A 0.527 kg metal cylinder is placed inside the top of a plastic tube, the lower end of which is sealed off by an adjustable plunger, and comes to rest some distance above the plunger. The plastic tube has an inner radius of 6.22 mm, and is frictionless. Neither the plunger nor the metal cylinder allow any air to flow around them. If the plunger is suddenly pushed upwards, increasing the pressure between the plunger and the metal cylinder by a factor of 3.03, what is the initial acceleration of the metal cylinder? Assume the pressure outside of the tube is 1.00 atm.

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To find the initial acceleration of the metal cylinder, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = m * a).

1. First, let's calculate the pressure difference between the plunger and the metal cylinder. The pressure difference (ΔP) is the increase in pressure caused by pushing the plunger upwards. It is given as a factor of 3.03 times the atmospheric pressure.

ΔP = (3.03 * 1.00 atm) - 1.00 atm
= 2.03 atm

2. The force acting on the metal cylinder is equal to the pressure difference multiplied by the area of contact between the plunger and the metal cylinder. The area of contact is equal to the area of the metal cylinder's circular cross-section.

A = π * r²
= π * (6.22 mm)^2
≈ 121.71 mm² (approximately)

Converting mm² to m²:
1 mm² = 1 * 10^-6 m²
Therefore,
A ≈ 121.71 * 10^-6 m² (approximately)

Now, the force (F) is calculated by:
F = ΔP * A
≈ 2.03 atm * 121.71 * 10^-6 m²
≈ 0.00024688 N (approximately)

3. We are given the mass (m) of the metal cylinder as 0.527 kg.

4. Using Newton's second law of motion, we can find the acceleration (a) of the metal cylinder:
F = m * a

Rearranging the equation, we have:
a = F / m
= 0.00024688 N / 0.527 kg
≈ 0.000469 m/s² (approximately)

Therefore, the initial acceleration of the metal cylinder is approximately 0.000469 m/s².

To find the initial acceleration of the metal cylinder, we can use the principles of fluid mechanics, specifically applying Pascal's law and Archimedes' principle.

Step 1: Calculate the initial pressure inside the tube
Since the pressure outside the tube is given as 1.00 atm, and the plunger increases the pressure by a factor of 3.03, we can calculate the initial pressure inside the tube.

Initial pressure inside the tube = 1.00 atm * 3.03 = 3.03 atm

Step 2: Calculate the net force acting on the metal cylinder
The net force on the metal cylinder is the difference between the downward force due to its weight and the upward force due to the pressure difference.

Downward force due to the weight of the metal cylinder = mass * acceleration due to gravity
= 0.527 kg * 9.8 m/s^2

Upward force due to pressure difference = pressure difference * area
= (initial pressure inside the tube - pressure outside the tube) * area of the cylinder

The area of the cylinder can be calculated using the radius provided. The radius given is for the inner part of the tube, but since the cylinder is placed inside, the radius of the cylinder will be the same.

Area of the cylinder = π * (radius)^2
= π * (6.22 x 10^(-3) m)^2

Now we can calculate the net force:

Net force = Upward force - Downward force

Step 3: Calculate the initial acceleration
After finding the net force, we can use Newton's second law of motion to find the acceleration.

Net force = mass * acceleration

By substituting the known values, we can solve for the initial acceleration.

Once we have followed all these steps, we should be able to find the initial acceleration of the metal cylinder.