Solve 2 Mass String Remote Control Problem

[AbsolutDreamcast]

Your roommates have lost the TV remote control, and no amount of searching can find it. Rather than buy a new one, you build a low-cost replacement. You attach one end of a small lever mechanism to the TV's channel-changing button. You plan to attach the other end of the lever to a 3.0-m long string that will run from the TV to the couch. When you pull the string tight and pluck your end of the string, a wave will travel down the string and trigger the lever, changing the channel. Your design assumes that you will disturb the string vertically by only 5.0 mm when you pluck it (so that using your "remote" will not require undue exertion) and that your wave will take only 0.20 s to travel horizontally along the string from your hand to the lever (to give quick channel changes). Unfortunately, your roommates have also taken most of your supply of string. All you can find around the apartment are two different pieces of 1.5-m long string; one piece has a mass of 90 g, while the other piece has a mass of just 10 g. You tie the two pieces together to make a combined 3.0-m long string and attach one end of the combined string to the lever mechanism. You then take the other end in hand and head for the couch.


so basically, I understand that the velocity will not be the same through the pieces of string, it will vary. I also understand v=sqrt(F/mu) and find the linear densities of each portion of the string. My question is can the average velocity be used to help find the tension, or should I try and use an equation for SHM, with Acos(kx-wt) or something to that extent? Any help would be greatly appreciated...

 

[AbsolutDreamcast]

nm, i figured it out

 

[M98Ranger]

There are a few different approaches you could take to solve this problem, but one way to approach it would be to use the equation for wave velocity, v = √(T/μ), where T is the tension in the string and μ is the linear density. Since the total length of the string is 3.0 m, we can set up an equation with the combined linear density and the average velocity (assuming the wave travels at a constant speed) to solve for the tension in the string.

Using the given information, we know that the average velocity is 0.20 m/s and the combined linear density is (90 g + 10 g)/3.0 m = 33.3 g/m. Plugging these values into the equation, we get:

0.20 m/s = √(T/33.3 g/m)

Squaring both sides and solving for T, we get T = 13.32 g⋅m/s².

Now, to find the tension in each individual piece of string, we can use the equation T = μv², where μ is the linear density of the specific piece of string and v is the average velocity. Plugging in the values for each piece of string, we get:

For the 90 g piece: T = (90 g/1.5 m)(0.20 m/s)² = 24 g⋅m/s²
For the 10 g piece: T = (10 g/1.5 m)(0.20 m/s)² = 2.67 g⋅m/s²

So, the tension in the 90 g piece is about 9 times greater than the tension in the 10 g piece. This means that the wave will travel faster through the 90 g piece, but since the wave is triggered by the plucking of the 10 g piece, the overall wave speed will be closer to the speed through the 10 g piece.

As for using an equation for simple harmonic motion, that could also be a viable approach. You could set up an equation for SHM with the amplitude of 5.0 mm and the period of 0.20 s, and then use the equation for wave velocity (v = ωA) to solve for the angular frequency, ω. From there, you could use the equations for tension in a string (T = μω²A²