Force / elastic potential energy of a rubber band

[Heirot]

Suppose we have a rubber band of some elasticity k and of unstreched radius r0 (the band is always kept in the shape of a circle). What work is necessary to strech it to some larger radius r? How do we apply Hooke's law in this situation?

Thanks

 

[Heirot]

I guess I misplaced my question... Sorry about that. This really isn't homework, but I'm just curious. Nevertheless... My problem is this. If we have an unstreched band of radius r0, and corresponding circumference (the length of the band) l0, and we strech it to some radius r (length l), what's the difference in potential energy? Is it 1/2 k (l - l0)^2. That would make sense due to Hooke's law. But what about the tension in the band? Is the tension dE/dr or dE/dl? The first expresion makes sense because of the circular shape. The second one follows strictly from Hooke's law. Note that the two forces are not the same. I hope this is not confusing to you as it is to me.

Thanks

 

[baba89]

for your question! When it comes to the force and elastic potential energy of a rubber band, we can use Hooke's law to help us understand the relationship between the force applied and the resulting stretch of the band. Hooke's law states that the force applied to an elastic material, such as a rubber band, is directly proportional to the amount of stretch or compression of the material. In other words, the more force we apply to the rubber band, the more it will stretch.

In this situation, we can apply Hooke's law by using the formula F = -kx, where F is the force applied, k is the elasticity constant of the rubber band, and x is the amount of stretch. So, the force required to stretch the rubber band from its unstretched radius r0 to a larger radius r would be F = -k(r-r0).

To find the work necessary to stretch the rubber band, we can use the formula W = 1/2kx^2, where W is the work done and x is the amount of stretch. So, the work required to stretch the rubber band from its unstretched radius r0 to a larger radius r would be W = 1/2k(r-r0)^2.

It's important to note that the rubber band will also have an elastic potential energy, which is the energy stored in the band due to its stretch. This can be calculated using the formula PE = 1/2kx^2, where PE is the elastic potential energy and x is the amount of stretch. So, as we stretch the rubber band, it will have an increasing amount of elastic potential energy.

In summary, Hooke's law allows us to understand the relationship between the force applied and the resulting stretch of a rubber band. By using this law, we can calculate the force and work necessary to stretch the rubber band, as well as the elastic potential energy stored in the band. This understanding is crucial in many applications, such as designing rubber band-powered machines or understanding the behavior of elastic materials.