Finding the Spring Constant of a mattress

[012anonymousx]

Homework Statement

A mattress manufacturer estimates that 20 springs are required to comfortable support a 100kg person. When supporting the person, the 20 springs are compressed 0.035m. Calculate the spring constant for one sprint.

Homework Equations

The answer is 1400N/M at the back of my textbook. How does one solve it?
Why does my solution not work?

The Attempt at a Solution

(20)Ee = Eg //The gravitational potential energy is transferred to the 20 springs.
20(0.5)(k)(x^2) = mgh
10(k)(0.035^2) = (100)(9.8)(0.035)
k = 2800N/M
So close yet so far...

 

[haruspex]

You have used conservation of energy. Think about the dynamics of that. Person let go from rest on uncompressed springs... what will happen?
You don't need to think about energy here. Just treat it as a statics question.

 

[012anonymousx]

Ah,
F = kx
k = F/x
k = mg/x
k = (100)(9.8) / 0.035 x 20
k = 1400

I still think my solution should have worked. I don't understand conceptually why energy is not conserved.

 

[haruspex]

I still think my solution should have worked. I don't understand conceptually why energy is not conserved.

As I said, you treated it as though the person was suspended horizontally, just touching the bed, then released. Suppose energy is conserved - what will happen? The person will fall, compressing the springs, but 'overshoot'. When the force from the springs matches the person's weight, the person will stop accelerating, but already has momentum and KE, so will continue to descend. When descent ceases, the springs will be applying a larger force and the person will bounce back up. If energy is conserved they will come all the way back up to starting position before descending again, and bounce forever.
The question required the point at which there was no acceleration, somewhere between the extremes of the bounce.

 

[Nickie]

I would like to point out that this problem is not solvable with the given information. In order to calculate the spring constant, we need to know the force exerted by each spring, which is not given in the problem. Additionally, the equation used in the attempt at a solution is incorrect. The equation for gravitational potential energy should be mgh, not mgh^2.

To accurately determine the spring constant, we would need to know the force exerted by each spring (which can be calculated using the given information about the number of springs and their compression) and the displacement of each spring. With this information, we can use Hooke's Law (F=-kx) to calculate the spring constant, where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.

Furthermore, the given answer of 1400N/M does not seem to be correct. Using the given information, the force exerted by each spring would be 490N (100kg x 9.8m/s^2 / 20 springs), and the displacement would be 0.035m. Plugging these values into Hooke's Law, we get a spring constant of 14,000N/M, not 1400N/M.

In conclusion, to accurately solve this problem, we would need more information and the correct equation. It is important to carefully check all equations and units to ensure an accurate solution.