Question 1

A 0.20-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.6 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.2 J.What is the maximum speed of the mass during its oscillation?

Accepted Answer

The total mechanical energy of the mass-spring system is the sum of the kinetic and potential energies:

E = K + U = 1.4 J + 2.2 J = 3.6 J

At the maximum displacement (amplitude) of the oscillation, all of the energy is in the form of kinetic energy, since the spring is momentarily at rest.

Thus,

E = K_max = (1/2)mv_max^2

Solving for v_max, we get

v_max = sqrt((2E)/m) = sqrt((2*3.6 J)/(0.20 kg))

v_max ≈ 6.0 m/s

Question 2

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

Accepted Answer

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

Question 3

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?

Accepted Answer

The maximum acceleration (a_max) of an object in simple harmonic motion is given by: 
a_max = 2πfA 
where f is the frequency of the motion and A is the amplitude.

Rearranging this formula gives: 
f = a_max / (2πA)

Substituting the given values, we get: 
f = 40 / (2π × 0.020) = 7.1 Hz

Therefore, the correct answer is C.

Question 4

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.

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 5

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

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 6

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

Accepted Answer

The period of simple harmonic motion of a spring-mass system is given by T=2π√(m/k), where m is the mass suspended from the spring and k is the spring constant. If we want to triple the period, we need to increase T by a factor of 3. Using the equation T=2π√(m/k), we can see that T is proportional to the square root of m/k. Therefore, we need to increase m/k by a factor of 9 in order to triple the period. Since k is a constant for a given spring, we need to increase the mass by a factor of 9. Hence, the weight suspended vertically from the spring coil must be increased by a factor of 9. Therefore, the correct answer is D.

Question 7

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?

Accepted Answer

The elastic potential energy stored in a spring can be calculated using the formula:

Elastic potential energy = (1/2) * spring constant * (compression or extension)^2

We are given that the mass on the spring has elastic potential energy of 1 J when the spring is compressed by 0.10 m. Therefore:

1 J = (1/2) * k * (0.10 m)^2

Solving for the spring constant k, 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 8

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?

Accepted Answer

The frequency of oscillation of a spring-mass system is inversely proportional to the square root of the mass. If the mass is increased by a factor of 4, the frequency is reduced by a factor of √4, which is 2.

Question 9

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?

Accepted Answer

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

Question 10

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?

Accepted Answer

The formula for the frequency of simple harmonic motion is:
f = 1/(2π) √(k/m)
where f is the frequency, k is the spring constant, and m is the mass.

We have f = 4.0 Hz, and m = 0.20 kg. Also, the amplitude A = 8.0 cm which can be converted to meters by dividing by 1000.

4.0 = 1/(2π) √(k/0.20)
16π^2 = k/0.20
k = 16π^2 x 0.20
k = 126 N/m

Therefore, the spring constant is 126 N/m. Answer choice C is correct.

Question 11

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 x 10^-26 kg has a vibrational energy of 1.6 x 10^-21 J and the effective spring constant is 50 N/m,then what is the amplitude of oscillation of each oxygen atom?

Accepted Answer

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

Question 12

A 0.20-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 4.0 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?

Accepted Answer

The answer of A 0.20-kg block rests on a frictionless...

Question 13

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:

Accepted Answer

The energy supplied to the spring is equal to the potential energy at maximum amplitude: $ E = \frac{1}{2} k A^2 $. Given $ E = 16 $ J, solve for $ A $ using the period $ T = 0.30 $ s to find $ k $ from $ T = 2\pi \sqrt{\frac{m}{k}} $.

Question 14

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?

Accepted Answer

The speed of the mass is zero when the displacement is at a maximum or minimum, which occurs at the endpoints of the motion. Since the timer starts when the displacement is maximum, the speed is zero at that instant.

Question 15

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?

Accepted Answer

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

Question 16

A 0.20-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 4.0 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 4.0-cm displacement point?

Accepted Answer

Explanation:The work done by the frictional force is:$$W_f = F_f d = (0.3 N)(0.04 m) = 0.012 J$$The initial potential energy of the spring is:$$U_i = \frac{1}{2}kx_i^2 = \frac{1}{2}(40 N/m)(0.04 m)^2 = 0.032 J$$The final kinetic energy of the block is:$$K_f = U_i - W_f = 0.032 J - 0.012 J = 0.02 J$$The speed of the block is:$$v = \sqrt{\frac{2K_f}{m}} = \sqrt{\frac{2(0.02 J)}{0.20 kg}} = 0.45 m/s$$

Question 17

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?

Accepted Answer

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

Question 18

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

Accepted Answer

When an object in simple harmonic motion is at its maximum displacement, its acceleration is also at a maximum. This is because the acceleration is directly proportional to the displacement and in the opposite direction of the displacement. Therefore, option B is the correct answer. Option A (speed) and Option C (kinetic energy) are not at a maximum at maximum displacement but at the equilibrium position. Option D (frequency) remains constant for a given system, regardless of the displacement.

Question 19

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?

Accepted Answer

The period of oscillation T is related to the mass m and the spring constant k by the formula T = 2π√(m/4k). Solving for k gives k = (4π²m)/(T²). Substituting m = 1500 kg and T = 1.50 s, we find k ≈ 6,580 N/m.

Question 20

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?

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).

Quiz 13: Vibrations and Waves

A 0.20-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.6 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.2 J.What is the maximum speed of the mass during its oscillation?

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

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?

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.

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

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

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?

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 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 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?

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 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 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?

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?

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

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?

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?

Quiz 13: Vibrations and Waves

A 0.20-kg mass is oscillating on a spring over a horizontal frictionless surface.When it is at a displacement of 2.6 cm for equilibrium it has a kinetic energy of 1.4 J and a spring potential energy of 2.2 J.What is the maximum speed of the mass during its oscillation?

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

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?

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.

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

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

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?

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 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 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?

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 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 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?

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?

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

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?

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?

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?