What Is the Linear Density of the String in a Standing Wave System?

[Puchinita5]

Homework Statement

In Fig. 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.The separation L between P and Q is 2.30 m, and the frequency f of the oscillator is fixed at 142 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 standing wave appears when the mass of the hanging block is 3503.5 g or 2574.0 g, but not for any intermediate mass. What is the linear density (in g/m) of the string?

Homework Equations

image: http://edugen.wiley.com/edugen/courses/crs1650/art/qb/qu/c16/pict_16_60.gif

The Attempt at a Solution

i'm confused because if two masses can make a standing wave, that would suggest two different tensions as well... and wouldn't this also suggest two different linear desnities?? i have no idea how to go about this problem...

if i were to guess, i would use f=v/wavelength to get v...but i don't know if I could assume the wavelength because the way the figure is drawn...because in the figure, the wavelength is obvious, but maybe the figure is just an example and not actually representative of what's happening...but if I do use the figure, then i can solve for v...then with v=(T/u)^.5...where T is tension and u is linear density, then i can solve for desnity, except I have two tensions to pick from...

other than that I'm lost...any help?

 

[rl.bhat]

If there are n loops in the standing wave, then L = n*λ/2. Or λ = 2L/n.
So v = f*λ = f*2L/n = ( T/μ )^0.5
Write two equations, one for T = 3503.5 g and other for 2574 g for the number of loops n and n + 1 respectively. Solve for n. Substitute it in one of the equations to find the linear density.

 

[Puchinita5]

okay, solving for N, doesn't that give you two results for N because i ended up with a quadratic equation, how do you know which N to use?

 

[Puchinita5]

oh wait nevermind, i did something silly