A cylinder is fitted with a piston that is connected to the bottom by a spring. The spring has a spring constant of k = 3600 N/m. The cylinder is open to air at the top and is frictionless. The radius of the cylinder is 25 mm and the piston has negligible mass. The air beneath the piston is completely pumped out very slowly so that it does not oscillate. (a) How far does the spring compress? (b) Find the work done by pressure in compressing the spring
To find the distance the spring compresses, we need to consider the equilibrium condition of forces acting on the piston.
(a) The equilibrium condition can be given by the equation:
Spring force = Atmospheric pressure × Area of piston
The spring force can be calculated using Hooke's Law:
F_spring = -kx
where k is the spring constant and x is the displacement of the spring.
The area of the piston is calculated as:
A = πr^2
where r is the radius of the piston.
The atmospheric pressure is the pressure acting on the top surface of the piston, which is atmospheric and equal to 1 atm.
Substituting the values into the equilibrium equation, we have:
-kx = 1 atm × π(0.025m)^2
Solving for x, we get:
x = (-1 atm × π(0.025m)^2) / k
Now we can calculate the distance the spring compresses.
(b) The work done by pressure in compressing the spring can be calculated using the equation:
Work = Force × Distance
The force acting on the piston due to pressure can be calculated as:
Pressure force = Atmospheric pressure × Area of piston
Substituting the values into the work equation, we have:
Work = (Atmospheric pressure × Area of piston) × Distance
Now we can calculate the work done by the pressure.
To solve this problem, we need to consider the equilibrium conditions of the system.
(a) To find how far the spring compresses, we need to determine the force exerted by the air pressure on the piston when it reaches equilibrium.
The force exerted by the air pressure on the piston can be determined using the formula:
Force = Pressure x Area
The pressure acting on the piston is the atmospheric pressure, which can be approximated as 101,325 N/m².
The area of the piston can be calculated using the formula for the area of a circle:
Area = π x radius²
Given that the radius of the cylinder is 25 mm, which is equivalent to 0.025 m, we can substitute this value into the equation to find the area.
Area = π x (0.025 m)²
Now, we can substitute the values we have into the formula:
Force = Pressure x Area
= 101,325 N/m² x (π x (0.025 m)²)
After calculating this expression, we obtain the force exerted by the air pressure on the piston. This force causes the spring to compress.
(b) To find the work done by pressure in compressing the spring, we can use the formula:
Work = (1/2) x k x (compression distance)²
Here, k represents the spring constant, and we are given that the spring constant is 3600 N/m. The compression distance represents the distance that the spring is compressed.
We need to find the compression distance. Since the system is at equilibrium, the force exerted by the spring must balance out the force exerted by the air pressure. Thus, we can set up the equation:
Force exerted by spring = Force exerted by air pressure
The force exerted by the spring can be calculated using Hooke's Law:
Force exerted by spring = k x compression distance
Equating this to the force exerted by the air pressure, we can solve for the compression distance:
k x compression distance = Force exerted by air pressure
Once we have the compression distance, we can substitute it into the equation for work to find the work done by pressure in compressing the spring.