Question 1

Standing waves on a string: A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 10³ kg/m³. What is the tension in the wire?

Accepted Answer

We can first find the wavelength of the standing wave using the formula:

wavelength = 2L/n

where L is the length of the wire and n is the harmonic number.

For the third harmonic (n=3), the wavelength is:

wavelength = 2(1.20 m)/3 = 0.80 m

Next, we can find the wave speed using the formula:

wave speed = frequency x wavelength

wave speed = 512 Hz x 0.80 m = 409.6 m/s

Finally, we can use the formula for the tension in a string under wave motion:

tension = (wave speed)^2 x mass per unit length

mass per unit length = density x cross-sectional area

cross-sectional area = πr^2

mass per unit length = (21.4 x 10^3 kg/m^3) x π(0.500 x 10^-6 m)^2 = 2.66 x 10^-7 kg/m

tension = (409.6 m/s)^2 x 2.66 x 10^-7 kg/m = 2.82 kN

Therefore, the answer is choice C.

Question 2

Standing waves on a string: A 2.0-m string is fixed at both ends and tightened until the wave speed is  What is the frequency of the standing wave shown in the figure?

Accepted Answer

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Question 3

Intensity: Two people are talking at a distance of 3.0 m from where you are and you measure the sound intensity as 1.1 × 10^-7 W/m². Another student is 4.0 m away from the talkers. What sound intensity does the other student measure? Assume that the sound spreads out uniformly and undergoes no significant reflections or absorption.

Accepted Answer

The intensity of sound decreases with the square of the distance from the source. Using the inverse square law: I2 = I1 * (r1/r2)^2, where I1 = 1.1 × 10^-7 W/m^2, r1 = 3.0 m, and r2 = 4.0 m. I2 = 1.1 × 10^-7 * (3/4)^2 = 6.2 × 10^-8 W/m^2.

Question 4

Intensity: The intensity of sunlight falling on the earth is about 1.4 kW/m² (before any gets absorbed by our atmosphere). At what rate does the sun emit light energy? (The earth-sun distance = 1.5 × 10⁸ km and the earth's radius = 6.4 × 10³ km.)

Accepted Answer

The rate at which the sun emits light energy can be calculated by considering the intensity of sunlight at the Earth's distance and the surface area of a sphere with a radius equal to the Earth-Sun distance. The intensity of sunlight is given as 1.4 kW/m^2. The surface area of a sphere is 4πr^2, where r is the Earth-Sun distance (1.5 × 10^11 m). Thus, the total power output of the sun is 1.4 kW/m^2 * 4π(1.5 × 10^11 m)^2 = 4.0 × 10^26 W.

Question 5

Speed of sound: A pipe that is 20.0 m long and 10.0 cm in diameter contains olive oil. The density of the olive oil is 890 kg/m³ and the bulk modulus is 1.3 × 10⁹Pa. A 3.4-Hz longitudinal wave is transmitted in the oil. How many milliseconds does it take for the wave to travel the length of the pipe in the oil?

Accepted Answer

The speed of sound in a fluid is given by:$$v = \sqrt{\frac{B}{ho}}$$where:* v is the speed of sound (m/s)* B is the bulk modulus (Pa)* ρ is the density (kg/m^3)In this problem, we are given the following information:* B = 1.3 × 10^9 Pa* ρ = 890 kg/m^3* L = 20.0 m* f = 3.4 HzWe can use the formula above to calculate the speed of sound in the olive oil:$$v = \sqrt{\frac{B}{ho}} = \sqrt{\frac{1.3 	imes 10^9 	ext{ Pa}}{890 	ext{ kg/m}^3}} = 143.4 	ext{ m/s}$$The time it takes for the wave to travel the length of the pipe is:$$t = \frac{L}{v} = \frac{20.0 	ext{ m}}{143.4 	ext{ m/s}} = 0.139 	ext{ s} = 139 	ext{ ms}$$Therefore, the correct answer is A) 17 ms.

Question 6

Standing waves on a string: A thin taut string is fixed at both ends and stretched along the horizontal x-axis with its left end at x = 0. It is vibrating in its third OVERTONE, and the equation for the vertical displacement of any point on the string is y(x,t) = (1.22 cm) sin[(14.4 m-1)x] cos[(166 rad/s)t].
(a) What are the frequency and wavelength of the fundamental mode of this string?
(b) How long is the string?
(c) How fast do waves travel on this string?

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Question 7

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

Accepted Answer

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Question 8

Speed of sound: The density of aluminum is 2.7 × 10³ kg/m³. The speed of longitudinal waves in an aluminum rod is measured to be 5.1 × 10³ m/s. What is the value of Young's modulus for this aluminum?

Accepted Answer

Young's modulus (E) can be calculated using the formula $E = \rho v^2$, where $\rho$ is the density of the material and $v$ is the speed of sound in the material. Substituting the given values, $\rho = 2.7 \times 10^3 \, \text{kg/m}^3$ and $v = 5.1 \times 10^3 \, \text{m/s}$, we get $E = 2.7 \times 10^3 \times (5.1 \times 10^3)^2 = 7.0 \times 10^{10} \, \text{N/m}^2$.

Question 9

Standing waves on a string: A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to

Accepted Answer

The frequency of the third harmonic (n=3) for a string fixed at both ends is given by $f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$, where $f_n$ is the frequency of the nth harmonic, $L$ is the length of the string, $T$ is the tension in the string, and $\mu$ is the mass per unit length of the string. Given that $f_3 = 100 \, \text{Hz}$, $L = 0.64 \, \text{m}$, and the mass of the string $m = 1.0 \, \text{g} = 0.001 \, \text{kg}$, we can find $\mu = \frac{m}{L} = \frac{0.001}{0.64} \, \text{kg/m}$. Plugging these values into the formula and solving for $T$ gives a tension close to 2.8 N.

Question 10

Standing waves on a string: A guitar string 0.650 m long has a tension of 61.0 N and a mass per unit length of 3.00 g/m.
(a) What is the speed of waves on the string when it is plucked?
(b) What is the string's fundamental frequency of vibration when plucked?

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Question 11

Standing waves on a string: A 2.00-m long piano wire with a mass per unit length of 12.0 g/m is under a tension of 8.00 kN. What is the frequency of the fundamental mode of vibration of this wire?

Accepted Answer

The fundamental frequency of a string is given by:f = (1/2L) * sqrt(T/m)where L is the length of the string, T is the tension, and m is the mass per unit length.Plugging in the given values, we get:f = (1/2 * 2.00 m) * sqrt(8.00 kN / 0.012 kg/m)= 204 HzTherefore, the correct answer is A.

Question 12

Intensity: Observer A is 3.0 m from a tiny light bulb and observer B is 12.0 m from the same bulb. Assume that the light spreads out uniformly and undergoes no significant reflections or absorption. If observer B sees a light of intensity I, the light intensity that A sees is

Accepted Answer

The intensity of light is inversely proportional to the square of the distance from the source. So, the intensity of light that A sees is 16 times the intensity of light that B sees.

Question 13

Standing waves on a string: A standing wave is oscillating at 690 Hz on a string, as shown in the figure. What is the speed of traveling waves on this string?

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Question 14

Speed of sound: The speed of sound in an alcohol at a temperature of 20°C is 1260 m/s. The density of the alcohol at that temperature is  At a temperature of 20°C, find the bulk modulus of the alcohol.

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Question 15

Standing waves on a string: A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the frequency of the sound it will produce?

Accepted Answer

The speed of the pulse in the wire can be calculated using the time it takes for the pulse to travel down and back, which is $7.84 \, \text{ms} = 0.00784 \, \text{s}$. Since the length of the wire is $65.0 \, \text{cm} = 0.65 \, \text{m}$, the total distance traveled by the pulse is twice this length, or $2 \times 0.65 \, \text{m} = 1.3 \, \text{m}$. The speed of the pulse ($v$) is then $v = \frac{distance}{time} = \frac{1.3 \, \text{m}}{0.00784 \, \text{s}} \approx 165.8 \, \text{m/s}$.For a string fixed at both ends, the fundamental frequency ($f_1$) is given by $f_1 = \frac{v}{2L}$, where $L$ is the length of the string. The frequencies of the overtones are multiples of the fundamental frequency. The second overtone corresponds to the third harmonic ($n = 3$), so its frequency ($f$) is given by $f = n \times f_1 = 3 \times \frac{v}{2L}$.Substituting the given values, $f = 3 \times \frac{165.8 \, \text{m/s}}{2 \times 0.65 \, \text{m}} \approx 382.5 \, \text{Hz}$, which rounds to $383 \, \text{Hz}$, making option D the correct answer.

Question 16

Standing waves on a string: Two violinists are trying to tune their instruments in an orchestra. One is producing the desired frequency of 440.0 Hz. The other is producing a frequency of 448.4 Hz. By what percentage should the out-of-tune musician change the tension in his string to bring his instrument into tune at 440.0 Hz?

Accepted Answer

The frequency produced by a vibrating string is directly proportional to the tension in the string. Therefore, the out-of-tune musician needs to adjust the tension in the string. The frequency ratio between the desired frequency of 440.0 Hz and the current frequency of 448.4 Hz is:

$$\frac{448.4}{440.0} = 1.019$$

To bring the frequency down to 440.0 Hz, the tension in the string needs to be decreased by the same factor of 1.019. The change in tension is therefore:

$$(1-1/1.019)	imes100\% \approx -3.7\%$$

Therefore, the musician needs to decrease the tension in his string by about 3.7% to bring his instrument into tune at 440.0 Hz. The correct answer is (D).

Question 17

Standing waves on a string: A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the wavelength of the sound it will produce?

Accepted Answer

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Question 18

Intensity: A tiny vibrating source sends waves uniformly in all directions. An area of 3.25 cm² on a sphere of radius 2.50 m centered on the source receives energy at a rate of 4.20 J/s. (a) What is the intensity of the waves at 2.50 m from the source and at 10.0 m from the source? (b) At what rate is energy leaving the vibrating source of the waves?

Accepted Answer

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Question 19

Intensity: Calculate the light intensity 1.51 m from a light bulb that emits 100 W of visible light, assuming that the light radiates uniformly in all directions.

Accepted Answer

The intensity (I) of light at a distance (r) from a point source is given by the formula I = P / (4πr^2), where P is the power of the light source. Substituting P = 100 W and r = 1.51 m into the formula gives I = 100 / (4π(1.51)^2) ≈ 3.49 W/m^2.

Question 20

Standing waves on a string: A thin 2.00-m string of mass 50.0 g is fixed at both ends and under a tension of 70.0 N. If it is set into small-amplitude oscillation, what is the frequency of the first harmonic mode?

Accepted Answer

The frequency of the first harmonic (fundamental frequency) is given by $ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} $, where $ L = 2.00 $ m, $ T = 70.0 $ N, and $ \mu = \frac{0.050 \text{ kg}}{2.00 \text{ m}} = 0.025 \text{ kg/m} $. Calculating gives $ f_1 = 13.2 $ Hz.

Quiz 14: Periodic Motion

Standing waves on a string: A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 10³ kg/m³. What is the tension in the wire?

Standing waves on a string: A 2.0-m string is fixed at both ends and tightened until the wave speed is What is the frequency of the standing wave shown in the figure?

Intensity: The intensity of sunlight falling on the earth is about 1.4 kW/m² (before any gets absorbed by our atmosphere) . At what rate does the sun emit light energy? (The earth-sun distance = 1.5 × 10⁸ km and the earth's radius = 6.4 × 10³ km.)

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

Speed of sound: The density of aluminum is 2.7 × 10³ kg/m³. The speed of longitudinal waves in an aluminum rod is measured to be 5.1 × 10³ m/s. What is the value of Young's modulus for this aluminum?

Standing waves on a string: A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to

Standing waves on a string: A guitar string 0.650 m long has a tension of 61.0 N and a mass per unit length of 3.00 g/m. (a) What is the speed of waves on the string when it is plucked? (b) What is the string's fundamental frequency of vibration when plucked?

Standing waves on a string: A 2.00-m long piano wire with a mass per unit length of 12.0 g/m is under a tension of 8.00 kN. What is the frequency of the fundamental mode of vibration of this wire?

Intensity: Observer A is 3.0 m from a tiny light bulb and observer B is 12.0 m from the same bulb. Assume that the light spreads out uniformly and undergoes no significant reflections or absorption. If observer B sees a light of intensity I, the light intensity that A sees is

Standing waves on a string: A standing wave is oscillating at 690 Hz on a string, as shown in the figure. What is the speed of traveling waves on this string?

Speed of sound: The speed of sound in an alcohol at a temperature of 20°C is 1260 m/s. The density of the alcohol at that temperature is At a temperature of 20°C, find the bulk modulus of the alcohol.

Intensity: Calculate the light intensity 1.51 m from a light bulb that emits 100 W of visible light, assuming that the light radiates uniformly in all directions.

Standing waves on a string: A thin 2.00-m string of mass 50.0 g is fixed at both ends and under a tension of 70.0 N. If it is set into small-amplitude oscillation, what is the frequency of the first harmonic mode?

Quiz 14: Periodic Motion

Standing waves on a string: A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 103 kg/m3. What is the tension in the wire?

Standing waves on a string: A 2.0-m string is fixed at both ends and tightened until the wave speed is What is the frequency of the standing wave shown in the figure?

Intensity: The intensity of sunlight falling on the earth is about 1.4 kW/m2 (before any gets absorbed by our atmosphere) . At what rate does the sun emit light energy? (The earth-sun distance = 1.5 × 108 km and the earth's radius = 6.4 × 103 km.)

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

Speed of sound: The density of aluminum is 2.7 × 103 kg/m3. The speed of longitudinal waves in an aluminum rod is measured to be 5.1 × 103 m/s. What is the value of Young's modulus for this aluminum?

Standing waves on a string: A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to

Standing waves on a string: A guitar string 0.650 m long has a tension of 61.0 N and a mass per unit length of 3.00 g/m. (a) What is the speed of waves on the string when it is plucked? (b) What is the string's fundamental frequency of vibration when plucked?

Standing waves on a string: A 2.00-m long piano wire with a mass per unit length of 12.0 g/m is under a tension of 8.00 kN. What is the frequency of the fundamental mode of vibration of this wire?

Intensity: Observer A is 3.0 m from a tiny light bulb and observer B is 12.0 m from the same bulb. Assume that the light spreads out uniformly and undergoes no significant reflections or absorption. If observer B sees a light of intensity I, the light intensity that A sees is

Standing waves on a string: A standing wave is oscillating at 690 Hz on a string, as shown in the figure. What is the speed of traveling waves on this string?

Speed of sound: The speed of sound in an alcohol at a temperature of 20°C is 1260 m/s. The density of the alcohol at that temperature is At a temperature of 20°C, find the bulk modulus of the alcohol.

Intensity: Calculate the light intensity 1.51 m from a light bulb that emits 100 W of visible light, assuming that the light radiates uniformly in all directions.

Standing waves on a string: A thin 2.00-m string of mass 50.0 g is fixed at both ends and under a tension of 70.0 N. If it is set into small-amplitude oscillation, what is the frequency of the first harmonic mode?

Standing waves on a string: A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 10³ kg/m³. What is the tension in the wire?

Intensity: The intensity of sunlight falling on the earth is about 1.4 kW/m² (before any gets absorbed by our atmosphere) . At what rate does the sun emit light energy? (The earth-sun distance = 1.5 × 10⁸ km and the earth's radius = 6.4 × 10³ km.)

Speed of sound: The density of aluminum is 2.7 × 10³ kg/m³. The speed of longitudinal waves in an aluminum rod is measured to be 5.1 × 10³ m/s. What is the value of Young's modulus for this aluminum?