- 2074 Bhadra · 6 marks
Cylinder with movable piston: specific volumes
Short answer
Initial volumes from length fractions: 0.5 m³ and 1.5 m³ with v=8 m³/kg, giving masses 0.0625 kg and 0.1875 kg. When piston is at middle, both volumes are 1 m³, so specific volumes are 16 m³/kg and 5.33 m³/kg.
A cylinder with a total volume of 2 m³ has a movable piston as shown in figure below, when the piston is at one fourth of the length, both sides have the same specific volume of 8 m³/kg. Determine the specific volumes of both sides when the piston is at middle of the cylinder. [Figure: horizontal cylinder of length L divided by a movable piston into side A (left) and side B (right); piston at x = L/4 from the left end.]
Given: total volume V = 2 m³; the piston at one fourth of the length (x = L/4) from side A; v_A = v_B = 8 m³/kg. The piston is free, so the masses on the two sides remain constant.
Step 1: Volumes at the initial position (constant cross-section, so volume ∝ length)
V_A1 = (1/4)(2) = 0.5 m³
V_B1 = (3/4)(2) = 1.5 m³
Step 2: Masses
m_A = V_A1 / v_A = 0.5/8 = 0.0625 kg
m_B = V_B1 / v_B = 1.5/8 = 0.1875 kg
Step 3: Piston at the middle (x = L/2)
V_A2 = V_B2 = 1 m³
Step 4: Specific volumes
v_A2 = V_A2/m_A = 1/0.0625 = 16 m³/kg
v_B2 = V_B2/m_B = 1/0.1875 = 5.33 m³/kg
Answer: v_A = 16 m³/kg and v_B = 5.33 m³/kg.
✓ Answer checked by Claude Opus 5.5 (high effort)
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