A piston of mass m and cross-sectional area A is in...

A piston of mass m and cross-sectional area A is in equilibrium under the pressure p at the center of a cylinder closed at both ends. Assuming that the piston is moved to the left a distance a/ 2 and released, and knowing that the pressure on each side of the piston varies inversely with the volume, determine the velocity of the piston as it again reaches the center of the cylinder. Neglect friction between the piston and the cylinder and express your answer in terms of m, a, p, and A.

Key Concepts

Equilibrium

Equilibrium refers to the condition in a physical system where all the forces acting on an object balance each other, resulting in a state of no net acceleration. In the context of a piston in a cylinder, equilibrium is the initial state where the pressure forces on both sides of the piston are equal, ensuring that the piston remains stationary until disturbed.

Pressure-Volume Relationship (Boyle's Law)

This concept describes the inverse relationship between the pressure and volume of a gas held at constant temperature. When the volume of a gas decreases, its pressure increases proportionally, and vice versa. This relationship is crucial for analyzing situations where the gas volume changes due to the movement of a piston within a closed chamber.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In the dynamics of the piston, the work done by the pressure forces as the piston moves is converted into kinetic energy. This conservation allows one to relate the work done by the pressure difference over a displacement to the kinetic energy gained by the piston.

Oscillatory Motion

Oscillatory motion describes the repetitive back and forth movement observed in systems that are displaced from their equilibrium position. In the piston scenario, once displaced from equilibrium, the piston undergoes motion influenced by the varying pressure forces acting symmetrically on either side, which ultimately can be analyzed as a form of oscillation under certain assumptions.

Transcript

00:01 Hi everybody, thanks for joining me today.

00:02 What we're going to be looking at is again the concept of work in kinetic energy.

00:06 And what we're given is we're given a piston of mass m and cross -sectional area of a.

00:12 That is at equilibrium in the cylinder shown.

00:16 It has a pressure of p on both sides.

00:20 And, you know, what we have to find is that this piston will be moved a distance away from the center line of a over 2.

00:29 And that pressure differential is going to drive the piston to move.

00:33 And we're asked to find, you know, when that piston crosses x equal to a, which is halfway down the cylinder, as shown here, what is the velocity? so we're given that the pressure on either side of the cylinder is proportional to volume.

00:52 So if we look at this further, we see that volume is simply the, cross -sectional area times x which is what we have defined here as x is the distance from the leftmost edge of the cylinder so this basically implies that pressure is proportional to 1 over a times x so if we expand this to the terms of we're working with the pressure on the left and the pressure on the right we know that the pressure on the left over the pressure that given to us in the beginning is going to be a times a over a over x right we're just genus in generic terms to figure out what we can how we can solve for terms that we don't know and we can just plug it in x so this gives us that p l is equal to p a over x similarly we're going to do a similar thing for the right side of the pressure so we're going to say p r over p is again equal to a because remember these two are related to each other however, for p on the right, given the way that we have defined where x is equal to 0, you can see that this is the distance that is going to be contributing to the volume portion.

02:16 So if you add those up, that's 2a minus x.

02:19 So we see that this right here will be 2a minus x, which is going to give us a relation for pr.

02:30 Is equal to p a over 2a minus x now this is important to define these because this the difference and pressure between these two quantities is going to be the force that is going to drive the motion of the piston so we know at t equals one at uh position one kinetic energy is zero because this cylinder is starting from rest so t1 is zero velocity is zero and the position that is starting at is x is a over 2 at position 2 x is equal to a and kinetic energy is 1 half m v squared so using the definition of work we can say that u from 1 to 2 is equal to to we're going from a over 2 to a of the pressure on the left minus the pressure on the right times area.

03:35 Remember this gives us the force, pressure times areas of force, dx...

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