Why Does the Textbook Solution Use Average Pressure for Piston Work Calculation?

In summary: So with initial and final pressures I can get two equations (using force balance) with two unknowns (m and k) and solve for both k and m?mg=Pi, the initial pressure. You have three unknowns, A, k and x(final), but A will cancel. Yes, that is correct.
  • #1
theBEAST
364
0

Homework Statement


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Here is the solution from the textbook:
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I don't understand how they can assume a linear relationship. In fact I don't even understand the math... Why does the solution take the average pressure? Isn't the relationship between pressure and volume PV^n = constant?
 
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  • #2
The pressure increases linearly with the volume because of the spring. If the cross section of the piston is A, and the spring gets shorter by x, the change of the volume is Ax, and the spring force kx balances the increment of pressure.

ehild
 
  • #3
ehild said:
The pressure increases linearly with the volume because of the spring. If the cross section of the piston is A, and the spring gets shorter by x, the change of the volume is Ax, and the spring force kx balances the increment of pressure.

ehild

Can this be represented mathematically? For example can you solve for x and A and have the math work out at the end? I am trying to do this because I am still a bit confused.
 
  • #4
theBEAST said:
Can this be represented mathematically? For example can you solve for x and A and have the math work out at the end? I am trying to do this because I am still a bit confused.

Yes, it can be done. Write up the volume in terms of x, change of length of the spring.
Write the equation of balance between the increment of pressure of the steam and the pressure corresponding to the spring force.

ehild
 
  • #5
ehild said:
Yes, it can be done. Write up the volume in terms of x, change of length of the spring.
Write the equation of balance between the increment of pressure of the steam and the pressure corresponding to the spring force.

ehild

My friend and I tried doing what you said but we are still pretty stuck at where to go next.

We did come to agreement on the fact that the spring is linear since you get some relationship P(x)*A = kx + mg.

So you see P(x) is linear.
 
  • #6
Yes, P(x) is linear in x. Find the work done by the steam. You need the P(V) function. How is the volume related to x?

ehild
 
  • #7
ehild said:
Yes, P(x) is linear in x. Find the work done by the steam. You need the P(V) function. How is the volume related to x?

ehild

So we know:

P(x)*A = kx + mg

W = ∫P(x)dV = = ∫P(x)Adx = ∫(kx + mg) dx

However what is k and m? We are not sure how to figure these two unknowns out.
 
  • #8
You know the initial and final volumes, so what are the limits of integration with respect to x?

You can get k from the initial and final pressures.

ehild
 
  • #9
ehild said:
You know the initial and final volumes, so what are the limits of integration with respect to x?

You can get k from the initial and final pressures.

ehild

So with initial and final pressures I can get two equations (using force balance) with two unknowns (m and k) and solve for both k and m?
 
  • #10
mg=Pi, the initial pressure. You have three unknowns, A, k and x(final), but A will cancel.

ehild
 

Related to Why Does the Textbook Solution Use Average Pressure for Piston Work Calculation?

1. What is work done on a piston by steam?

The work done on a piston by steam is the amount of energy transferred to the piston through the expansion of the steam. It is a measure of the force applied by the steam on the piston multiplied by the distance the piston moves.

2. How is work done on a piston by steam calculated?

The work done on a piston by steam can be calculated by multiplying the pressure of the steam by the change in volume of the piston. This is known as the formula for work, W = PΔV, where W is work, P is pressure, and ΔV is change in volume.

3. What is the relationship between work done and heat in a piston?

The work done on a piston by steam is directly related to the heat transferred to the steam. This is because the heat energy added to the steam is converted into work as the steam expands and pushes on the piston.

4. Why is the work done on a piston by steam important?

The work done on a piston by steam is important because it is a measure of the power and efficiency of an engine. The more work that can be done by the steam, the more energy that can be converted into useful work, such as turning a turbine.

5. How does the work done on a piston by steam affect the performance of an engine?

The work done on a piston by steam is a crucial factor in determining the performance of an engine. A higher amount of work done on the piston means more power and efficiency, resulting in a more powerful and efficient engine. It is also important for maintaining the desired speed and torque of the engine.

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