Various properties of waves

[MathewsMD]

I have a few waves questions I would really like to have addressed. I'll post my logic/solution for each question and any feedback is welcome.

Q. 10: This one seems pretty straight forward but shouldn't the units be 3.0 rad/m? Since x is in meters and we're not changing the order of magnitude for any other variables, I don't quite see how the units cancel out properly...I'd just like to confirm this.

Q. 20 and 21: This seems also pretty simply but I may be missing something here. To find instantaneous velocity is to take the derivative (find slope of tangent) at a point. Looking at the question like this, my answer for 20 would be E and 21 would be D. Am I missing something fundamental here?

Q. 31: This question also didn't make complete sense to me. Looking at 3, the tension appears to be the greatest since there's one string and 2 masses for which it must be keep up. 2 looks like the lowest and 1 is intermediate, since F_T ~ v². Once again, am I missing something big here?

Any help with these questions would be great! Thanks!

 

[kontejnjer]

For Q10, you're right, the units should be rad/m.

For Q20 and Q21, try to imagine what happens to point P as the wave moves to the right. The waves are traverse in both cases, so a point "on" the wave cannot by definition have a longitudinal velocity component, just a traverse component, hence it can only go up or down.

As for Q31, I don't see any sort of picture so I can't really help out much.

 

[MathewsMD]

for q10, you're right, the units should be rad/m.

For q20 and q21, try to imagine what happens to point p as the wave moves to the right. The waves are traverse in both cases, so a point "on" the wave cannot by definition have a longitudinal velocity component, just a traverse component, hence it can only go up or down.

As for q31, i don't see any sort of picture so i can't really help out much.

q. 31

 

[kontejnjer]

If you haven't done so already, draw free body diagrams for all the cases depicted. You can then figure out the tensions in all of them. Notice that case 1 and 3 are equivalent because the forces on both ends of the rope are the same (in the first case, the wall must exert a force equal in magnitude to that of the weight of the block, otherwise it wouldn't be in equilibrium anymore).

 

[HalfLostAndSearching]

A: Hello! Thank you for reaching out with your questions about waves. I am happy to provide some clarification and feedback on your solutions.

Q. 10: You are correct in your thinking that the units should be 3.0 rad/m. When dealing with waves, we often use radians as the unit for phase, and it is important to keep track of the units in our calculations to ensure they are consistent. In this case, the units do not cancel out properly because we are not dealing with a simple multiplication or division, but rather a calculation involving trigonometric functions. So it is important to keep the units in mind and make sure they are consistent throughout the calculation.

Q. 20 and 21: Your understanding of finding instantaneous velocity by taking the derivative is correct. For question 20, the correct answer is indeed E. However, for question 21, the correct answer is actually C. This is because the velocity is the derivative of displacement, and in this case, the displacement is changing with respect to time (since the wave is moving), not with respect to position. So the correct equation would be v = -Aωsin(ωt), where ω is the angular frequency of the wave.

Q. 31: This question is asking about the tension in the string, not the force of tension. The tension in a string is directly proportional to the velocity of the wave, and inversely proportional to the mass per unit length of the string. So, in this case, the tension would be greatest in scenario 2, where the velocity is highest and the mass per unit length is lowest. In scenario 1, the tension would be intermediate, and in scenario 3, it would be the lowest. I hope this helps clarify things!

Overall, it seems like you have a good understanding of the properties of waves. Keep up the good work and don't be afraid to ask for clarification when needed!