- #1

reed2100

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## Homework Statement

The force generated by a spring is given by F=kx where k is the spring constant and x is the displacement of the spring. The spring in the below figure has a constant of 8 kN/cm. The pressures are P1=5000 kPa, P2=10,000 kPa, and P3=1000 kPa. The piston diameters are D1=8cm and D2=3cm.

**Calculate the spring displacement.**

## Homework Equations

Pressure = Force / Area

F = kx --> x = F/k

## The Attempt at a Solution

I uploaded the supplied diagram, hopefully it worked.

So I first approached this by thinking that each pressure would supply a different force by acting on a different area. P1 would push the piston plate upwards, P2 would push downwards on the spring and thus downwards on the plate. P3 would push downwards on the piston plate but not with the same area as P1 because of the space taken up by the tube I'm assuming the spring is in.

APPROACH 1 - Then I would find the total net force and it's direction, and say "

*well for the system to be in equilibrium the spring must be applying an equal force in the opposite direction, and plugging that force into the F=kx equation would allow me to find the displacement of the spring necessary to cause said force*."

Doing this I arrived at the following -

P1 supplies an upward force of 25,132.74 Newtons

P2 supplies a downward force of 7068.58 Newtons

P3 supplies a downward force of 4319.68 Newtons

The resulting net force would be 13,744.48 Newtons in the upward direction

Then, for the system to be motionless and at equilibrium, the spring must be supplying 13,744.48 Newtons in the downward direction. Plugging 13,744.48 Newtons into x = F / k gives me a

**displacement x of 1.718 cm**.

So my big issue is that I'm unsure if I'm understanding how to do this correctly, and I have another "method" of solving the problem. I'm not sure which is the correct understanding.

APPROACH 2 - The 2nd approach involved solving for the displacement x three times, once for each force supplied by its corresponding pressure. My idea here was that each force would cause a displacement of the spring, supplying an opposing force. P2 would compress the spring, P3 would elongate the spring as it's pushing on the plate it's attached to, and P1 would compress the spring by pushing upwards on the plate. I could then take the net displacement as the displacement x,

**3.484 cm**.

I'm leaning towards the first approach, but I'm unsure. I feel like anxiety is getting the better of me and I should know this easily, but this is an engineering thermodynamics class notorious for ruining semesters so I'd like to start strong no matter what. Any and all help / advice is greatly appreciated, thank you.