Reflection and Transmission (Waves in strings)

[trusean]

Homework Statement

Two copper wires, one 1.0 mm in diameter and 1.0 m long, the other 2.0 mm in diameter
and 2.0 m long, are joined together end-to-end and hung vertically. In order to tension
this compound wire, a block is suspended from it. It is found that a transverse pulse takes
50 ms to travel the length of the joined wires. What is the mass of the block? The density
of copper is 8920 kg/m3. (Hint: Given the volume mass density and diameters of the
wires, how can we find their linear mass densities?)

Homework Equations

v=square root (T/Mu)
Sum of All Forces=0
V=Pir^2h
d=m/V
Where:
v=wave speed
T=Tension of string/wire
Mu=mass/length of string in units kg/m
V=volume of cylinder
d=density

The Attempt at a Solution

We started by finding Mu with the above equations to be 7.0e-3 kg/m for wire one and 2.8e-2 kg/m for wire two, respectably. What we do not understand is the relationship between the pulse time (50 milli-seconds), length of wire (3m) and the wavespeed. We think that after finding the wavespeed of the pulse we can use Newtons law to find the mass attached to the tensile forces. Any suggestions?

 

[trusean]

Perhaps using the a sine function and v(wavespeed)=lambda/T(period) we could determine the wavespeed by inputing Pi/2 for T and 50ms for lambda?

 

[ogb p]

I would approach this problem by first analyzing the given information and identifying the key variables and equations that can be used to solve for the mass of the block. From the given information, we know that the wires are joined end-to-end, which means they are connected in series and the tension in the wires will be the same. We also know the diameter and length of each wire, as well as the density of copper.

To solve for the mass of the block, we need to find the tension in the wires. This can be done using the equation v=square root (T/Mu), where v is the wave speed, T is the tension, and Mu is the mass per unit length of the wire. We can rearrange this equation to solve for T: T=v^2*Mu.

Next, we can use the equation V=Pir^2h to find the volume of each wire, where V is the volume, Pi is the constant 3.14, r is the radius, and h is the length. From the given information, we can calculate the volume of each wire to be 3.14*(0.5mm)^2*1m = 7.85e-7 m^3 for the first wire, and 3.14*(1mm)^2*2m = 6.28e-6 m^3 for the second wire.

Using the equation d=m/V, we can then find the mass of each wire by multiplying the volume by the density of copper. This gives us a mass of 7.0e-3 kg for the first wire and 2.8e-2 kg for the second wire.

Now, we can plug these values into the equation T=v^2*Mu to solve for the tension in the wires. We know that the wave speed is equal to the length of the wire (3m) divided by the time it takes for the pulse to travel that length (50 ms = 0.05 s). This gives us a wave speed of 60 m/s. Plugging in the values, we get T=60^2*(7.0e-3 + 2.8e-2) = 144 N.

Finally, we can use Newton's Second Law, where the sum of all forces is equal to the mass times the acceleration, to find the mass of the block. The only force acting on the block is the tension