A gas-filled vertical cylinder, closed at the bottom end, is fitted at the top with a piston that can move freely. The mass of the piston is 10.0 kg, and the initial height of the piston above the bottom of the cylinder is 25 cm. A mass of 8.0 kg is placed on the piston. What is the resulting height of the piston, assuming that the temperature of the ideal gas is kept constant?
A) 16 cm
B) 12 cm
C) 14 cm
D) 15 cm
E) 13 cm

Question

A gas-filled vertical cylinder, closed at the bottom end, is fitted at the top with a piston that can move freely. The mass of the piston is  10.0 kg, and the initial height of the piston above the bottom of the cylinder is  25 cm. A mass of  8.0 kg is placed on the piston. What is the resulting height of the piston, assuming that the temperature of the ideal gas is kept constant?
A)  16 cm
B)  12 cm
C)  14 cm
D)  15 cm
E)  13 cm

Quizplus · Accepted Answer

According to Boyle's Law (P1V1 = P2V2) for an ideal gas at constant temperature, the product of pressure and volume remains constant. Initially, the pressure exerted by the piston on the gas is due to its own weight plus atmospheric pressure (P1). When an additional mass is placed on the piston, the pressure increases (P2). Assuming atmospheric pressure remains constant and is equal on both sides of the equation, the increase in pressure is directly proportional to the total mass on the piston. Initially, the force due to the piston's weight is $10.0 \, \text{kg} \times 9.8 \, \text{m/s}^2$, and after adding the mass, the total force is $(10.0 \, \text{kg} + 8.0 \, \text{kg}) \times 9.8 \, \text{m/s}^2$. The initial and final heights of the piston can be related by the inverse proportion of the total forces (or masses, since acceleration due to gravity is constant) acting on it. Thus, the final height $h_2$ can be found using the relation $\frac{h_1}{h_2} = \frac{m_2}{m_1}$, where $h_1 = 25 \, \text{cm}$, $m_1 = 10.0 \, \text{kg}$, and $m_2 = 10.0 \, \text{kg} + 8.0 \, \text{kg}$. Solving gives $h_2 = 14 \, \text{cm}$.

A Gas-Filled Vertical Cylinder, Closed at the Bottom End, Is