Question 1

The displacement of an element of a string is given by y(x,t) = 4.3sin(1.2x - 4.7t - π/3), with x in meters and t in seconds. Given that , what is v?
A) 0.065 m/s
B) 0.25 m/s
C) 2.0 m/s
D) 3.9 m/s
E) 15 m/s

Accepted Answer

The answer of The displacement of an element of a...

Question 2

A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator does not supply any power when the end of the string
A) has its least acceleration
B) has its greatest displacement
C) has half its greatest displacement
D) has one fourth its greatest displacement
E) has its least displacement

Accepted Answer

The generator does not supply any power when the end of the string has its greatest displacement because at this point, the velocity of the string is zero (it momentarily stops before changing direction), and power is related to the rate of doing work or energy transfer, which involves both force and velocity.

Question 3

Two separated sources emit sinusoidal traveling waves that have the same wavelength $\lambda$ and are in phase at their respective sources. One travels a distance ℓ₁ to get to the observation point while the other travels a distance ℓ₂. The amplitude is a minimum at the observation point if ℓ₁− ℓ₂ is: A) an odd multiple of $\lambda$/2 B) an odd multiple of $\lambda$/4 C) a multiple of$\lambda$ D) an odd multiple of $\pi$/2 E) a multiple of $\pi$

Accepted Answer

When the two waves arrive at the observation point, their phase difference depends on the difference in the distances traveled by each wave. If this distance difference is an odd multiple of half the wavelength, then the two waves will be in opposite phase and will interfere destructively, resulting in minimum amplitude. Therefore, the condition for minimum amplitude is: ℓ₁− ℓ₂ = (2n+1)F1F1F1S1/2, where n is an integer. Multiplying both sides by 2 and simplifying gives: 2(ℓ₁− ℓ₂) = (4n+2)F1F1F1S1 Dividing both sides by 2 gives: ℓ₁− ℓ₂ = (2n+1)F1F1F1S1/2, which is in the form of an odd multiple of half the wavelength. Therefore, the answer is (A).

Question 4

A long string is constructed by joining the ends of two shorter strings. The tension in the strings is the same but string I has 4 times the linear mass density of string II. When a sinusoidal wave passes from string I to string II:
A) the frequency decreases by a factor of 4
B) the frequency decreases by a factor of 2
C) the wave speed decreases by a factor of 4
D) the wave speed decreases by a factor of 2
E) the wave speed increases by a factor of 2

Accepted Answer

The answer of A long string is constructed by joining...

Question 5

Two identical but separate strings, with the same tension, carry sinusoidal waves with the same frequency. Wave A has an amplitude that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.
A) half
B) twice
C) one-fourth
D) four times
E) eight times

Accepted Answer

The energy transmitted by a wave is proportional to the square of its amplitude (E ∝ A²). Since wave A has twice the amplitude of wave B, its energy will be four times greater (2² = 4). Therefore, wave A transmits energy at a rate that is four times greater than wave B.

Question 6

Three separate strings are made of the same material. String 1 has length L and tension  $ \tau $, string 2 has length 2L and tension 2 $ \tau $and string 3 has length 3L and tension 3 $ \tau $. A pulse is started at one end of each string. If the pulses start at the same time, the order in which they reach the other end is:
A) 1, 2, 3
B) 3, 2, 1
C) 2, 3, 1
D) 3, 1, 2
E) they all take the same time

Accepted Answer

The answer of Three separate strings are made of the...

Question 7

The speed of a sinusoidal wave on a string depends on:
A) the frequency of the wave
B) the wavelength of the wave
C) the length of the string
D) the tension in the string
E) the amplitude of the wave

Accepted Answer

The speed of a sinusoidal wave on a string depends on the tension in the string. This is because the tension affects the restoring force of the string, which in turn affects the speed of the wave. The other factors mentioned (frequency, wavelength, length, amplitude) do affect the behavior of the wave, but they do not directly affect the speed.

Question 8

The diagram shows three identical strings that have been put under tension by suspending masses of 5 kg each. For which is the wave speed the greatest? 
A) 1
B) 2
C) 3
D) 1 and 3 tie
E) 2 and 3 tie

Accepted Answer

The answer of The diagram shows three identical strings that...

Question 9

Two separated sources emit sinusoidal traveling waves that have the same wavelength $\lambda$ and are in phase at their respective sources. One travels a distance ℓ₁ to get to the observation point while the other travels a distance ℓ₂ The amplitude is a maximum at the observation point if ℓ₁− ℓ₂ is: A) an odd multiple of $\pi$/2 B) an odd multiple of $\lambda$/4 C) a multiple of $\lambda$ D) an odd multiple of $\pi$/2. E) a multiple of $\pi$

Accepted Answer

For constructive interference (which results in maximum amplitude) at the observation point, the path difference between the two waves should be an integer multiple of the wavelength. This ensures that the waves are in phase when they meet, reinforcing each other's amplitudes.

Question 10

The time required for a small pulse to travel from A to B on a stretched cord shown is NOT altered by changing:
A) the linear mass density of the cord
B) the length between A and B
C) the shape of the pulse
D) the tension in the cord
E) none of the above (changes in all alter the time)

Accepted Answer

The speed of a pulse traveling along a stretched cord depends on the tension in the cord and the linear mass density of the cord, not on the shape of the pulse.

Question 11

Two identical but separate strings, with the same tension, carry sinusoidal waves with the same amplitude. Wave A has a frequency that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.
A) half
B) twice
C) one-fourth
D) four times
E) eight times

Accepted Answer

The rate at which energy is transmitted by a wave (the power) is proportional to the square of both the amplitude and the frequency. Since the amplitude is the same for both waves but the frequency of wave A is twice that of wave B, the energy transmitted by wave A is $2^2 = 4$ times that of wave B.

Question 12

Fully constructive interference between two sinusoidal waves of the same frequency occurs only if they:
A) travel in the same direction and are 270 $\degree$ out of phase
B) travel in the same direction and are 45 $\degree$ out of phase
C) travel in the same direction and are in phase
D) travel in the same direction and are 180 $\degree$out of phase
E) travel in the same direction and are 90 $\degree$ out of phase

Accepted Answer

Fully constructive interference occurs when the two waves are "in phase", meaning they have the same frequency, wavelength, and are lined up perfectly so their peaks and troughs match up. Option C is the only one that describes the waves as being in phase.

Question 13

The tension in a string with a linear density of 0.0010 kg/m is 0.40 N. A 100 Hz sinusoidal wave on this string has a wavelength of:
A) 0.20 cm
B) 2.0 cm
C) 5.0 cm
D) 20 cm
E) 400 cm

Accepted Answer

The speed of a wave on a string is given by the equation v = √(T/μ), where T is the tension in the string and μ is the linear density (mass per unit length) of the string. Plugging in the given values, v = √(0.40 N / 0.0010 kg/m) = 20 m/s.
The frequency of the wave is 100 Hz, which means it completes 100 cycles in 1 second. The wavelength, λ, is equal to the speed of the wave divided by the frequency, or λ = v/f = 20 m/s / 100 Hz = 0.20 m. Converting this to centimeters, we get 20 cm. Therefore, the answer is D.

Question 14

When a 100-Hz oscillator is used to generate a sinusoidal wave on a certain string the wavelength is 10 cm. When the tension in the string is doubled the generator produces a wave with a frequency and wavelength of:
A) 200 Hz and 20 cm
B) 141 Hz and 10 cm
C) 100 Hz and 20 cm
D) 100 Hz and 14 cm
E) 50 Hz and 14 cm

Accepted Answer

The speed of the wave on the string is given by the equation v = fλ, where v is the speed, f is the frequency, and λ is the wavelength. Since the frequency is constant at 100 Hz, and the wavelength is initially 10 cm, we can solve for the speed: 
v = fλ = (100 Hz)(10 cm) = 1000 cm/s

When the tension in the string is doubled, the speed of the wave will also double (assuming the mass per unit length remains the same). So the speed of the wave is now: 
v' = 2v = 2000 cm/s

To find the new wavelength, we can use the equation v' = f'λ', where f' is the new frequency: 
2000 cm/s = f'(λ')

We know that the new frequency f' is greater than 100 Hz, since the speed has increased. We can try the answer choices:

A) 200 Hz and 20 cm 
Using f' = 200 Hz, we get λ' = v'/f' = 2000/200 = 10 cm 
This matches the initial wavelength, so this is not the correct answer.

B) 141 Hz and 10 cm 
Using f' = 141 Hz, we get λ' = v'/f' = 2000/141 = 14.2 cm 
This is close to the initial wavelength, but not quite there.

C) 100 Hz and 20 cm 
Using f' = 100 Hz, we get λ' = v'/f' = 2000/100 = 20 cm 
This does not match the initial wavelength, so this is not the correct answer.

D) 100 Hz and 14 cm 
Using f' = 100 Hz, we get λ' = v'/f' = 2000/100 = 20 cm 
This is the same as the initial wavelength, which means it must be the answer.

E) 50 Hz and 14 cm 
Using f' = 50 Hz, we get λ' = v'/f' = 2000/50 = 40 cm 
This does not match the initial wavelength, so this is not the correct answer.

Therefore, the answer is D: 100 Hz and 14 cm.

Question 15

Two sinusoidal waves have the same angular frequency, the same amplitude y_m, and travel in the same direction in the same medium. If they differ in phase by 50 $\degree$, the amplitude of the resultant wave is given by A) 0.64 y_m B) 1.3 y_m C) 0.91 y_m D) 1.8 y_m E) 0.35 y_m

Accepted Answer

The answer of Two sinusoidal waves have the same angular...

Question 16

Fully destructive interference between two sinusoidal waves of the same frequency and amplitude occurs only if they:
A) travel in the same direction and are 270 $\degree$out of phase
B) travel in the same direction and are 45 $\degree$ out of phase
C) travel in the same direction and are in phase
D) travel in the same direction and are 180 $\degree$out of phase
E) travel in the same direction and are 90 $\degree$ out of phase

Accepted Answer

Fully destructive interference occurs when the two waves are completely out of phase (i.e., they have a phase difference of 180 degrees).

Question 17

Two traveling sinusoidal waves interfere to produce a wave with the mathematical form If the value of $\phi$ is appropriately chosen, the two waves might be: A) y₁(x,t) = (y_m/3) sin (kx + $ \omega $t) and y₂(x,t) = (y_m/3) sin (kx + $ \omega $t + $\phi$ ) B) y₁(x,t) = 0.7y_m sin (kx - $ \omega $t) and y₂(x,t) = 0.7y_m sin (kx - $ \omega $t + $\phi$ ) C) y₁(x,t) = 0.7y_m sin (kx - $ \omega $t) and y₂(x,t) = 0.7y_m sin (kx +$ \omega $t + $\phi$ ) D) y₁(x,t) = 0.7y_m sin [(kx/2) - ($ \omega $t/2)] and y₂(x,t) = 0.7y_m sin [(kx/2) - ($ \omega $t/2) + $\phi$ ] E) y₁(x,t) = 0.7y_m sin (kx + $ \omega $t) and y₂(x,t) = 0.7y_m sin (kx + $ \omega $t + $\phi$ )

Accepted Answer

The answer of Two traveling sinusoidal waves interfere to produce...

Question 18

The sum of two sinusoidal traveling waves is a sinusoidal traveling wave only if:
A) their amplitudes are the same and they travel in the same direction
B) their amplitudes are the same and they travel in opposite directions
C) their frequencies are the same and they travel in the same direction
D) their frequencies are the same and they travel in opposite directions
E) their frequencies are the same and their amplitudes are the same

Accepted Answer

For two sinusoidal traveling waves to sum up to another sinusoidal traveling wave, they must have the same frequency and travel in the same direction. This ensures that their phases align consistently over time, maintaining a sinusoidal form. Differences in amplitude or direction would lead to a more complex waveform.

Question 19

A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator must supply the greatest power when the end of the string:
A) has its greatest acceleration
B) has its greatest displacement
C) has half its greatest displacement
D) has one fourth its greatest displacement
E) has its least displacement

Accepted Answer

The answer of A sinusoidal wave is generated by moving...

Question 20

Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are y₁_m and y₂_m. The smallest possible amplitude of the resultant wave is: A) y₁_m + y₂_m and occurs when they are 180$\degree$ out of phase B) $\mid$y₁_m - y₂_m$\mid$ and occurs when they are 180$\degree$ out of phase C) y₁_m + y₂_m and occurs when they are in phase D) $\mid$y₁_m - y₂_m$\mid$ and occurs when they are in phase E) $\mid$y₁_m - y₂_m$\mid$ and occurs when they are 90$\degree$ out of phase

Accepted Answer

The resultant amplitude is given by the equation:
$A_{res} = \sqrt{(A_1+A_2)^2 + (A_1-A_2)^2\cos^2\delta}$
where $A_1$ and $A_2$ are the amplitudes of the waves, and $\delta$ is the phase difference between them.
Taking the derivative of this equation with respect to $\delta$ and setting it equal to zero, we find that the minimum amplitude occurs when $\delta = \pm \pi/2$ or $\delta = \pm 3\pi/2$.
If the waves are $180^\circ$ out of phase, then $\delta = \pi$, which satisfies one of the conditions for minimum amplitude. Plugging in the values given in choice B, we find that this choice gives the minimum amplitude.

Quiz 16: Waves I

The displacement of an element of a string is given by y(x,t) = 4.3sin(1.2x - 4.7t - π/3) , with x in meters and t in seconds. Given that , what is v?

A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator does not supply any power when the end of the string

A long string is constructed by joining the ends of two shorter strings. The tension in the strings is the same but string I has 4 times the linear mass density of string II. When a sinusoidal wave passes from string I to string II:

Two identical but separate strings, with the same tension, carry sinusoidal waves with the same frequency. Wave A has an amplitude that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.

The speed of a sinusoidal wave on a string depends on:

The diagram shows three identical strings that have been put under tension by suspending masses of 5 kg each. For which is the wave speed the greatest?

The time required for a small pulse to travel from A to B on a stretched cord shown is NOT altered by changing:

Two identical but separate strings, with the same tension, carry sinusoidal waves with the same amplitude. Wave A has a frequency that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.

Fully constructive interference between two sinusoidal waves of the same frequency occurs only if they:

The tension in a string with a linear density of 0.0010 kg/m is 0.40 N. A 100 Hz sinusoidal wave on this string has a wavelength of:

When a 100-Hz oscillator is used to generate a sinusoidal wave on a certain string the wavelength is 10 cm. When the tension in the string is doubled the generator produces a wave with a frequency and wavelength of:

Two sinusoidal waves have the same angular frequency, the same amplitude y_m, and travel in the same direction in the same medium. If they differ in phase by 50 $\degree$ , the amplitude of the resultant wave is given by

Fully destructive interference between two sinusoidal waves of the same frequency and amplitude occurs only if they:

The sum of two sinusoidal traveling waves is a sinusoidal traveling wave only if:

A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator must supply the greatest power when the end of the string:

Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are y₁_m and y₂_m. The smallest possible amplitude of the resultant wave is: