+502 a rubber rope of mass m and length x is stretched by a tension force T. When the rope is moved up and down a wave moves along the rope with a velocity v. Which equation is homogeneous and thus could be correct?
A) v2 = T mx -1
B) v2 = T m-1 x-1
C) v2 = T m-1 x
D) v2 = T m x
+267 One way to think about this type of questions is to make sure that the units on both sides are equal.
Consider the left hand side of the equation which is always velocity squared. Since [v] = m^1 * s^(-1) we know that the unit for v^2 is m^2 * s(-2)
Consider the right hand side, all of them have a force T which has the unit kg^1 * m^1 * s^(-2)
Now let's look closer at the available options. In order for the right hand side be equal to the left hand side we need to get rid of one unit of kg and get one more unit of m.
If you look at the options m happens to be the mass and x the length. Therefore we can see that dividing by one m we can balance the kg part and by multiplying one x we will get an additional m(meters) in the equation. Therefore the answer C has to be correct, we can double check this by putting in the units of all values to check:
\(\frac{\text{m}^2}{\text{s}^2}= \frac{\text{kg}^1 \space \text{m}^1}{\text{s}^2} \cdot \text{kg}^{-1} \cdot \text{m}^1\)
We can see that it works out and gives us the correct units!
