displayed

Part A.2

Find the amplitude

Part not displayed

Part A.3

Find the angular frequency

Part not displayed

Express your answer as a function of . Express numerical constants to three significant figures.

ANSWER:

   =

   

Part B

What is , the position of mass II as a function of time? Assume that position is measured in meters and time is measured in seconds.Part B.1

How to approach the problem

Part not displayed

Part B.2

Find the amplitude

Part not displayed

Part B.3

Find the angular frequency

Part not displayed

Express your answer as a function of . Express numerical constants to three significant figures.

ANSWER:

   =

   

 

Problem 14.14

The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A

The amplitude.

ANSWER:

 2.00 

 cm

Part B

The period.

ANSWER:

 0.628 

 s

Part C

The spring constant.

ANSWER:

 5.00 

 N/m

Part D

The phase constant.

ANSWER:

 -0.785 

 rad

Part E

The initial coordinate of the mass.

ANSWER:

 1.41 

 cm

Part F

The initial velocity.

ANSWER:

 14.1 

 cm/s

Part G

The maximum speed.

ANSWER:

 20.0 

 cm/s

Part H

The total energy.

ANSWER:

 1.0 

 mJ

Part I

The velocity at .

ANSWER:

 1.46 

 cm/s

 

Problem 14.22

A mass on a string of unknown length oscillates as a pendulum with a period of 4.0 s. Parts a to d are independent questions, each referring to the initial situation. What is the period if Part A

The mass is doubled?

ANSWER:

 4.00 

 s

Part B

The string length is doubled?

ANSWER:

 5.66 

 s

Part C

The string length is halved?

ANSWER:

 2.83 

 s

Part D

The amplitude is doubled?

ANSWER:

 4.00 

 s

 

Problem 14.58

Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. Part A

What is the Martian acceleration due to gravity?

ANSWE 3.67   

R:

 

Problem 14.19

A spring with spring constant 13.1  hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 7.50  and released. The ball makes 20.0  oscillations in 19.0  seconds. Part A

What is its the mass of the ball?

ANSWER:

 299 

 g

Part B

What is its maximum speed?

ANSWER:

 49.6 

 cm/s

  Problem 14.40

Part A

When the displacement of a mass on a spring is the half of the amplitude, what fraction of the energy is kinetic energy?

ANSWER:

 75.0 

 %

Part B

At what displacement, as a fraction of , is the energy half kinetic and half potential?

ANSWER:

 0.707 

 

Problem 14.63

A 0.880  block is attached to a horizontal spring with spring constant 1800  . The block is at rest on a frictionless surface. A 11.7  bullet is fired into the block, in the face opposite the

spring, and sticks. Part A

What was the bullet's speed if the subsequent oscillations have an amplitude of 7.90  ?

ANSWER:

 271 

 m/s

 

Problem 14.34

The figure is the velocity-versus-time graph of a particle in simple harmonic motion. Part A

What is the amplitude of the oscillation?

ANSWER:

 115 

 cm

Part B

What is the phase constant?

ANSWER:

 2.62 

 rad

Part C

What is the position at

ANSWER:

 -99.3 

 cm

Summary 13 of 13 problems complete (101.26% avg. score)96.33 of 95 points

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