How to Calculate Time, Weight, and Wavelength in a String with Weight System?

[Vanessa Avila]

Homework Statement

A 1.60-m string of weight 1.30 N is tied to the ceiling at its upper end, and the lower end supports a weight W. When you pluck the string slightly, the waves traveling up the string obey the equation y(x,t)=(8.50mm)cos(172rad⋅m⁻¹(x)−2730rad⋅s⁻¹(t))

a) How much time does it take a pulse to travel the full length of the string?
b) What is the weight W?
c) How many wavelengths are on the string at any instant of time?

Homework Equations

Acos2π(x/λ- t/T)
k = 2π/λ
w = 2π/T

v =√(F/μ)

The Attempt at a Solution

I found the mass of the string by using F = mg and got 0.133 kg. With this mass, I was able to find the density (m/L) which is 0.0831m/kg

for the time, I thought i should do
2730rad/s = 2π/T
T = 2π/2730 rad/s

for how many wavelengths, I thought of doing:
172rad/m = 2π/λ
λ = 2π/172 rad/m, but that was wrong.

 

[TSny]

I found the mass of the string by using F = mg and got 0.133 kg. With this mass, I was able to find the density (m/L) which is 0.0831m/kg

OK, except for the units.

for the time, I thought i should do
2730rad/s = 2π/T
T = 2π/2730 rad/s

You calculated a time here. But is it the time that is asked for in part (b)? What does the symbol T stand for in your calculation?

for how many wavelengths, I thought of doing:
172rad/m = 2π/λ
λ = 2π/172 rad/m, but that was wrong.

I think you calculated λ correctly. But part (c) is asking for "how many wavelengths are on the string".

 

[Vanessa Avila]

OK, except for the units.

You calculated a time here. But is it the time that is asked for in part (b)? What does the symbol T stand for in your calculation?

I think you calculated λ correctly. But part (c) is asking for "how many wavelengths are on the string".

Oh T stands for the period. How do we use the period to find small t? and how do we find how many wavelengths are on the string? and my bad! I mixed up the units for density. It should be kg/m. Thanks for that!

 

[TSny]

Can you find the speed of a wave pulse using the information given? Hint: the speed of a wave pulse is the same as the wave speed of the wave y(x,t) that is given in the problem.

I think "how many wavelengths are on the string" means how many wavelengths fit into the length of the string.

 

[Vanessa Avila]

Can you find the speed of a wave pulse using the information given? Hint: the speed of a wave pulse is the same as the wave speed of the wave y(x,t) that is given in the problem.

I think "how many wavelengths are on the string" means how many wavelengths fit into the length of the string.

So it's v = λ/T ?

T = 2.302E-3
λ = 0.0365 m

v = (0.0365/2.302E-3) = 15.9 m/s

 

[TSny]

Your answer for v looks right.

 

[Vanessa Avila]

Your answer for v looks right.

Would I then solve for time by doing v = distance/time by using the length of the string as distance?
15.9 m/s = 1.60m/t
t = 1.60 /15.9 = 0.101s

 

[TSny]

Would I then solve for time by doing v = distance/time by using the length of the string as distance?
15.9 m/s = 1.60m/t
t = 1.60 /15.9 = 0.101s

OK

 

[Vanessa Avila]

OK

:) My answer for t was correct for that one. yay!

And now for the weight, I'm not sure where to start.

 

[TSny]

The weight W probably has something to do with the tension in the string.

 

[Vanessa Avila]

The weight W probably has something to do with the tension in the string.

would it be 1.30 N as well since it's pulling the weight up?

 

[TSny]

Consider a free body diagram for the object hanging on the end of the string.

 

[Vanessa Avila]

Consider a free body diagram for the object hanging on the end of the string.

Oh woop. It has Tension up and mg down. I don't have T, but i can solve that by v = √(F_t/μ)

So my F_t is 21 N which means W is also 21 N. That answr was correct! :)

Now for finding how many wavelengths, do I use the λ I solved?

 

[TSny]

Oh woop. It has Tension up and mg down. I don't have T, but i can solve that by v = √(F_t/μ)

So my F_t is 21 N which means W is also 21 N. That answr was correct! :)

Great!

Now for finding how many wavelengths, do I use the λ I solved?

Sure.

 

[Vanessa Avila]

Great!

Sure.

But what equation do I use to solve for the n numbers of wavelenghts?

 

[TSny]

Suppose the wave y(x, t) that is given in the problem exists on the string. If you took a picture of the string at a particular instant of time, how many wavelengths would you count along the string?

 

[TSny]

The problem is a little confusing. If you "pluck" the string to get a "pulse", it would not have the form y(x, t) that is given in the problem (which represents a "sinusoidal wave" spread all along the string).

 

[Vanessa Avila]

The problem is a little confusing. If you "pluck" the string to get a "pulse", it would not have the form y(x, t) that is given in the problem (which represents a "sinusoidal wave" spread all along the string).

So if it's just a pulse then it's just 1?

 

[Vanessa Avila]

The problem is a little confusing. If you "pluck" the string to get a "pulse", it would not have the form y(x, t) that is given in the problem (which represents a "sinusoidal wave" spread all along the string).

oh wait. So i have a string 1.60 m long, and each wavelength is 0.0365
should it then be
1.60m/0.0365 to find how many there are?

 

[TSny]

So if it's just a pulse then it's just 1?

But, the function y(x, t) given in the problem is not really a pulse. It is a sinusoidal wave spread along the string. So, there are a certain number of wavelengths that would occupy the string at a given instant of time.

 

[TSny]

oh wait. So i have a string 1.60 m long, and each wavelength is 0.0365
should it then be
1.60m/0.0365 to find how many there are?

Yes!

 

[Vanessa Avila]

Yes!

alright! Awesome! Thank you so much :)

 

[TSny]

You are welcome. Good work.