Waves on a String: Mass & Standing Wave Mode

[danago]

[SOLVED] Waves On A String

In the figure below, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L=1.2m, linear density 1.6 g/m, and the oscillator frequency 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m=1kg?

http://edugen.wiley.com/edugen/courses/crs1650/art/images/halliday8019c16/image_t/tfg044.gif

Part A
For the string to be vibrating in its fourth harmonic, the wavelength of the wave must be half of the string length so that two full waves can fit on the piece of string simultaneously.

$$\lambda = \frac{L}{2} = 0.6m$$


We can then use this to calculate the velocity of a wave in the string:

$$<br /> v = \lambda f = 0.6(120) = 72ms^{ - 1}$$

Which allows the strings tension to be calculated:

$$v = \sqrt {\frac{T}{\mu }} \Rightarrow T = v^2 \mu = 8.2994N$$


The mass of the block is therefore 0.85kg.

Part B
Knowing the mass of the block (thus the tension), the velocity of a wave in the string can be calculated:

$$v = \sqrt {\frac{T}{\mu }} = \sqrt {\frac{{9.81}}{{0.0016}}} \approx 78.3ms^{ - 1}$$



The wavelength of a wave with frequency of 120Hz can then be calculated:

$$<br /> \lambda = \frac{v}{f} = \frac{{78.3}}{{120}} = 0.6525m$$


This wavelength does not corrospond to a specific harmonic number, so would this therefore mean that the 120Hz wave does not create any standing wave in the string?

 

[physixguru]

What is Your take on this:
A specific harmonic number is not necessary for resonance.

 

[danago]

What is Your take on this:
A specific harmonic number is not necessary for resonance.

I guess that's where I am a little confused.

From what i understand, a specific harmonic number is required for resonance.

 

[Doc Al]

In the figure below, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation , linear density , and the oscillator frequency . The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if ?

It's difficult to comment on your work since essential information is missing from the problem statement. (When you cut and pasted, some numbers must have gotten lost.)

 

[danago]

It's difficult to comment on your work since essential information is missing from the problem statement. (When you cut and pasted, some numbers must have gotten lost.)

Oh sorry, i didnt realize I've fixed it now.

 

[Doc Al]

This wavelength does not corrospond to a specific harmonic number, so would this therefore mean that the 120Hz wave does not create any standing wave in the string?

Sounds right to me.

 

[danago]

Sounds right to me.

Ok, good to hear :)

Thanks for the confirmation.