Displacement of a spring within a piston

[reed2100]

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.

 

[TSny]

Your second approach should also work. Check to make sure you correctly decided if the spring tends to elongate or compress due to each force individually.

 

[reed2100]

Your second approach should also work. Check to make sure you correctly decided if the spring tends to elongate or compress due to each force individually.

So going off your suggestion -

If I go back to APPROACH 2 and look at the amounts of displacement per pressure - I originally wanted to say that P2 at the top of the system pushes downward, P3 in the middle of the system also pushes downward, and P1 at the bottom of the system pushes upward.

If I take these displacements purely in the direction of the force and find their sum I get a net result of
3.14 - .539 - .883 = 1.718 cm, the same as in the first method.

This is reassuring, but does it just mean that I interpreted the spring's "displacement" incorrectly? I thought that P2 pushing downward would compress the spring, causing a displacement from it's normal form, and P3 would pushing downward would elongate the spring, stretching it. I'm assuming I imagined it incorrectly - the spring compresses and it's "net" location in 3d space has shifted downward due to P2, then P3 elongates it and drags the spring's "net" location down further. P1 then pushes back upward, compressing the spring, and pushes it's location upward. When you sum the displacements, you get 1.718 cm...

I think I've got it, does that all sound correct?

 

[TSny]

That sounds correct. The drawing was not real clear. I assume that the lower end of the spring presses against (or is attached to) the piston while the upper end of the spring presses against (or is attached to) the container.

 

[parash shrestha]

i am clear with the equation setup you used to find F3. Could you post the equation

 

[parash shrestha]

i meant i am not clear with the equation setup for finding F3

 

[TSny]

I'm not sure what you are asking. Could you show your attempt at finding the equation for F3 so that I know what you mean?

 

[SimenFoyn]

I think this method makes a lot of sense, but not sure if its the intended way or not.

 

[Chestermiller]

So going off your suggestion -

If I go back to APPROACH 2 and look at the amounts of displacement per pressure - I originally wanted to say that P2 at the top of the system pushes downward, P3 in the middle of the system also pushes downward, and P1 at the bottom of the system pushes upward.

If I take these displacements purely in the direction of the force and find their sum I get a net result of
3.14 - .539 - .883 = 1.718 cm, the same as in the first method.

This is reassuring, but does it just mean that I interpreted the spring's "displacement" incorrectly? I thought that P2 pushing downward would compress the spring, causing a displacement from it's normal form, and P3 would pushing downward would elongate the spring, stretching it. I'm assuming I imagined it incorrectly - the spring compresses and it's "net" location in 3d space has shifted downward due to P2, then P3 elongates it and drags the spring's "net" location down further. P1 then pushes back upward, compressing the spring, and pushes it's location upward. When you sum the displacements, you get 1.718 cm...

I think I've got it, does that all sound correct?

Where is the spring anchored? My assessment is that it is anchored to the case at its top end.