Tension in elastic cord at low point

[Opus_723]

Problem:

Consider the following scenario:
An elastic band is used to suspend a metal cylinder vertically. The cylinder is given an initial
downward motion so that it moves down, reaches a low point, and then moves back up again.

Your challenge task is to determine the tension in the band at the instant that the cylinder is at its low point. You may use a motion sensor.

Attempt at a solution:

I know that the tension is greater than the weight of the cylinder at the low point, but I'm not sure how to determine by how much. I thought about using kinematic equations to find the acceleration at the bottom and from that the force, but we've only learned how to deal with constant acceleration and I'm sure that the tension in the band varies with the position of the cylinder. The only thing I've thought of is to use the motion sensor to make an acceleration graph somehow and find the acceleration at the lowest x point. But that seems more numerical than what we usually do in this class, so I figure there's probably a mathematical way.

Keep in mind that this is only my first physics class. We've just finished projectile motion and moved on to forces, so we haven't gotten to any special rules for springs or anything.

I appreciate any nudges to get me on the right track, if there is another way to think about this.

 

[grzz]

Think about the forces acting on the cylinder at the lowest point.

 

[Opus_723]

I have. Weight and tension, with the tension being greater than the weight. I just don't see how that helps me determine the tension. I don't know how much greater the tension is, unless I'm missing something.

 

[grzz]

If you manage to measure the acceleration at the lowest point, then you can use
Fnet = ma to find the tension because the net force is given in terms of Tension and Weight as you correctly said.

 

[asteriatic]

Hi there,

Thank you for reaching out for help with this problem. I can provide some guidance and suggestions for approaching this problem.

Firstly, it's great that you have identified that the tension in the elastic band varies with the position of the cylinder. This is an important concept to keep in mind when solving this problem.

One approach you can take is to use the concept of conservation of energy. At the low point, the potential energy of the cylinder is at its minimum since it is at its lowest position. This means that all of the initial potential energy has been converted into kinetic energy. Therefore, you can use the equation for kinetic energy, KE = 1/2mv^2, to determine the velocity of the cylinder at the low point.

From there, you can use the relationship between force, mass, and acceleration (F=ma) to determine the acceleration of the cylinder at the low point. This acceleration will be equal to the tension in the elastic band divided by the mass of the cylinder.

Another approach you can take is to use Hooke's Law, which states that the force exerted by a spring or elastic band is directly proportional to the displacement or stretch of the spring. In this case, you can use the displacement of the elastic band at the low point to determine the tension using the equation T = kx, where T is the tension, k is the spring constant, and x is the displacement.

I hope these suggestions help you get on the right track. Keep in mind that as a scientist, it's important to think critically and explore different approaches to problem-solving. Good luck!