Question

A vertical piston-cylinder device contains a gas at a pressure of 100 kPa. The piston has a mass of 5 kg and a diameter of \(12 \mathrm{cm} .\) Pressure of the gas is to be increased by placing some weights on the piston. Determine the local atmospheric pressure and the mass of the weights that will double the pressure of the gas inside the cylinder.

Short answer

Answer: To determine the mass of the weights required to double the pressure, follow these steps: 1. Calculate the initial force exerted by the piston on the gas by finding the area of the piston and multiplying it by the initial pressure of the gas. 2. Calculate the local atmospheric pressure by considering the weight of the piston and assuming it is balanced by the atmospheric pressure acting on the piston. 3. Calculate the additional pressure needed to double the pressure inside the cylinder by finding the difference between the initial and final pressures. 4. Calculate the mass of the weights needed to increase the pressure by using the additional force needed and the relation between force and mass.

Step-by-step solution

Calculate the initial force exerted by the piston

To find the initial force exerted by the piston on the gas, we need to calculate the area of the piston and multiply it by the pressure of the gas inside the cylinder. The area A can be calculated using the formula \(A = \pi r^2\), where r is the radius of the piston, and the force F can be calculated using the formula \(F = P \cdot A\), where P is the initial pressure of the gas. The radius of the piston can be calculated as \(r = \dfrac{d}{2}\), where d is the diameter of the piston.

Calculate the atmospheric pressure

In order to find the atmospheric pressure, we need to consider the force acting on the top of the piston due to the weight of the piston and assume that this force is balanced by the atmospheric pressure acting on the top of the piston. The weight of the piston is given by \(W = m \cdot g\), where m is the mass of the piston and g is the acceleration due to gravity (approximately \(9.81 \, \mathrm{m/s^2}\)). The atmospheric pressure can be calculated as \(P_\mathrm{atm} = \dfrac{W}{A}\).

Calculate the pressure required to double the pressure inside the cylinder

We are given that the pressure inside the cylinder must be doubled. Therefore, the final pressure should be \(P_\mathrm{final} = 2 \cdot P\). The difference in pressure between the initial and final pressures is the additional pressure needed. This can be calculated as \(\Delta P = P_\mathrm{final} - P\).

Calculate the mass of the weights needed

To increase the pressure inside the cylinder by \(\Delta P\), we need to add more weight on top of the piston, which will increase the force acting on the piston. The additional force needed can be calculated as \(\Delta F = \Delta P \cdot A\). To find the mass of the weights required, we can use the relation \(\Delta F = \Delta m \cdot g\), where \(\Delta m\) is the mass of the weights needed. Solving for \(\Delta m\), we get \(\Delta m = \dfrac{\Delta F}{g}\). By following these steps, we can determine the local atmospheric pressure and the mass of the weights required to double the pressure of the gas inside the piston-cylinder device.