Question 1

Suppose a 0.7-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.5 J.How much further must the spring be compressed to triple the elastic potential energy?
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A) 0.17 m
B) 0.27 m
C) 0.3 m
D) 0.07 m
E) 0.2 m

Accepted Answer

The answer of Suppose a 0.7-kg mass on a spring...

Question 2

By what factor must one change the weight suspended vertically from a spring coil in order to five times its period of simple harmonic motion?
​
A) 1/25
B) 0.2
C) 5.0
D) 25.0
E) ​1.0

Accepted Answer

The period of simple harmonic motion is given by T=2π√(m/k), where m is the mass of the object and k is the spring constant. If we want to increase the period by a factor of 5, we need to solve for the new mass that will give us the new period. Let's call this mass m'.

5T=2π√(m'/k)
(5T/2π)^2=m'/k
m'=k(5T/2π)^2

Now, since we are changing the weight (and keeping the spring constant the same), we need to find the ratio of the new weight to the old weight that will give us this new mass:

m'/m=(k(5T/2π)^2)/(km/1)=(5T/2π)^2

So, we need to find the square root of this ratio to find the factor by which we need to change the weight:

√(m'/m)=√((5T/2π)^2)=25

Therefore, we need to increase the weight by a factor of 25 to increase the period of simple harmonic motion by a factor of 5. Answer: D) 25.0.

Question 3

The motion of a piston in an automobile engine is nearly simple harmonic.If the 1-kg piston travels back and forth over a total distance of 11.9 cm,what is its maximum speed when the engine is running at 3,540 rpm?
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A) 22.06 m/s
B) 11.03 m/s
C) 44.11 m/s
D) 5.51 m/s
E) ​2.76 m/s

Accepted Answer

Given, 
mass of piston, m = 1 kg
total distance travelled by the piston, d = 11.9 cm = 0.119 m (one way)
frequency of motion, f = 3,540 rpm = 59 Hz  (one cycle in 1/59 sec)
We can find the amplitude, A using the total distance travelled as:
d = 2A
Therefore, amplitude, A = d/2 = 0.0595 m

The maximum velocity, Vmax is given by:
Vmax = 2πAf = 2π(0.0595)(59) = 22.06 m/s

Therefore, the best choice is B) 11.03 m/s, which is half of the calculated maximum velocity since the piston has harmonic motion and its velocity changes from maximum to zero at the end points of its motion.

Question 4

For a mass suspended on a spring in the vertical direction,the time for one complete oscillation will depend on:
A) the value for g (the acceleration due to gravity).
B) the distance the mass was originally pulled down.
C) the maximum speed of the oscillating mass.
D) the time doesn't depend on any of the above.

Accepted Answer

The time for one complete oscillation of a mass suspended on a spring (the period) depends on the mass of the object and the spring constant (Hooke's Law), not on the acceleration due to gravity, the initial displacement, or the maximum speed of the oscillation.

Question 5

Which one of the following quantities is at a maximum when an object in simple harmonic motion is at its maximum displacement?
A) speed
B) acceleration
C) kinetic energy
D) frequency

Accepted Answer

When an object in simple harmonic motion is at its maximum displacement, the acceleration is at a maximum as well. This is because the net force on the object is at a maximum (either in the positive or negative direction, depending on the direction of displacement), and acceleration is directly proportional to net force. Therefore, choice B is the correct answer. Choices A, C, and D are not necessarily at a maximum at the point of maximum displacement. The speed and kinetic energy are both zero at the point of maximum displacement, and the frequency is determined by the properties of the system, not the displacement.

Question 6

Suppose a 0.9-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.1 J.What is the spring constant?
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A) 198 N/m
B) 244 N/m
C) 220 N/m
D) 22 N/m
E) ​90 N/m

Accepted Answer

We know that elastic potential energy (PE) and spring constant (k) are related by the formula: PE = 1/2kx^2, where x is the displacement of the spring from its equilibrium position. 
Substituting the given values, we get:
1.1 J = 1/2 k (0.1 m)^2
Solving for k, we get:
k = 2.2 J/m^2
Converting to N/m, we get:
k = 220 N/m 
Therefore, the answer is C) 220 N/m.

Question 7

A 0.34-kg block rests on a frictionless level surface and is attached to a horizontally aligned spring with a spring constant of 55 N/m.The block is initially displaced 4.1 cm from the equilibrium point and then released to set up a simple harmonic motion.What is the speed of the block when it passes through the equilibrium point?
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A) 2.58 m/s
B) 0.74 m/s
C) 1.82 m/s
D) 0.52 m/s
E) ​2.35 m/s

Accepted Answer

The answer of A 0.34-kg block rests on a frictionless...

Question 8

A mass on a spring vibrates in simple harmonic motion at a frequency of 3.9 Hz and an amplitude of 7.8 cm.If a timer is started when its displacement is a maximum (hence x = 7.8 cm when t = 0),what is the displacement of the mass when t = 3.4 s?
​
A) zero
B) -0.005 m
C) 0.079 m
D) -1.239 m
E) ​-0.01 m

Accepted Answer

The displacement in simple harmonic motion is given by x(t) = A * cos(2πft). Substituting A = 0.078 m, f = 3.9 Hz, and t = 3.4 s, we find x(3.4) ≈ -0.005 m.

Question 9

A mass on a spring vibrates in simple harmonic motion at a frequency of 2 Hz and an amplitude of 7 cm.If a timer is started when its displacement is a maximum (hence x = 7 cm when t = 0),what is the speed of the mass when t = 7 s?
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A) zero
B) 87.965 m/s
C) 9.016 m/s
D) 7 m/s
E) ​43.982 m/s

Accepted Answer

The speed of the mass is zero when the displacement is at a maximum, which occurs at t = 7 s since the motion is periodic with a frequency of 2 Hz.

Question 10

A 0.36-kg block rests on a frictionless level surface and is attached to a horizontally aligned spring with a spring constant of 40 N/m.The block is initially displaced 6 cm from the equilibrium point and then released to set up a simple harmonic motion.A frictional force of 0.3 N exists between the block and surface.What is the speed of the block when it passes through the equilibrium point after being released from the 6-cm displacement point?
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A) 0.55 m/s
B) 0.39 m/s
C) 0.77 m/s
D) 0.3 m/s
E) 2.19 m/s

Accepted Answer

Explanation:The mechanical energy of the block is conserved, so:$$1/2kA^2=1/2mv^2+f_kd$$$$v=\sqrt{\frac{kA^2-2f_kd}{m}}=\sqrt{\frac{(40	ext{ N/m})(0.06	ext{ m})^2-2(0.3	ext{ N})(0.06	ext{ m})}{0.36	ext{ kg}}}=0.55	ext{ m/s}$$

Question 11

Suppose a 0.36-kg mass on a spring-loaded gun that has been compressed 0.2 m has elastic potential energy of 1.9 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up?
​
A) 0.54 m
B) 0.88 m
C) 0.34 m
D) 5.08 m
E) 51.52 m

Accepted Answer

The answer of Suppose a 0.36-kg mass on a spring-loaded...

Question 12

The oxygen molecule (O₂)may be regarded as two masses connected by a spring.In vibrational motion,each oxygen atom alternately approaches,then moves away from the center of mass of the system.If each oxygen atom of mass m = 2.67 × 10−²⁶ kg has a vibrational energy of 1.6 × 10−²¹ J and the effective spring constant is 100 N/m,then what is the amplitude of oscillation of each oxygen atom? A) 3.2 × 10−¹¹ m B) 1.6 × 10−¹¹ m C) 1.1 × 10−¹¹ m D) 5.7 × 10−¹² m E) 4 × 10−¹² m

Accepted Answer

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Question 13

A car with bad shocks bounces up and down with a period of 1.62 s after hitting a bump.The car has a mass of 1,600 kg and is supported by four springs of force constant k.What is k for each spring?
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A) 6,017 N/m
B) 1,504 N/m
C) 1,551 N/m
D) 3,009 N/m
E) ​788 N/m

Accepted Answer

The period of oscillation T is related to the mass m and the spring constant k by the formula T = 2π√(m/k). Solving for k gives k = (4π²m)/(T²). Substituting m = 1600 kg and T = 1.62 s, we find the total k for all springs. Since there are four springs, divide the total k by 4 to find k for each spring, which results in approximately 6,017 N/m.

Question 14

A 0.25-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.8 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.3 J.What is the maximum speed of the mass during its oscillation?
​
A) 3.8 m/s
B) 0.2 m/s
C) 5.4 m/s
D) 4.7 m/s
E) ​3.4 m/s

Accepted Answer

We can use the conservation of mechanical energy to solve for the maximum speed of the mass during its oscillation.

At the equilibrium position, the system has a total mechanical energy of:

E = KE + PE
E = 1.4 J + 2.3 J
E = 3.7 J

At the maximum displacement, all the energy is in the form of kinetic energy, so:

E = KE_max
KE_max = 3.7 J

We can now solve for the speed using the formula for kinetic energy:

KE = 0.5mv^2

Plugging in the values, we get:

3.7 J = 0.5(0.25 kg)v^2
v^2 = 29.6
v = 5.4 m/s

Therefore, the maximum speed of the mass during its oscillation is 5.4 m/s, which is answer choice C.

Question 15

An object is attached to a spring and its frequency of oscillation is measured.Then another object is connected to the first object,and the resulting mass is three times the original value.By what factor is the frequency of oscillation changed?
​
A) 1/3
B) 0.58
C) 1/9
D) 3
E) ​9

Accepted Answer

The answer of An object is attached to a spring...

Question 16

An ore car of mass 4,700 kg rolls downhill on tracks from a mine.At the end of the tracks,11.1 m lower in elevation,is a spring with k = 410,000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction.
​
A) 1.12 m
B) 0.79 m
C) 1.58 m
D) 0.5 m
E) ​0.47 m

Accepted Answer

The potential energy of the ore car is converted to spring potential energy when the car reaches the spring. So, we can use the conservation of energy principle to solve this problem. 
Initial potential energy = mgh1 
Final potential energy = 0 (at the spring's equilibrium position) and spring potential energy = 1/2 kx^2
Equating both: 
mgh1 = 1/2 kx^2 
x^2 = 2mgh1/k
x = sqrt(2mgh1/k) = sqrt(2*4700*9.81*11.1/410000) = 1.58 m 
Therefore, the spring is compressed by 1.58 meters in stopping the ore car. Hence, the answer is C.

Question 17

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.2 Hz and an amplitude of 4.2 cm.If a timer is started when its displacement is a maximum (hence x = 4.2 cm when t = 0),what is the acceleration magnitude when t = 4 s? A) zero B) 0.3 m/s² C) 18.1 m/s² D) 9 m/s² E) 11.8 m/s²

Accepted Answer

The answer of A mass on a spring vibrates in...

Question 18

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.4 Hz and an amplitude of 8.6 cm.If the mass of the object is 0.4 kg,what is the spring constant?
​
A) 11 N/m
B) 764 N/m
C) 306 N/m
D) 36 N/m
E) ​657 N/m

Accepted Answer

The spring constant k can be found using the formula for the frequency of simple harmonic motion: f = (1/2π)√(k/m). Rearranging for k gives k = (2πf)² * m. Substituting f = 4.4 Hz and m = 0.4 kg, we get k ≈ 306 N/m.

Question 19

An object moving in simple harmonic motion has an amplitude of 0.05 m and a maximum acceleration of 40 m/s².What is the frequency of the system? A) 14.1 Hz B) 177.7 Hz C) 4.5 Hz D) 11.3 Hz E) 2 Hz

Accepted Answer

Maximum acceleration in simple harmonic motion is given by a = ω^2A, where ω is the angular frequency and A is the amplitude.

Rearranging the equation, we get:

ω = √(a/A)

From the given data, maximum acceleration (a) = 40 m/s^2 and amplitude (A) = 0.05 m.

Substituting these values, we get:

ω = √(40/0.05) = 89.44 rad/s

The frequency (f) of the system is given by:

f = ω/2π

Substituting the value of ω, we get:

f = 89.44/2π ≈ 14.1 Hz

Therefore, the answer is C) 4.5 Hz.

Question 20

I attach a 2.9-kg block to a spring that obeys Hooke's Law and supply 19 J of energy to stretch the spring.I release the block;it oscillates with period 0.35 s.The amplitude is:
​
A) 14 cm.
B) 20 cm.
C) 10 cm.
D) 85 cm.
E) ​29 cm.

Accepted Answer

The answer of I attach a 2.9-kg block to a...

Quiz 13: Vibrations and Waves

Suppose a 0.7-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.5 J.How much further must the spring be compressed to triple the elastic potential energy?

By what factor must one change the weight suspended vertically from a spring coil in order to five times its period of simple harmonic motion?

The motion of a piston in an automobile engine is nearly simple harmonic.If the 1-kg piston travels back and forth over a total distance of 11.9 cm,what is its maximum speed when the engine is running at 3,540 rpm?

For a mass suspended on a spring in the vertical direction,the time for one complete oscillation will depend on:

Which one of the following quantities is at a maximum when an object in simple harmonic motion is at its maximum displacement?

Suppose a 0.9-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.1 J.What is the spring constant?

A mass on a spring vibrates in simple harmonic motion at a frequency of 3.9 Hz and an amplitude of 7.8 cm.If a timer is started when its displacement is a maximum (hence x = 7.8 cm when t = 0) ,what is the displacement of the mass when t = 3.4 s?

A mass on a spring vibrates in simple harmonic motion at a frequency of 2 Hz and an amplitude of 7 cm.If a timer is started when its displacement is a maximum (hence x = 7 cm when t = 0) ,what is the speed of the mass when t = 7 s?

Suppose a 0.36-kg mass on a spring-loaded gun that has been compressed 0.2 m has elastic potential energy of 1.9 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up?

A car with bad shocks bounces up and down with a period of 1.62 s after hitting a bump.The car has a mass of 1,600 kg and is supported by four springs of force constant k.What is k for each spring?

A 0.25-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.8 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.3 J.What is the maximum speed of the mass during its oscillation?

An object is attached to a spring and its frequency of oscillation is measured.Then another object is connected to the first object,and the resulting mass is three times the original value.By what factor is the frequency of oscillation changed?

An ore car of mass 4,700 kg rolls downhill on tracks from a mine.At the end of the tracks,11.1 m lower in elevation,is a spring with k = 410,000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction.

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.2 Hz and an amplitude of 4.2 cm.If a timer is started when its displacement is a maximum (hence x = 4.2 cm when t = 0) ,what is the acceleration magnitude when t = 4 s?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.4 Hz and an amplitude of 8.6 cm.If the mass of the object is 0.4 kg,what is the spring constant?

An object moving in simple harmonic motion has an amplitude of 0.05 m and a maximum acceleration of 40 m/s².What is the frequency of the system?

I attach a 2.9-kg block to a spring that obeys Hooke's Law and supply 19 J of energy to stretch the spring.I release the block;it oscillates with period 0.35 s.The amplitude is:

Quiz 13: Vibrations and Waves

Suppose a 0.7-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.5 J.How much further must the spring be compressed to triple the elastic potential energy? ​

By what factor must one change the weight suspended vertically from a spring coil in order to five times its period of simple harmonic motion? ​

The motion of a piston in an automobile engine is nearly simple harmonic.If the 1-kg piston travels back and forth over a total distance of 11.9 cm,what is its maximum speed when the engine is running at 3,540 rpm? ​

For a mass suspended on a spring in the vertical direction,the time for one complete oscillation will depend on:

Which one of the following quantities is at a maximum when an object in simple harmonic motion is at its maximum displacement?

Suppose a 0.9-kg mass on a spring that has been compressed 0.1 m has elastic potential energy of 1.1 J.What is the spring constant? ​

A mass on a spring vibrates in simple harmonic motion at a frequency of 3.9 Hz and an amplitude of 7.8 cm.If a timer is started when its displacement is a maximum (hence x = 7.8 cm when t = 0) ,what is the displacement of the mass when t = 3.4 s? ​

A mass on a spring vibrates in simple harmonic motion at a frequency of 2 Hz and an amplitude of 7 cm.If a timer is started when its displacement is a maximum (hence x = 7 cm when t = 0) ,what is the speed of the mass when t = 7 s? ​

Suppose a 0.36-kg mass on a spring-loaded gun that has been compressed 0.2 m has elastic potential energy of 1.9 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up? ​

A car with bad shocks bounces up and down with a period of 1.62 s after hitting a bump.The car has a mass of 1,600 kg and is supported by four springs of force constant k.What is k for each spring? ​

A 0.25-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.8 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.3 J.What is the maximum speed of the mass during its oscillation? ​

An object is attached to a spring and its frequency of oscillation is measured.Then another object is connected to the first object,and the resulting mass is three times the original value.By what factor is the frequency of oscillation changed? ​

An ore car of mass 4,700 kg rolls downhill on tracks from a mine.At the end of the tracks,11.1 m lower in elevation,is a spring with k = 410,000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction. ​

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.2 Hz and an amplitude of 4.2 cm.If a timer is started when its displacement is a maximum (hence x = 4.2 cm when t = 0) ,what is the acceleration magnitude when t = 4 s? ​

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.4 Hz and an amplitude of 8.6 cm.If the mass of the object is 0.4 kg,what is the spring constant? ​

An object moving in simple harmonic motion has an amplitude of 0.05 m and a maximum acceleration of 40 m/s2.What is the frequency of the system? ​

I attach a 2.9-kg block to a spring that obeys Hooke's Law and supply 19 J of energy to stretch the spring.I release the block;it oscillates with period 0.35 s.The amplitude is: ​