Question 1

I attach a 2.0-kg block to a spring that obeys Hooke's Law and supply 16 J of energy to stretch the spring.I release the block; it oscillates with period 0.30 s.The amplitude is:
A)38 cm.
B)19 cm.
C)9.5 cm.
D)4.3 cm.

Accepted Answer

We can use the formula for the period of a simple harmonic oscillator:
$T = 2\pi\sqrt{\frac{m}{k}}$
where m is the mass of the block, k is the spring constant, and T is the period of oscillation.

First, we need to find the spring constant k:
$U = \frac{1}{2}kx^2$
where U is the energy supplied to stretch the spring, and x is the displacement of the block from its equilibrium position. Rearranging, we get:
$k = \frac{2U}{x^2}$

Substituting the given values, we get:
$k = \frac{2(16	ext{ J})}{x^2} = \frac{32	ext{ J}}{x^2}$

Next, we can use the formula for the period T, and solve for the amplitude A:
$T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{m}{\frac{32	ext{ J}}{x^2}}} = 2\pi\sqrt{\frac{mx^2}{32	ext{ J}}}$
$0.30	ext{ s} = 2\pi\sqrt{\frac{(2	ext{ kg})(x^2)}{32	ext{ J}}}$
Solving for x, we get:
$x \approx 0.19	ext{ m} = 19	ext{ cm}$

Therefore, the amplitude is 19 cm, so the correct answer is B.

Question 2

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 four times the original value.By what factor is the frequency of oscillation changed?
A)1/4
B)1/2
C)1/16
D)4

Accepted Answer

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

Question 3

A car with bad shocks bounces up and down with a period of 1.50 s after hitting a bump.The car has a mass of 1 500 kg and is supported by four springs of force constant k.What is k for each spring?
A)6 580 N/m
B)5 850 N/m
C)4 440 N/m
D)3 630 N/m

Accepted Answer

The answer of A car with bad shocks bounces up...

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

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If the mass of the object is 0.20 kg, what is the spring constant?
A)40 N/m
B)87 N/m
C)126 N/m
D)160 N/m

Accepted Answer

The frequency of a mass on a spring in simple harmonic motion is given by:

$f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$

where f is the frequency in hertz, k is the spring constant in newtons per meter, and m is the mass in kilograms.

We are given that f = 4.0 Hz, a = 8.0 cm = 0.08 m, and m = 0.20 kg.

We can use the formula for the period of a mass on a spring in simple harmonic motion to find the value of k:

$T=\frac{2\pi}{\sqrt{k/m}}$

$4.0	ext{ Hz}=\frac{1}{T}=\frac{2\pi}{\sqrt{k/m}}$

Solving for k, we get:

$k=\frac{4\pi^2m}{T^2}$

$k=\frac{4\pi^2(0.20	ext{ kg})}{(1/4)^2\pi^2}=125.66\approx126	ext{ N/m}$

Therefore, the spring constant is 126 N/m (Choice C).

Question 6

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0), what is the acceleration magnitude when t = 3 s? A)zero B)8.13 m/s² C)14.3 m/s² D)25.3 m/s²

Accepted Answer

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

Question 7

An ore car of mass 4 000 kg rolls downhill on tracks from a mine.At the end of the tracks, 10.0 m lower in elevation, is a spring with k = 400 000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction.
A)0.14 m
B)0.56 m
C)1.40 m
D)1.96 m

Accepted Answer

The potential energy of the ore car at the top of the hill is converted to the potential energy of the compressed spring at the bottom of the hill.

Using conservation of energy, we can calculate the initial potential energy of the ore car:

PE_initial = mgh
PE_initial = (4000 kg)(9.81 m/s^2)(h)
PE_initial = 392040 J

where h is the height of the hill.

At the bottom of the hill, all of this potential energy is stored in the compressed spring:

PE_final = (1/2)kx^2

where x is the compression of the spring.

Setting PE_initial equal to PE_final, we can solve for x:

392040 J = (1/2)(400000 N/m)(x^2)
x^2 = 1.96
x = 1.4 m

Therefore, the spring is compressed by 1.4 m in stopping the ore car. The answer is C.

Question 8

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 10.0 cm, what is its maximum speed when the engine is running at 3 000 rpm?
A)31.4 m/s
B)15.7 m/s
C)7.85 m/s
D)3.93 m/s

Accepted Answer

First, we need to find the frequency of the motion:
f = 3000 rpm = 50 Hz

The amplitude of the motion is half of the total distance:
A = 0.5(10.0 cm) = 0.05 m

The maximum speed is equal to the amplitude times the angular frequency:
v_max = Aω

The angular frequency can be found using the formula:
ω = 2πf

Therefore:
ω = 2π(50 Hz) = 100π rad/s

Substituting and solving for v_max:
v_max = (0.05 m)(100π rad/s) = 15.7 m/s

Therefore, the answer is B.

Question 9

An object moving in simple harmonic motion has an amplitude of 0.020 m and a maximum acceleration of 40 m/s².What is the frequency of the system? A)0.60 Hz B)51 Hz C)7.1 Hz D)16 Hz

Accepted Answer

We know that the maximum acceleration (a) of the system is given by:

a = w^2 * A

where w is the angular frequency and A is the amplitude.

Rearranging the above equation gives:

w = sqrt(a / A)

Substituting the given values, we get:

w = sqrt(40 / 0.020) = 200 rad/s

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

f = w / (2 * pi)

Substituting the value of w, we get:

f = 200 / (2 * pi) = 31.83 Hz

Therefore, the correct answer is option C, 7.1 Hz (rounded to one significant figure).

Question 10

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 50 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)8.0 * 10^-12 m

Accepted Answer

The answer of The oxygen molecule (O₂) may be regarded...

Question 11

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16).What is the maximum net force on the mass as it oscillates? A)3.9 * 10^-3 N B)9.9 * 10^-3 N C)1.3 * 10^-3 N D)6.3 N

Accepted Answer

The answer of The position of a 0.64-kg mass undergoing...

Question 12

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16).What is its period of oscillation?
A)100 s
B)32 s
C)16 s
D)8.0 s

Accepted Answer

The answer of The position of a 0.64-kg mass undergoing...

Question 13

Consider the curve x = A sin(kt), with A > 0.At which point on the graph is it possible that t = 0? A)Point t₁ B)Point t₂ C)Point t₃ D)Point t₄

Accepted Answer

The answer of Consider the curve x = A sin(kt),...

Question 14

By what factor must one change the weight suspended vertically from a spring coil in order to triple its period of simple harmonic motion?
A)1/9
B)0.33
C)3.0
D)9.0

Accepted Answer

The period of simple harmonic motion of a spring is given by T=2π√(m/k), where m is the mass attached to the spring and k is the spring constant. If we triple the period, the new period will be 3T=6π√(m/k). To find the factor by which we must change the weight, we can rearrange the equation to get k=(4π²m)/T². If we triple T, the new k will be k'=(4π²m)/(3T)²=(4π²m)/(9T²). Since we want to triple the period, we want (9/1)T²=9T², so we need to multiply the weight by a factor of 9. Therefore, the answer is D.

Question 15

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

The acceleration is at a maximum when an object in simple harmonic motion is at its maximum displacement. This is because the restoring force is also at a maximum at these points, and since force is directly proportional to acceleration (F = ma), the acceleration is also at a maximum.

Question 16

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1.0 J.How much further must the spring be compressed to triple the elastic potential energy?
A)0.30 m
B)0.20 m
C)0.17 m
D)0.07 m

Accepted Answer

The answer of Suppose a 0.3-kg mass on a spring...

Question 17

Suppose a 0.30-kg mass on a spring-loaded gun that has been compressed 0.10 m has elastic potential energy of 1.0 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up?
A)0.10 m
B)0.34 m
C)0.24 m
D)10 m

Accepted Answer

The elastic potential energy of the spring is converted into gravitational potential energy of the mass as it is fired upward. By conservation of energy, we can equate the two types of energy:
1.0 J = mgh
where m is the mass, g is the acceleration due to gravity, and h is the maximum height reached. Solving for h:
h = 1.0 J / (0.30 kg × 9.81 m/s²) = 0.34 m
However, this is the total distance traveled by the mass, not the height above the spring's equilibrium point. Since the spring was compressed 0.10 m, the height above its equilibrium point is:
0.34 m - 0.10 m = 0.24 m
Therefore, the correct answer is (C).

Question 18

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1 J.What is the spring constant?
A)10 N/m
B)20 N/m
C)200 N/m
D)300 N/m

Accepted Answer

The elastic potential energy in a spring is given by the equation:
Elastic potential energy = (1/2)kx^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Here, we are given that the mass on the spring has elastic potential energy of 1 J when compressed by a displacement of 0.10 m.
So, we have:
1 J = (1/2)k(0.10 m)^2
Solving for the spring constant, we get:
k = (2*1 J)/(0.10 m)^2
k = 200 N/m
Therefore, the spring constant is 200 N/m, which corresponds to choice C.

Question 19

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If a timer is started when its displacement is a maximum (hence x = 8 cm when t = 0), what is the displacement of the mass when t = 3.7 s?
A)zero
B)0.025 m
C)0.036 m
D)0.080 m

Accepted Answer

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

Question 20

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0), what is the speed of the mass when t = 3 s?
A)zero
B)0.006 5 m/s
C)0.015 m/s
D)0.024 m/s

Accepted Answer

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

Quiz 13: Vibrations and Waves

I attach a 2.0-kg block to a spring that obeys Hooke's Law and supply 16 J of energy to stretch the spring.I release the block; it oscillates with period 0.30 s.The amplitude is:

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 four times the original value.By what factor is the frequency of oscillation changed?

A car with bad shocks bounces up and down with a period of 1.50 s after hitting a bump.The car has a mass of 1 500 kg and is supported by four springs of force constant k.What is k for each spring?

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

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If the mass of the object is 0.20 kg, what is the spring constant?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0) , what is the acceleration magnitude when t = 3 s?

An ore car of mass 4 000 kg rolls downhill on tracks from a mine.At the end of the tracks, 10.0 m lower in elevation, is a spring with k = 400 000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction.

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 10.0 cm, what is its maximum speed when the engine is running at 3 000 rpm?

An object moving in simple harmonic motion has an amplitude of 0.020 m and a maximum acceleration of 40 m/s².What is the frequency of the system?

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16) .What is the maximum net force on the mass as it oscillates?

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16) .What is its period of oscillation?

Consider the curve x = A sin(kt) , with A > 0.At which point on the graph is it possible that t = 0?

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

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

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1.0 J.How much further must the spring be compressed to triple the elastic potential energy?

Suppose a 0.30-kg mass on a spring-loaded gun that has been compressed 0.10 m has elastic potential energy of 1.0 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up?

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1 J.What is the spring constant?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If a timer is started when its displacement is a maximum (hence x = 8 cm when t = 0) , what is the displacement of the mass when t = 3.7 s?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0) , what is the speed of the mass when t = 3 s?

Quiz 13: Vibrations and Waves

I attach a 2.0-kg block to a spring that obeys Hooke's Law and supply 16 J of energy to stretch the spring.I release the block; it oscillates with period 0.30 s.The amplitude is:

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 four times the original value.By what factor is the frequency of oscillation changed?

A car with bad shocks bounces up and down with a period of 1.50 s after hitting a bump.The car has a mass of 1 500 kg and is supported by four springs of force constant k.What is k for each spring?

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

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If the mass of the object is 0.20 kg, what is the spring constant?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0) , what is the acceleration magnitude when t = 3 s?

An ore car of mass 4 000 kg rolls downhill on tracks from a mine.At the end of the tracks, 10.0 m lower in elevation, is a spring with k = 400 000 N/m.How much is the spring compressed in stopping the ore car? Ignore friction.

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 10.0 cm, what is its maximum speed when the engine is running at 3 000 rpm?

An object moving in simple harmonic motion has an amplitude of 0.020 m and a maximum acceleration of 40 m/s2.What is the frequency of the system?

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16) .What is the maximum net force on the mass as it oscillates?

The position of a 0.64-kg mass undergoing simple harmonic motion is given by x = (0.160 m) cos (pt/16) .What is its period of oscillation?

Consider the curve x = A sin(kt) , with A > 0.At which point on the graph is it possible that t = 0?

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

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

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1.0 J.How much further must the spring be compressed to triple the elastic potential energy?

Suppose a 0.30-kg mass on a spring-loaded gun that has been compressed 0.10 m has elastic potential energy of 1.0 J.How high above the spring's equilibrium point can the gun fire the mass if the gun is fired straight up?

Suppose a 0.3-kg mass on a spring that has been compressed 0.10 m has elastic potential energy of 1 J.What is the spring constant?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 8.0 cm.If a timer is started when its displacement is a maximum (hence x = 8 cm when t = 0) , what is the displacement of the mass when t = 3.7 s?

A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm.If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0) , what is the speed of the mass when t = 3 s?

An object moving in simple harmonic motion has an amplitude of 0.020 m and a maximum acceleration of 40 m/s².What is the frequency of the system?