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 7.07 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 2.79, what is the initial acceleration of the metal cylinder? Assume the pressure outside of the tube is 1.00 atm.

To find the initial acceleration of the metal cylinder, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

First, we need to determine the net force acting on the metal cylinder. The net force is the difference between the upward force applied by the plunger and the downward force due to the pressure acting on the metal cylinder.

The upward force applied by the plunger can be found using the increased pressure and the area over which the pressure is applied. The area is given by the cross-sectional area of the cylinder, which can be calculated using the inner radius of the tube.

The downward force due to the pressure acting on the metal cylinder can be calculated by multiplying the pressure outside the tube by the area of the cylinder.

Once we have the net force, we can divide it by the mass of the cylinder to find the initial acceleration.

Let's perform the calculations step by step:

1. Calculate the cross-sectional area of the cylinder:
- The inner radius of the tube is given as 7.07 mm, which is equivalent to 0.00707 m.
- The cross-sectional area of the cylinder is given by the formula: π * radius^2.
- Therefore, the area is equal to π * (0.00707)^2.

2. Calculate the upward force applied by the plunger:
- The increased pressure is given as a factor of 2.79 multiplied by the atmospheric pressure.
- Therefore, the upward force is equal to the increased pressure multiplied by the cross-sectional area.

3. Calculate the downward force due to the pressure acting on the metal cylinder:
- The pressure outside the tube is given as 1.00 atm, which is equivalent to 101325 Pa (standard atmospheric pressure).
- Therefore, the downward force is equal to the pressure outside multiplied by the cross-sectional area.

4. Calculate the net force:
- The net force is equal to the upward force minus the downward force.

5. Calculate the initial acceleration:
- The initial acceleration is equal to the net force divided by the mass of the metal cylinder.

By following these steps and performing the calculations, you can find the initial acceleration of the metal cylinder based on the given information.