A cylinder is fitted with a piston, beneath which is a spring, as in the drawing. The cylinder is open 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 0.023 m.

(a) When 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?

A) Pressure=Force/Area

101,300=F/pi(.023)^2
Solve for f, then plug F into
F=kx.
k=4100
solve for X

B)work=change in enegy,
measure SPE (spring potential energy)
SPE=.5Kx^2

X is the answer from part A

(a) Well, this sounds like quite a "pressing" question! To determine how much the atmospheric pressure compresses the spring, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement. The formula for the force exerted by a spring is F = kx, where F is the force, k is the spring constant, and x is the displacement.

Now, when the air beneath the piston is completely pumped out, the only force acting on the spring is the atmospheric pressure pushing down on the piston. This force can be calculated using the formula P = F/A, where P is the pressure, F is the force, and A is the area.

Since the only force acting on the spring is the atmospheric pressure, we can equate the force from Hooke's law to the force from the atmospheric pressure: kx = P * A.

Rearranging the equation, we can solve for the displacement x: x = P * A / k.

To find the pressure caused by the atmosphere, we can use the standard atmospheric pressure, which is approximately 101325 pascals.

The area A can be calculated using the formula A = π * r^2, where r is the radius of the piston given as 0.023 m.

By substituting the known values into the equation, we get: x = (101325 * π * (0.023^2)) / 4100.

Calculating that, we find that the atmospheric pressure causes the spring to compress by approximately x = 0.041 m.

(b) Now that we know how much the spring compresses, we can determine the work done by the atmospheric pressure. The work done by a force is given by the formula W = F * d, where W is the work, F is the force, and d is the displacement.

Again, the force we're looking for is the atmospheric pressure, P = F / A. The work done can be calculated using W = P * A * x, where x is the displacement that we calculated in part (a).

By substituting the known values into the equation, we get: W = (101325 * π * (0.023^2)) * (0.041).

Calculating that, we find that the atmospheric pressure does approximately W = 25 joules of work in compressing the spring.

So, the atmospheric pressure is quite the hard worker, compressing the spring and doing some impressive work!

To find the answers to these questions, we need to use the concept of equilibrium. In equilibrium, the net force on the piston is zero, which means the spring force exactly balances the atmospheric pressure force.

(a) When the air beneath the piston is completely pumped out, the only force acting on the piston is the atmospheric pressure pushing downwards. The force due to the atmospheric pressure can be calculated using the formula:

Force = pressure * area

Here, the area of the piston is given by:

Area = π * (radius^2)

Given that the radius of the piston is 0.023 m, we can find the area as follows:

Area = π * (0.023^2) = 0.00166 m^2

The atmospheric pressure is approximately 101,325 Pa. Therefore, the force exerted by the atmospheric pressure is:

Force = 101,325 Pa * 0.00166 m^2 = 168.1119 N

Since the spring force and atmospheric pressure force balance each other, the amount of compression of the spring can be determined using Hooke's Law:

F = k * Δx

Where F is the force applied to the spring, k is the spring constant, and Δx is the displacement or compression of the spring. Rearranging the equation, we can solve for Δx:

Δx = F / k

Δx = 168.1119 N / 4100 N/m = 0.041 m (rounded to 3 decimal places)

Therefore, the atmospheric pressure causes the spring to compress by approximately 0.041 m.

(b) To find the work done by the atmospheric pressure in compressing the spring, we can use the formula:

Work = force * displacement

The force exerted by the atmospheric pressure is 168.1119 N, and the displacement or compression of the spring is 0.041 m. Therefore:

Work = 168.1119 N * 0.041 m = 6.8863 J (rounded to 4 decimal places)

Therefore, the atmospheric pressure does approximately 6.8863 Joules of work in compressing the spring.

To solve this problem, we can use the principles of equilibrium and the relationship between force, pressure, and area.

(a) To find the compression of the spring caused by the atmospheric pressure, we need to calculate the force exerted by the atmospheric pressure on the piston.

1. Determine the area of the piston:
The area of a circle can be calculated using the formula A = πr^2, where r is the radius of the piston.
Given the radius of the piston is 0.023 m, we can calculate the area as follows:
A = π(0.023)^2 = 0.00166 m^2

2. Calculate the force exerted by the atmospheric pressure:
The force exerted by the atmospheric pressure is given by the formula F = P × A, where P is the pressure and A is the area.
The pressure exerted by the atmosphere is typically around 101,325 Pa.
Substituting the values, we can calculate the force as follows:
F = 101,325 Pa × 0.00166 m^2 = 168.04 N

3. Calculate the compression of the spring:
The force exerted by the spring can be calculated using Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from its equilibrium position.
The formula for the force exerted by a spring is given by F = kx, where k is the spring constant and x is the displacement.
Rearranging the formula, we can solve for x:
x = F / k

Substituting the values, we can calculate the compression of the spring as follows:
x = 168.04 N / 4100 N/m = 0.041 m (rounded to three decimal places)

Therefore, when the air beneath the piston is completely pumped out, the atmospheric pressure causes the spring to compress by approximately 0.041 m.

(b) To calculate the work done by the atmospheric pressure in compressing the spring, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.

The work done by a force can be calculated using the formula W = F × d, where W is the work done, F is the force, and d is the displacement.

In this case, the atmospheric pressure force causes the spring to compress by a distance of 0.041 m, as calculated in part (a).
Substituting the values, we can calculate the work done by the atmospheric pressure as follows:
W = 168.04 N × 0.041 m = 6.887 N·m (also known as Joules)

Therefore, the atmospheric pressure does approximately 6.887 Joules of work in compressing the spring.