4) A 10 cm cylinder chamber has a 5cm diameter piston attached to one end. The piston is connected to an ideal spring with a spring constant of 10N/cm. Initially the spring is not compressed but is latched in place so that it cannot move. The cylinder is filled with a gas to a pressure of 5x105 Pa. Once the gas cylinder is at this pressure, the spring is unlatched. Find the compression in the spring Δx taking the temperature of the gas remains constant.
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To find the compression in the spring (Δx), we can use the equation for the force exerted by a gas:
Force = Pressure * Area
First, we need to find the area of the piston. The diameter of the piston is 5 cm, so the radius (r) is half of that, which is 2.5 cm or 0.025 m.
Area = π * (radius)^2
Area = π * (0.025)^2
Area ≈ 0.00196 m^2
Now, we can calculate the force exerted by the gas:
Force = Pressure * Area
Force = 5x10^5 Pa * 0.00196 m^2
Force ≈ 980 N
The force exerted by the gas corresponds to the force exerted by the spring when it is compressed. We can use Hooke's Law to find the compression in the spring:
Force = Spring constant * Compression
980 N = 10 N/cm * Compression
Since the spring constant is in N/cm and we want the force in N, we need to convert the spring constant to N/m:
Spring constant = 10 N/cm * (1 m/100 cm)
Spring constant = 0.1 N/m
Now we can solve for the compression (Δx):
Compression = Force / Spring constant
Compression = 980 N / 0.1 N/m
Compression = 9800 m
Therefore, the compression in the spring is 9800 m.