Falling string: Force and momentum changing over time

In summary: The acceleration of the string when it is first released is equal to g, the acceleration due to gravity. The acceleration of the string (portion above the floor) just prior to the last part hitting the floor is also equal to g. At this point, the velocity of the string is equal to the velocity of the last part, which is equal to the speed at which the entire string was falling. I hope this helps to clarify the questions and provide some guidance for your solutions. Remember to always carefully consider the variables involved and use the appropriate equations to solve the problem. Good luck! In summary, the conversation discusses a flexible string of length L and mass M
  • #1
a4rino
3
0

Homework Statement



Consider a flexible string of length L and mass M that is held vertically so that its bottom just touches the floor. The string is then dropped. Let the position of the top of the string be y and the position of the floor be y = 0.

1.1 Is every piece (for y > 0) of the string moving at the same speed when it falls (you might want to consider whether the string remains straight)?


1.2 Write the mass of the string above the floor in terms of y, L, M and g.


1.3 Write the momentum of the string at time t in terms of y, dy/dt, L, M and g.


1.4 Write the momentum of the string at a time t + dt using the variables above. Note that both the length of the string and its velocity are changed from what they are at time t.


1.5 Write external force on the string at time t in terms of y, L, M and g.


1.6 Using the time rate of change of momentum determined above in Newton's 2nd law, write the equation for the second derivative of y with respect to time, d2y/dt2.


1.7 What is the normal force exerted by the floor on the rope (note that it depends on time)?


1.8 What is the acceleration of the string when it is first released? What is the acceleration of the string (portion above the floor) just prior to the last part hitting the floor? What is the velocity of the string at this point?



Homework Equations



p=mv
F=ma
v=dy/dt
F=dp/dt

The Attempt at a Solution



1.1 Yes, every piece is moving at the same speed as it falls. The string stays straight.

1.2 This seems easy enough:

m=M*(y/L)

where M is the total mass of the string, and y/L is the ratio of the string's height above the floor to its total length.

1.3 I'm not so sure about this one, but it seems like since p=mv, we would have:

p=M*(y/L)*(dy/dt)

Taking this a step farther, we can deduce that since dy/dt=v, and v=v_0+a*t, then dy/dt=g*t, where g is acceleration due to gravity. I'm not sure if that's right, but it seemed to make sense, and it gave me this:

p=g*t*M*(y/L)

1.4 I'm confused about this question. Do they just want me to substitute t+dt in for t and show that y changes by some dy as well? I don't understand...

1.5 Well I think F=dp/dt is involved here. Would I just differentiate my solution for 1.3 with respect to time? Also, would I have to do anything special with y, since it is changing also, or can I leave it alone?

Then at this point I don't think I can even speculate on parts 1.6 through 1.8 until I get the first five parts taken care of. Thanks for the help.
 
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  • #2


Hello and thank you for your post. I will attempt to answer your questions and provide some guidance for your solutions.

1.1 You are correct, every piece of the string is moving at the same speed as it falls. This is due to the fact that the string remains straight and there is no external force acting on it to cause it to accelerate or decelerate.

1.2 Your solution is correct. The mass of the string above the floor can be written as M*(y/L), where M is the total mass of the string and y/L is the ratio of the string's height above the floor to its total length.

1.3 Your solution is also correct. The momentum of the string at time t can be written as p=M*(y/L)*(dy/dt), where M is the total mass of the string, y/L is the ratio of the string's height above the floor to its total length, and dy/dt is the velocity of the string at time t.

1.4 This question is asking you to consider the change in momentum of the string at a time t+dt. This can be written as p(t+dt)=M*(y+dy)/(L+dl)*(dy/dt+dv/dt), where dy is the change in the string's height and dv is the change in the string's velocity. This equation is showing that both the length and velocity of the string change from what they were at time t.

1.5 Your solution is correct. The external force on the string can be written as F=dp/dt=M*(y/L)*(g*t), where M is the total mass of the string, y/L is the ratio of the string's height above the floor to its total length, and g is the acceleration due to gravity.

1.6 Using Newton's second law, F=ma, we can write the equation for the second derivative of y with respect to time as d2y/dt2=F/m=M*(y/L)*(g*t)/M*(y/L)=g*t. This shows that the acceleration of the string is constant and equal to g.

1.7 The normal force exerted by the floor on the rope can be written as N=M*(y/L)*g, where M is the total mass of the string, y/L is the ratio of the string's height above the floor to its total length, and g is the acceleration due to gravity. This force is dependent
 
  • #3


Hello,

I would like to provide some clarification and guidance to help you with your questions.

1.1 Yes, every piece of the string is moving at the same speed as it falls. This is because the string is falling in a straight line, and all points along a straight line have the same velocity.

1.2 You are correct in your answer for the mass of the string above the floor. It is important to note that this mass will change as the string falls, since the length y of the string above the floor is decreasing.

1.3 You are on the right track with your answer, but there are a few things to consider. Firstly, the momentum of the string is a vector quantity, so you will need to include a direction in your equation. Secondly, the velocity of the string is not constant as it falls, so you will need to take this into account. The correct equation for the momentum of the string at time t is:

p = M * (y/L) * (dy/dt) * (-j)

where j is a unit vector pointing in the negative y direction. This takes into account the changing velocity of the string as it falls.

1.4 This question is asking you to write the momentum of the string at a time t + dt, so you will need to use your answer from 1.3 and substitute in t + dt for t. This will give you the momentum of the string at a slightly later time, taking into account the change in both the length and velocity of the string.

1.5 You are correct that F = dp/dt, but in this case, the external force on the string is simply its weight, which is given by mg. So the external force on the string at time t is:

F = M * (y/L) * g * (-j)

1.6 This question is asking for the equation for the second derivative of y with respect to time, which is the acceleration of the string. To find this, you can use Newton's second law, F = ma. Substitute in your answer for the external force in terms of y and t, and use your answer from 1.3 for the momentum of the string.

1.7 The normal force exerted by the floor on the rope will depend on the weight of the string at that particular time. So you can use the equation for the weight of the string, mg, and substitute
 

1. How does the force on a falling string change over time?

The force on a falling string changes over time due to the acceleration of gravity. As the string falls, the force of gravity pulling it downwards increases, resulting in an increase in the force on the string.

2. Does the momentum of a falling string change over time?

Yes, the momentum of a falling string changes over time due to the changing force acting on the string. As the force increases, the velocity and momentum of the string also increase.

3. How does the length of the string affect the force and momentum of a falling string?

The length of the string does not directly affect the force and momentum of a falling string. However, a longer string may experience more air resistance, which can affect the speed and momentum of the string as it falls.

4. What factors can influence the force and momentum of a falling string?

The main factors that can influence the force and momentum of a falling string include the mass of the string, the acceleration of gravity, and any external forces such as air resistance or friction.

5. How can we measure the force and momentum of a falling string?

The force and momentum of a falling string can be measured using various tools such as force sensors, accelerometers, and high-speed cameras. These tools can help record the acceleration, velocity, and displacement of the string to calculate the force and momentum over time.

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