Mastering physics Problem 20.58 tension and wave velocity

[Second_addition]

Homework Statement

The figure shows two masses hanging from a steel wire. The mass of the wire is 60.5 g . A wave pulse travels along the wire from point 1 to point 2 in 24.0 ms .
https://session.masteringphysics.com/problemAsset/1383975/6/knight_Figure_20_80.jpg
//What is mass m?

Homework Equations

mass/(length of wire)=linear density
velocity=sqrt(Tension/Linear density)
sin(theta)*tension=downward force of weight
distance/time=velocity

The Attempt at a Solution

.0605kg/8meters=.00756 linear density
4m/24ms=166.7m/s
166.7m/s=sqrt(Tension/.00756)
27777.8=tension/.00756
210=tension now I get stuck here...
Sin(40)*210=135
135/9.8=14 which is not the right answer is it 7 each and 14 combined Thank you for any assistance I know I'm not super far off just missing how to translate tension force in the whole string into downward force of both those weights

 

[Nathanael]

It's basically two separate problems:
1.) Find the tension supposing you know the speed the waves travel (which you've done)
2.) Find the tension supposing you know the mass m
Then you can put the two together to find m.

You're going to need to attack this second problem with a little more care.
If you still agree with your method, then explain it in more detail; all you've said about this second problem is "Sin(40)*210=135"

(And if you can, please use letters, for example type "sin(40)*T=mg" instead of "sin(40)*210=135")

 

[Second_addition]

I reviewed tension problems diagrams some and am still not figuring out 2. Tension=sin(40)*mass*gravity if the rope was hanging from the ceiling. well the tension in the rope is the reaction to the two masses and because the angles are the same and the masses are the same and the distances are the same the some of the tension force must be the sum of the forces due to gravity of the weights. which would mean Tension total which was 210/2 105n is the force from one weight 105/sin(40)=m*g of weight 140.918N/9.8m/s^2=14kg but 14 kg didn't work I'm not sure if I need to Subtract the tension caused by the the force of gravity on the rope itself or what. method is normally used. I also tried pretending the masses were combined 2m*sin(40)=tension still 14

 

[Nathanael]

Consider the net force on one of those two points where the three ropes meet. There is one tension pulling it to the side, there is another tension pulling it upwards-and-to-the-other-side, and there is a third tension (from the weight of m) pulling it downwards.

If this point is not moving (more specifically, not accelerating) then what must these three forces combine to?

Try to use this to find a relationship between the weight of m and the tension in the middle part of the rope.