Mechanical Waves and linear Density

[plzNOCarribbean]

So I read a few posts on mechanical waves and they stated that:

1) mechanical waves can be transverse or longitudinal
2) transfer energy and momentum

However, I thought mechanical waves were only waves on a rope/string? is this incorrect?

Specifically, when I think of a mechanical wave, I think about how in the EK books it talks about how you can take a rope, tie one end to a tree and hold the other end in your hand, and my moving your hand up and down you create a mechanical wave the propagates forwards, while the movement of the medium is perpendicular to the direction of the movement (and thus illustrates that ropes are transverse waves). Is this a good way to think about it or are there other kinds of medium besides a rope that could be considered mechanical waves.

Lastly, the equation V= sqr. root Tension/u shows that as tension increases the speed of a wave increases. However, if linear density increases the velocity decreases. With this in mind, what if the mass of the rope increased, would it slow down? what if the length of the rope increased, would it speed up?

just want to double check and make sure I understand this right. Does the equation basically say that lighter, longer ropes travel faster than shorter heavier ropes? Thanks for the help guys!

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[MT Headed]

So I read a few posts on mechanical waves and they stated that:

1) mechanical waves can be transverse or longitudinal
2) transfer energy and momentum

However, I thought mechanical waves were only waves on a rope/string? is this incorrect?

Specifically, when I think of a mechanical wave, I think about how in the EK books it talks about how you can take a rope, tie one end to a tree and hold the other end in your hand, and my moving your hand up and down you create a mechanical wave the propagates forwards, while the movement of the medium is perpendicular to the direction of the movement (and thus illustrates that ropes are transverse waves). Is this a good way to think about it or are there other kinds of medium besides a rope that could be considered mechanical waves.

Lastly, the equation V= sqr. root Tension/u shows that as tension increases the speed of a wave increases. However, if linear density increases the velocity decreases. With this in mind, what if the mass of the rope increased, would it slow down? what if the length of the rope increased, would it speed up?

just want to double check and make sure I understand this right. Does the equation basically say that lighter, longer ropes travel faster than shorter heavier ropes? Thanks for the help guys!

A mechanical wave can be transverse like a rope on a string, longitudinal like a sound wave, or a surface wave like water.

The general form of the wave speed equation is speed=sqrt(restoring/resistance). This applies to pendulums, sounds, ropes, water, IR spectroscopy, and a hundred other phenomena found on the MCAT. Know this equation, it is high yield. In particular, the restoring force of a rope is its tension, and the resistance to movement of a rope is its mass per unit length.

If the mass of the rope increased by making it more dense, then waves would travel slower because (mass/length) went up. If the mass of the rope increased because you have twice as much rope, then the wave speed would not change.

If the length of the rope increased because you have twice as much of the same rope, the wave speed will not change. If the length of the rope increased because you unwound its fibers and reassembled a rope of the same mass but twice the length, then waves will travel faster because now (mass/length) went down.

I am being somewhat of a pedant to cover my bases on rope length and mass. Don't get too caught up on a rope's mass or a rope's length. The important concept is (mass/length), which appears in the denominator of the wave speed equation for ropes.

 

[plzNOCarribbean]

A mechanical wave can be transverse like a rope on a string, longitudinal like a sound wave, or a surface wave like water.

The general form of the wave speed equation is speed=sqrt(restoring/resistance). This applies to pendulums, sounds, ropes, water, IR spectroscopy, and a hundred other phenomena found on the MCAT. Know this equation, it is high yield. In particular, the restoring force of a rope is its tension, and the resistance to movement of a rope is its mass per unit length.

If the mass of the rope increased by making it more dense, then waves would travel slower because (mass/length) went up. If the mass of the rope increased because you have twice as much rope, then the wave speed would not change.

If the length of the rope increased because you have twice as much of the same rope, the wave speed will not change. If the length of the rope increased because you unwound its fibers and reassembled a rope of the same mass but twice the length, then waves will travel faster because now (mass/length) went down.

I am being somewhat of a pedant to cover my bases on rope length and mass. Don't get too caught up on a rope's mass or a rope's length. The important concept is (mass/length), which appears in the denominator of the wave speed equation for ropes.

Perfect! in your example above, where you state, if the length of the rope increased because you have twice as much of the same rope, the wave speed will not change. Is this because your increasing length by a factor of 2, but at the same time your assuming that "twice as much rope" means mass x 2, and so the two cancel out and we have the same speed?

Also, thanks for clarifying the sqr root (restoring/resistance). you said this applies to hundreds of phenomena and you listed some. Basically, this applies to anything that oscillates and has wave/spring like like behavior (anything that moves from A to -A and back to A where A=max amplitude and -A=min amplitude?

the only thing i think of when I hear restoring is the restoring force F= -Kx. Are you basically saying that the velocity of a any type of mechanical wave, which results due to a disturbance in the medium will be proportional to the restoring force required to get the medium back to its original length/position?

and for resistance, this can be anything that impedes the motion? like, if we had a pendulum, the resistance in the equation would be air resistance imposing the direction of motion?

 

[MT Headed]

Perfect! in your example above, where you state, if the length of the rope increased because you have twice as much of the same rope, the wave speed will not change. Is this because your increasing length by a factor of 2, but at the same time your assuming that "twice as much rope" means mass x 2, and so the two cancel out and we have the same speed?

Also, thanks for clarifying the sqr root (restoring/resistance). you said this applies to hundreds of phenomena and you listed some. Basically, this applies to anything that oscillates and has wave/spring like like behavior (anything that moves from A to -A and back to A where A=max amplitude and -A=min amplitude?

That's a good way of thinking about it for the MCAT. While it certainly doesn't describe every wiggly phenomena, it describes every wiggly system you would encounter in an algebra based physics class.

the only thing i think of when I hear restoring is the restoring force F= -Kx. Are you basically saying that the velocity of a any type of mechanical wave, which results due to a disturbance in the medium will be proportional to the restoring force required to get the medium back to its original length/position?

Well, proportional to the square root of the restoring force. Of course, a mass wiggling back and forth on a spring isn't a mechanical wave. It doesn't transmit energy or momentum anywhere. But, (ready for something mind blowing?) remember the formula for the angular velocity of a mass wiggling on a spring? w = sqrt(k/m)? Well, the "k" of the spring is the restoring component, and the mass is what resists restoration... it's all tied together, man.

and for resistance, this can be anything that impedes the motion? like, if we had a pendulum, the resistance in the equation would be air resistance imposing the direction of motion?

The restoring component of a pendulum is gravity, but the resistance to being restored is the length of the pendulum. Angular velocity = sqrt(restoring/resistance) = sqrt(g/L).


Angular velocity = 2 x pi x frequency = 2 x pi / period. That's the source of these equations with which you might be more familiar:

period of pendulum = 2 pi sqrt(L/g)
period of mass on spring = 2 pi sqrt(m/k)

 

[plzNOCarribbean]

Perfect, thanks!

I read a lot of post recently on SDN about how we don't need to know angular velocity, angular acceleration, and moment of inertia (basically, everything associated with rotational dynamics) because its not on the AAMC topics list. Is this true? were you just giving me this info so I could understand the concept more in depth. Thanks for the explanation btw it really helped

 

[MD Odyssey]

Perfect, thanks!

I read a lot of post recently on SDN about how we don't need to know angular velocity, angular acceleration, and moment of inertia (basically, everything associated with rotational dynamics) because its not on the AAMC topics list. Is this true? were you just giving me this info so I could understand the concept more in depth. Thanks for the explanation btw it really helped

I've yet to see anything at all in any of the practice materials about it.