A cart attached to a spring is displaced from equilibrium and then

released. A graph of displacement as a function of time for the cart is shown below. There is no friction. Eight points are labeled A – H in the graph.

Time, s

Displacement, cm4

2

0

–2

–4

2 4 6 8 10 12 14 16

A B

C

D

G

EF

H

For each question below, choose from the labeled points above or state “none.”

1. At which point or points is the acceleration positive?Explain.

2. At which point or points does the cart have zero velocity but nonzero net force?Explain.

3. At which point or points is the net force on the cart equal to zero?Explain.

4. At which point or points are the acceleration, velocity, and displacement all positive?Explain.

5. At which point or points is the acceleration nonzero and opposite in sign to the position?Explain.

6. At which point or points is the velocity nonzero and opposite in sign to the acceleration?Explain.

+– 0

A mass-spring system consists of a spring with a spring constant (or stiffness) k and unstretched length L, connected to a cart of mass M resting on a horizontal frictionless surface as shown. If the cart is pulled to one side a small distance and released, it will oscillate back and forth with amplitude A and frequency f.

kM

Amplitude A

Identify from choices (a) – (d) how each change described below will affect the frequency of the oscillating mass-spring system.

Compared to the case above, this change will: (a) increase the frequency of the system. (b) decrease the frequency of the system. (c) have no effect on the frequency of the system. (d) have an indeterminate effect on the frequency of the system.

Each of these modifications is the only change to the initial situation described above.

(1) The mass is increased. ______

Explain.

(2) The spring constant or stiffness is increased. ______

Explain.

(3) The mass is pulled a little farther and then released. ______

Explain.

(4) The spring constant is doubled to 2k and the mass is reduced to M/2. ______

Explain.

(5) The amplitude is increased and the mass is increased. ______

Explain.

(6) The mass-spring system is oscillating on the moon (smaller gravitational forces). ______

Explain.

A simple pendulum consists of a mass M attached to a massless string of length L as shown in figure A below. If the mass is pulled to one side a small distance and released, it will swing back and forth with a frequency fo and swing amplitude A. Cases B, C, and D shown below are variations of this system.

L

M

C

Swingamplitude2A

L

M

A

SwingamplitudeA

L

MSwingamplitudeA

2L

2M

B

SwingamplitudeA

D

Rank these cases on the basis of the oscillation frequency of the masses.

Greatest 1 ______ 2 ______ 3 ______ 4 ______ Least

OR, The frequencies are the same for all these cases. ___

OR, We cannot determine the ranking for the frequencies. ___

Please explain your reasoning.

The figures below show six metal pieces attached to identical springs and hung vertically. The metal pieces are pulled down various distances and released. In each figure the kinetic energy of the metal piece as it passes through the equilibrium point and the mass of the piece are given.

A

m= 130 gKE = 12 J

B

m = 140 gKE = 15 J

C

m= 140 gKE = 10 J

D

m = 130 gKE = 15 J

E

m= 150 gKE = 12 J

F

m = 150 gKE = 10 J

Rank these situations on the basis of the amplitude of the oscillation, i.e., the maximum distance the metal pieces move down from the equilibrium point.

Greatest 1 _______ 2 _______ 3 _______ 4 _______ 5 _______ 6 _______ Least

OR, The amplitudes are the same for all these situations. ___

OR, The amplitudes are zero for all these situations. ___

OR, We cannot determine the ranking for these situations. ___

Please explain your reasoning.

A mass attached to a spring is suspended as shown. When the mass is at rest the spring is 12 centimeters long. The unstretched length of the spring is 8 centimeters. The mass is pulled down so that the spring length is 16 centimeters and then released. A graph of the spring length as a function of time is shown. Eight points are labeled A – H in the graph.

Time, s

Spring length, cm8

10

12

14

16

2 4 6 8 10 12

A B

C

D

G

EF

H

For each question below, choose from the labeled points above or state “none.”

1. At which point or points is the spring potential energy at its maximum value?

Explain.

2. At which point or points is the gravitational potential energy at its maximum value?

Explain.

3. At which point or points is the kinetic energy at its maximum value?

Explain.

4. At which point or points is the total energy at its maximum value?

Explain.

5. At which point or points is the spring potential energy zero?

Explain.

6. At which point or points is the kinetic energy positive?

Explain.

7. At which point or points is the kinetic energy at its maximum value and the gravitationalpotential energy at its minimum value?

Explain.

1. The graph below shows a snapshot of a wave that is traveling to the right. Seven points are labeled A through G.

Suppose this represented a (transverse) wave on a string. What is the direction of motion of each of the labeled points in the figure? 2. Sound waves are longitudinal waves. The image below is a snapshot of a sound wave traveling to the right.

Suppose the graph at the top of the page represents the longitudinal displacement from equilibrium of air while a sound wave travels through it, towards the right, on the vertical axis with the equilibrium position of the air along the horizontal axis. Assume that if a labeled point is above the horizontal axis the air at that location has been displaced to the right (the +x direction) and if a labeled point is below the horizontal axis the air at that location has been displaced to the left (the -x direction) then answer the following questions. (a) At which of the labeled points is the density of the air greater than the equilibrium density? (b) At which of the labeled points is the density of the air equal to the equilibrium density? (c) At which of the labeled points is the density of the air less than the equilibrium density?

3. A mechanical wave is described by the equation below, where the displacement of the medium from equilibrium is represented by 𝑦 𝑥, 𝑡 :

𝑦 𝑥, 𝑡 = 1.0  𝑚𝑚  sin    2𝜋2.0

𝑟𝑎𝑑𝑐𝑚  𝑥 −

2𝜋3.0

𝑟𝑎𝑑𝑠  𝑡  

How long would it take this wave to travel a distance of 2.0 cm? (a) 2.0 s (b) 3.0 s (c) 6.0 s (d) 4.0 s (e) 5.0 s 4. The equations for 5 waves are given below. Determine which of the waves could, in principle, describe waves traveling in the same media under identical conditions. L. 2.0  x  10!!  m  sin  (30 !"#

!  𝑥 − 90   !"#

!  𝑡)

M. 5.0  x  10!!  m  sin  (5 !"#

!  𝑥 − 32   !"#

!  𝑡)

N. 2.0  x  10!!  m  sin  (91 !"#!  𝑥 − 273   !"#

!  𝑡)

O. 4.0  x  10!!  m  sin  (34 !"#!  𝑥 − 68   !"#

!  𝑡)

P. 2.0  x  10!!  m  sin  (91 !"#!  𝑥 − 364   !"#

!  𝑡)

(a) O, M (b) N, P (c) L, O (d) L, N (e) O, N

5. An oscillating object creates periodic waves on two strings made of the same material. The tension is the same in both strings. I. If the strings have different thicknesses, which of the following parameters, if any will be different in the two strings? (a) wave frequency (b) wave speed (c) wavelength (d) none of the above II. If the strings have the same thickness but different lengths, which of the following parameters, if any, will be different in the two strings? (a) wave frequency (b) wave speed (c) wavelength (d) none of the above

Shown below are six different pairs of rectangular transverse wave pulses that vary in height (H) andlength (L). Specific values for the heights and lengths are given in each figure for each pulse. In each pairthe pulses are moving toward each other. At some point in time the pulses meet and interact (interfere)with each other.

Left Right Left Right Left RightH = 2 cm H = 4 cm H = 2 cm H = 2 cm H = 2 cm H = 4

cmL = 2 cm L = 2 cm L =4 cm L = 2 cm L =4 cm L = 2 cm

D E F

A B C

Left Right Left Right Left RightH = -2 cm H = 4 cm H = -2 cm H = 2 cm H = -2 cm H = 4

cmL = 4 cm L = 2 cm L = 2 cm L = 2 cm L = 2 cm L = 2 cm

Rank these pairs, from highest to lowest, on the basis of the maximum (peak) height of the combined(resultant) pulse at the center of the combined waves at the time when the centers of the pairs coincide.

Greatest 1 _____ 2 _____ 3 _____ 4 _____ 5 _____ 6 _____ Least

Or, all of these pairs will have the same maximum height._____

Please carefully explain your reasoning.

The figures below show systems of standing waves set up in strings, fixed at both ends, under tension. Allof the strings are identical except for their lengths and are under the same tension. The variables in thesesituations, in addition to the lengths (L) of the strings, are the amplitudes (A) at the antinodes and thenumber of nodes.

Rank these systems, from greatest to least, on the basis of the wavelengths of the waves.

A A = 12 cm L = 25 cm B A = 12 cm L = 28 cm

C A = 18 cm L = 27 cm D A = 16 cm L = 28 cm

E A = 24 cm L = 20 cm F A = 36 cm L = 30 cm

G A = 36 cm L = 50 cm

Greatest 1_______ 2_______ 3______ 4_______ 5_______ 6______ 7 ________ Least

Or, all of these systems have the same wavelength. _________

Please carefully explain your reasoning.

Shown below are six standing wave systems in strings. These systems vary in frequency of oscillation,tension in the strings, and number of nodes. The systems are also set up in various strings. The specificvalues for the string tensions and the frequencies of oscillation are given in each figure. All of the stringshave the same length.

Rank these systems, from greatest to least, on the basis of the speeds of the waves in the strings. That is,put first the system whose waves have the greatest speed in their string and put last the system whosewaves are traveling slowest in their string.

A T = 2 N f = 500 Hz B T = 8 N f = 500 Hz

C T = 5 N f = 300 Hz D T = 6 N f = 300 Hz

E T = 2 N f = 4 00 Hz F T = 5 N f = 200 Hz

Greatest 1________ 2________ 3________ 4________ 5________ 6_______ Least

Or, all of the waves in these systems have the same speed. _______

Please carefully explain your reasoning.

A large 75-kg lighting fixture can be hung from wires of identical size and shape made of aluminum, brass, or copper. The values of Young's modulus for these metals are 0.70 x 1011 Pa (aluminum), 0.91 x 1011 (brass), and 1.1 x 1011 (copper). Which wire would stretch the least distance? (a) copper (b) aluminum (c) brass (d) They will all stretch the same distance. A 25-kg piece of equipment can be hung by steel wires of length 1.00 m, 2.00 m, or 3.00 m. If all the wires have the same diameter, which one will stretch the greatest distance? (a) the 2.00-m wire (b) the 3.00-m wire (c) the 1.00-m wire (d) They will all stretch by the same distance. A 25-kg piece of equipment can be hung by steel wires of length 1.00 m, 2.00 m, or 3.00 m. If all the wires have the same diameter, which one will stretch the greatest percent? (a) the 2.00-m wire (b) the 3.00-m wire (c) the 1.00-m wire (d) They will all stretch by the same percent. Two wires are made out of the same metal, but wire is twice as long as the other wire. Which wire will have the greatest elastic modulus? (a) the longer wire (b) the shorter wire (c) it will be the same for both wires. If a metal wire has its length doubled and its diameter tripled, by what factor does its Young’s modulus change? A metal wire of diameter D stretches by 0.100 mm when supporting a weight W. If the same-length wire is used to support a weight three times as heavy, what would its diameter have to be (in terms of D) so it still stretches only 0.100 mm?

The table lists the compressibilities of 5 fluids. Rank the 5 from greatest bulk modulus to least bulk modulus.

Suppose sound waves of equal wavelength and amplitude are traveling through the liquids listed in the table above. Rank the pressure amplitudes from greatest to least. If the pressure amplitude of a sound wave is halved, by what factor does the does the intensity of the wave decrease? By what factor must the pressure amplitude of a sound wave be increased in order to increase the intensity by a factor of 16?

1. Does the sound intensity level 𝛽 obey the inverse square law? Why? 2. The figure below displays a standing wave pattern in a pipe with one-open and one-closed end. The entire pipe is shown in the figure; represent its length by the variable, L, the horizontal distance between the ends of the pipe.

Determine the ratio of the frequency of the wave shown above to the fundamental frequency of a standing wave in this pipe. (a) 7 (b) 4 (c) 1 (d) 9 (e) 5 3. The figure below shows the displacement of the air in a standing wave in a pipe with both ends open. Which harmonic is shown?

(a) 4th (b) 1st (c) 3rd (d) 5th (e) 2nd 4. The figure displays the displacement of air in a standing wave in a column with one end closed. Determine the wavelength of the standing wave pictured below if the column is 112 cm long.

(a) 56 cm (b) 64 cm (c) 28 cm (d) 14 cm (e) 112 cm

5. Consider a standing wave in a 1.5-meter long column that has one end closed and one end open. Take the closed end to be located at x = 0 and the open end to be located at x = 1.5 m. Suppose the standing wave is the 3rd harmonic. Rank the amplitude of the standing wave, from least to greatest, at the following locations along the column: P: x = 1.0 m, Q: x = 1.25 m, R: x = 0 m, S: x = 0.5 m, T: x = 1.5 m (a) P = R, Q, S = T (b) P = S, T, R = Q (c) S = Q, R, T = P (d) Q = T, P, R = S (e) T = R, S, Q = P 6. An open organ pipe (i.e., a pipe open at both ends) of length L0 has a fundamental frequency f0. Part 1. If the organ pipe is cut in half, what is the new fundamental frequency? (a) 4f0 (b) 2f0 (c) f0 (d) f0/2 (e) f0/4 Part 2. After being cut in half in Part A, the organ pipe is closed off at one end. What is the new fundamental frequency? (a) 4f0 (b) 2f0 (c) f0 (d) f0/2 (e) f0/4 Part 3. The air from the pipe in Part B (i.e., the original pipe after being cut in half and closed off at one end) is replaced with helium. (The speed of sound in helium is about three times faster than in air.). What is the approximate new fundamental frequency? (a) 3f0 (b) 2f0 (c) f0 (d) f0/2 (e) f0/3

At the instant shown, three asteroids are in a line, and the distance between A and B is the same as the distance between B and C. Asteroids B and C have the same mass, while asteroid A has twice the mass.

A Bd

Mass 2M Mass M

C

Mass M

d

Rank the magnitude of the net force on each asteroid due to the other two asteroids.

Greatest1

OR

Allthe same

Allzero

Cannotdetermine

2 3Least

Explain your reasoning.

Two planets have the same mass density, but one has twice the radius of the other. What is the ratio of the acceleration due to gravity on the larger planet to that on the smaller planet? (a) 1/ 4 (b) 1/ 2 (c) 1 (d) 2 (e) 4 Two planets have the same mass density, but one has twice the radius of the other. What is the ratio of the potential energy (relative to infinity) of a mass m on the surface of the larger planet to that on the surface of the smaller planet? (a) 1/ 4 (b) 1/ 2 (c) 1 (d) 2 (e) 4 Let FA and UA be, respectively, the magnitudes of the gravitational force on, and the potential energy of, a mass M due to the two masses m shown in Setup A in the figure below.

Likewise for FB and UB in Setup B. If all of the m’s are the same distance from the M’s, then (a) FA = FB and UA = UB (b) FA = FB and UA < UB (c) FA < FB and UA = UB (d) FA < FB and UA < UB (e) FA > FB and UA < UB

Wood blocks that have different masses and different volumes are floating in water. On top of these blocks are additional masses as shown.

A B C D

M = 10 gV = 40 cm3

M = 20 gV = 80 cm3

M = 10 gV = 40 cm3

M = 30 gV = 80 cm3

20 g 10 g 10 g 20 g

Rank the buoyant force exerted by the water on the wood blocks.

Greatest1

ORAll

the sameAllzero

Cannotdetermine

2 3 4Least

Explain your reasoning.

Two equal-sized cubes that have different masses are held by strings so that they are submerged in water at different depths.

1 kg

3 kg

(a) Is the buoyant force exerted by the water on the 3 kg cube (i) greater than, (ii) less than, or (iii) equal to the buoyant force on the 1 kg cube? _____ Explain your reasoning.

(b) Is the tension in the sting holding the 3 kg cube (i) greater than, (ii) less than, or (iii) equal to the tension in the string holding the 1 kg cube? _____ Explain your reasoning.

(c) Is the pressure exerted on the bottom surface of the 3 kg cube by the water (i) greater than, (ii) less than, or (iii) equal to the pressure on the bottom surface of the 1 kg cube? _____ Explain your reasoning.

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The picture shows a man with mass M = 75 kg and an elephant supported on a hydraulic lift. Thediameter of the piston where the man is standing is 0.5 m and the density of oil is 900 kg/m3. If the elephant is to be raised 2.0 meters how much additional force has to be applied on the piston where the man is standing?

(a) 8.5 x 105 N (b) 6.9 x 102 N (c) 1.1 x 103 N (d) 5.9 x 104 N

The diagram shows a fluid in a pipe with varying widths and heights. We want to compare the pressuresat points 1, 2, 3 and 4. Point 1 is at the highest elevation, points 2 and 3 are at the same elevation and point 4 is at the lowest elevation. The widths of the pipe at points 1 and 2 are the same, the width at 3 is greater than at points 1 and 2 and the width at point 4 is greater than the width at point 3.

How do the pressures at points 2 and 3 compare when the fluid is not flowing, to the situation when the fluid has a constant, non-zero, flow rate?

(a) The pressures at 2 and 3 are equal when the fluid is not flowing and the pressure is higher at 2 when it is flowing.

(b) The pressure at 2 is larger when the fluid is not flowing, but is equal to the pressure at 3 when the fluid is flowing.

(c) The pressures at 2 and 3 are equal when the fluid is not flowing and the pressure is higher at 3 when it is flowing.

(d) The pressure at 3 is larger when the fluid is not flowing, but is equal to the pressure at 2 when the fluid is flowing.