Question

In: Physics

Cousin Throckmorton is playing with the clothesline. One end of the clothesline is attached to a vertical post.

 

Cousin Throckmorton is playing with the clothesline. One end of the clothesline is attached to a vertical post. Throcky holds the other end loosely in his hand, so that the speed of waves on the clothesline is a relatively slow 0.740m/s. He finds several frequencies at which he can oscillate his end of the clothesline so that a light clothespin 43.0cm from the post doesn't move.

 

What are these frequencies? 

 (0.860 kHz)n, n=1, 2, 3...

 (0.860Hz )n, n=1, 2, 3...

 (1.72 kHz)n, n= 1, 2, 3...

 (1.72Hz )n, n= 1, 2, 3...


Solutions

Expert Solution

Concepts and reason

Use the concepts of vibrations in a stretched string to solve the given problem.

Consider the vibrations in a stretched string and find the fundamental frequency using the frequency formula. And then, find the few other frequencies (first overtone and second overtone, etc.) by multiplying the fundamental frequency with number of loops.

Fundamentals

The frequency ff expression is,

f=vλf = \frac{v}{\lambda }

Here, λ\lambda is the wavelength and vv is the speed of the wave.

When the string vibrates with fundamental frequency, the string forms only one loop.

So, the length L of the string between the two fixed points is,

L=λ2λ=2L\begin{array}{c}\\L = \frac{\lambda }{2}\\\\\lambda = 2L\\\end{array}

The fundamental frequency is,

f1=vλ{f_1} = \frac{v}{\lambda }

Replace λ\lambda with 2L2L .

f1=v2L{f_1} = \frac{v}{{2L}}

When the two loops are formed between the two fixed points in the stretched string, then the length of the string is,

L=(2)λ2λ=L\begin{array}{c}\\L = \left( 2 \right)\frac{\lambda }{2}\\\\\lambda = L\\\end{array}

The first overtone frequency is,

f2=vλ{f_2} = \frac{v}{\lambda }

Replace λ\lambda with LL .

f2=vLf2=(2)v2L\begin{array}{l}\\{f_2} = \frac{v}{L}\\\\{f_2} = \left( 2 \right)\frac{v}{{2L}}\\\end{array}

Similarly, when the string vibrates with three loops, then

L=(3)λ2λ=2L3\begin{array}{c}\\L = \left( 3 \right)\frac{\lambda }{2}\\\\\lambda = \frac{{2L}}{3}\\\end{array}

The second overtone frequency is,

f3=vλ{f_3} = \frac{v}{\lambda }

Replace λ\lambda with 2L3\frac{{2L}}{3} .

f3=v2L3f3=(3)v2L\begin{array}{l}\\{f_3} = \frac{v}{{\frac{{2L}}{3}}}\\\\{f_3} = \left( 3 \right)\frac{v}{{2L}}\\\end{array}

Thus, from the above discussion, it can be concluded that the harmonic frequencies of the vibrating string can be expressed as integral nn multiple of fundamental frequency. Thus,

f=nv2Lf = n\frac{v}{{2L}}

Substitute 0.740 m/s for v and 0.430 m for L.

f=n0.740m/s2(0.430m)f=n(0.86Hz)\begin{array}{c}\\f = n\frac{{0.740{\rm{ m/s}}}}{{2\left( {0.430{\rm{ m}}} \right)}}\\\\f = n\left( {0.86{\rm{ Hz}}} \right)\\\end{array}

Here, n is equal to integer values from 1, 2, 3, ……n.

Ans:

Thus, the frequencies observed are n(0.86Hz)n=1,2,3,....n\left( {0.86{\rm{ Hz}}} \right){\rm{ }}n = 1,2,3,.... .


Related Solutions

A 100 kg uniform beam is attached to a vertical wall at one end and is...
A 100 kg uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. Calculate the magnitude of the vertical component of the force that the wall exerts on the left end of the beam if the angle between the cable and horizontal is θ = 43°. The angle between the horizontal and the beam is 30 degrees.
You are pulling a rope that is attached to a sled that your cousin is riding...
You are pulling a rope that is attached to a sled that your cousin is riding on. The rope that you are pulling on is at an angle of 30 degrees above the surface of the snow. Your cousin is holding a second rope that is attached to a second sled that is carrying firewood. The rope that your cousin is pulling is parallel to the surface of the snow. The mass of your cousin and her sled is 45...
A ball is attached to one end of a wire, the other end being fastened to...
A ball is attached to one end of a wire, the other end being fastened to the ceiling. The wire is held horizontal, and the ball is released from rest (see the drawing). It swings downward and strikes a block initially at rest on a horizontal frictionless surface. Air resistance is negligible, and the collision is elastic. The masses of the ball and block are, respectively, 1.6 kg and 2.3 kg, and the length of the wire is 1.21 m....
You have a vertical metal rod, a spring with one end closed and the other end...
You have a vertical metal rod, a spring with one end closed and the other end open so that it can slide down over the rod and then fly up when released, and a meter stick. The mass of the spring is 15g and its spring constant k is 6 N/m Your group decides to test the work-energy principle by stretching the spring a distance Dy = 30 cm, releasing it from rest and measuring how high it goes. Use...
One end of a horizontal rope is attached to a prong of an electrically driven tuning...
One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at 125 Hz. The other end passes over a pulley and supports a 1.50 kg mass. The linear mass density of the rope is 5.50×10-2 kg/m .a.What is the speed of a transverse wave on the rope?b. What is the wavelength?c. How would your answers to part (A) change if the mass were increased to 3.00 kg?d. How would your answers...
One end of a cord is fixed and a small 0.400-kg object is attached to the...
One end of a cord is fixed and a small 0.400-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 1.00 m, as shown in the figure below. When θ = 18.0°, the speed of the object is 7.50 m/s. An object is swinging to the right and upward from the end of a cord attached to a horizontal surface. The cord makes an angle θ with the vertical. An...
A 16.3-kg block rests on a horizontal table and is attached to one end of a...
A 16.3-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 5.99 m/s in 1.37 s. In the process, the spring is stretched by 0.180 m. The block is then pulled at a constant speed of 5.99 m/s, during which time the spring is stretched by only 0.0584 m. Find...
A 11.4-kg block rests on a horizontal table and is attached to one end of a...
A 11.4-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 4.08 m/s in 1.13 s. In the process, the spring is stretched by 0.231 m. The block is then pulled at a constant speed of 4.08 m/s, during which time the spring is stretched by only 0.0543 m. Find...
A mass resting on a horizontal, frictionless surface is attached to one end of a spring;...
A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.7 J of work to compress the spring by 0.14 m . If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 12 m/s2. Find the value of the spring constant. Find the value of the mass.
One end of a cord is fixed and a small 0.550-kg object is attached to the...
One end of a cord is fixed and a small 0.550-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 3.00 m, as shown in the figure below. When θ = 17.0°, the speed of the object is 9.00 m/s. An object is swinging to the right and upward from the end of a cord attached to a horizontal surface. The cord makes an angle θ with the vertical. An...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT