Question

A 0.550 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 7.41 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 2.55, what is the initial acceleration of the metal cylinder? Assume the pressure outside of the tube is 1.00 atm.

Answer 1

To solve this we need to understand and remember some basic concepts, first of all let us revise some formulas:-

Pressure= Force/ Area
Force= Mass X Acceleration

Now the radius of the cylinder is nearly the same as that of the plastic tube since it is fixed inside the plastic tube isn't it.
Now let us begin, If the cylinder came to rest, it would mean that the force acting on the cylinder from outside would be equal to the force acting on the cylinder from inside, correct, so since pressure is the force per unit area, we would first need to find the area on which the force was acting to calculate the pressure.

The top of the cylinder experiences the force due to the air since no air flows through the sides, the force on the top of the cylinder must be equal to that at the bottom when it is at rest, so lets calculate that force, Force= Pressure X Area (In this case the top of the cylinder which is circular)

Now 1 atm is 101325 Pa (SI unit of pressure)

So,
101325 (Pi X (7.41 X 10^-3)^2) We used Pi x Radius ^2 formula.

Now that gives us 16.2 N

So now the lower pressure would increase by 2.55

So that means the new pressure would be 2.55 X 1 atm (101325 Pa)
=258378 Pa

So now lets find the force on the lower end of the cylinder

258378 X (Pi X (7.14 X 10^-3) ^2 = 41.38 N

This is the force now acting from the bottom, so to find the NET force which causes the acceleration we should subtract the force acting in one way from the other, so the force acting from above is subtracted.

41.38 - 16.2= 25.16N

Force= Mass X Acceleration
Acceleration= Force/ Mass

25.16/0.55= 45.74 m/s^2