A system of cylinder(one end open) and piston is held in horizontal position inside a vacuum chamber. the mass of the cylinder and that of the piston are m1, m2, respectively. The piston which has negligible thickness dividesthe volume of the cylinder into 2 equal parts.

The closed section of the cylinder contains n moles of helium which may be considered as an ideal monoatomic gas at T0=273 K. The piston is held at rest in the mid position and suddenly released. Determine the final velocity of the CYLINDER. Assume that the heat transfer between cylinder and piston, and the motion of the centre of gravity are negligible.

T0=273K

m1=0.6 kg, m2=0.3 kg

n=25 moles

C_v of He =12.6 J/mol.K

C_p/C_v=5/3

To determine the final velocity of the cylinder after the piston is released, we can apply the principles of conservation of momentum and the ideal gas laws. Since the system is initially at rest and the piston divides the cylinder into two equal parts, we can analyze the forces acting on the piston and the cylinder when the piston moves. Let's break down the problem step by step.

Understanding the System Dynamics

Initially, the piston is held in place, and the gas inside the closed section of the cylinder is at a temperature of T0 = 273 K. When the piston is released, the gas will expand, pushing the piston outward. This expansion will exert a force on the piston, which in turn will affect the cylinder.

Applying the Ideal Gas Law

For an ideal gas, we can use the ideal gas law, which states:

• PV = nRT

Where:

• P = pressure of the gas
• V = volume of the gas
• n = number of moles of gas
• R = ideal gas constant (approximately 8.314 J/(mol·K))
• T = temperature in Kelvin

In our case, the gas is helium, and we have n = 25 moles. The initial pressure can be calculated using the volume of the gas and the temperature. Since the piston divides the cylinder into two equal parts, the volume of the gas can be expressed as:

• V = V_cylinder / 2

Calculating Initial Pressure

Assuming the total volume of the cylinder is V_cylinder, the initial pressure (P_initial) can be calculated as:

• P_initial = (nRT) / (V_cylinder / 2)

Substituting the values:

• P_initial = (25 moles * 8.314 J/(mol·K) * 273 K) / (V_cylinder / 2)

Force on the Piston

When the piston is released, the gas expands and exerts a force on the piston. The force (F) can be calculated using the pressure and the area (A) of the piston:

• F = P * A

As the piston moves, it will accelerate due to this force. According to Newton's second law, the acceleration (a) of the piston can be expressed as:

• a = F / m2

Conservation of Momentum

Since the system is closed and initially at rest, the total momentum before and after the release must be conserved. Let v_cylinder be the final velocity of the cylinder and v_piston be the final velocity of the piston. The conservation of momentum gives us:

• m1 * v_cylinder + m2 * v_piston = 0

From this, we can express the velocity of the cylinder in terms of the velocity of the piston:

• v_cylinder = - (m2 / m1) * v_piston

Final Velocity Calculation

To find the final velocities, we need to relate the acceleration of the piston to its velocity. The relationship between acceleration and velocity can be derived from kinematic equations, but for simplicity, we can assume that the piston moves a small distance d in a short time t, leading to:

• v_piston = a * t

Substituting the expression for acceleration:

• v_piston = (F / m2) * t

Now, substituting this back into the momentum equation gives us:

• v_cylinder = - (m2 / m1) * (F / m2) * t

After simplification, we can express the final velocity of the cylinder in terms of the force exerted by the gas and the masses involved. However, to find the exact numerical value, we would need to determine the force from the pressure and area of the piston, and the time of motion, which can be complex without specific values for the area and distance moved.

Final Thoughts

In summary, the final velocity of the cylinder can be derived from the conservation of momentum and the dynamics of the gas expansion. The exact numerical value would require additional information about the dimensions of the cylinder and piston. However, the principles outlined here provide a solid foundation for understanding the mechanics at play in this system.