A cylinder is fitted with a piston, beneath which is a spring, as in the drawing. The cylinder is open to the air at the top. Friction is absent. The spring constant of the spring is 4100 N/m. The piston has a negligible mass and a radius of 0.010 m. (a) When the air beneath the piston is completely pumped out, how much does the atmospheric pressure cause the spring to compress? (b) How much work does the atmospheric pressure do in compressing the spring?
To determine the answers to these questions, we need to consider the forces acting on the system and apply principles of equilibrium and work.
(a) When the air beneath the piston is completely pumped out, the only force acting on the piston is the atmospheric pressure above it. This force can be calculated using the formula for pressure:
Pressure = Force / Area
The force exerted by the atmospheric pressure can be calculated by multiplying the pressure by the area of the piston:
Force = Pressure * Area
The area of the piston can be obtained using the formula for the area of a circle:
Area = π * (radius)^2
Substituting the given values, we have:
Area = π * (0.010 m)^2
Next, we need to calculate the atmospheric pressure, which is typically around 101,325 Pa or 101.3 kPa.
Now, we can calculate the force exerted by the atmospheric pressure:
Force = Pressure * Area
Finally, using Hooke's Law for springs, we can determine the amount of compression on the spring based on the force applied:
Force = Spring Constant * Compression
Rearranging the equation, we have:
Compression = Force / Spring Constant
Substituting the values we have calculated, we can find the compression of the spring.
(b) To calculate the work done by the atmospheric pressure in compressing the spring, we use the formula for work:
Work = Force * Distance
The distance in this case is the compression of the spring, which we calculated in part (a). Multiplying the force and the distance, we can determine the work done by the atmospheric pressure.