Question

The closed cylinder of FIGURE CP18.74 has a tight-fitting but frictionless piston of mass M. The piston is in equilibrium when the left chamber has pressure p₀ and length L₀ while the spring on the right is compressed by ΔL. Suppose the piston is moved a small distance x to the right. Find an expression for the net force (Fₓ)_neton the piston. Assume all motions are slow enough for the gas to remain at the same temperature as its surroundings.

Accepted Answer

Step 1: Begin by analyzing the forces acting on the piston. The piston is influenced by the pressure exerted by the gas in the left chamber, the force exerted by the spring in the right chamber, and the gravitational force due to its mass. Since the piston is frictionless, we do not need to account for frictional forces. Step 2: Write the expression for the pressure force exerted by the gas in the left chamber. The force due to pressure is given by $$ F_{	ext{gas}} = p \cdot A $$, where $$ p  is $$the pressure in the left chamber and $$ A  is $$the cross-sectional area of the piston. The pressure $$ p $$ changes as the piston moves, and can be related to the initial pressure $$ p_0 $$ using the ideal gas law: $$ p \cdot V = p_0 \cdot V_0 $$, where $$ V  is $$the volume of the left chamber. Step 3: Write the expression for the spring force exerted by the spring in the right chamber. The spring force is given by $$ F_{	ext{spring}} = -k \cdot (\Delta L + x) $$, where $$ k  is $$the spring constant, $$ \Delta L  is $$the initial compression of the spring, and $$ x  is $$the additional displacement of the piston to the right. Step 4: Combine the forces to find the net force $$ F_x $$ acting on the piston. The net force is the sum of the pressure force from the left chamber and the spring force from the right chamber, minus the gravitational force: $$ F_x = F_{	ext{gas}} - F_{	ext{spring}} - Mg . $$Substitute the expressions for $$ F_{	ext{gas}} $$ and $$ F_{	ext{spring}} $$ into this equation. Step 5: Simplify the expression for $$ F_x  by $$substituting $$ p  in $$terms of $$ p_0 $$, $$ L_0 $$, and $$ x $$ using the ideal gas law. The final expression for $$ F_x $$ will depend on $$ p_0 $$, $$ L_0 $$, $$ \Delta L $$, $$ x $$, $$ k $$, $$ A $$, and $$ M . $$Ensure all terms are clearly defined and consistent with the problem setup.

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주요 개념

Equilibrium in Fluid Mechanics

Ideal Gas Law

Net Force and Newton's Second Law