A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is 20°C, the piston floats 20 cm above the bottom of the cylinder. a. What is the gas pressure?

Video transcript

Hey, everyone. So this problem is dealing with gasses and forces. Let's see what it's asking us. A 10 centimeter diameter cylinder with a movable piston contains helium gas at a temperature of 25 °C and a pressure of one atmosphere. The cylinder is oriented vertically and the air above the piston is removed. The movable piston is made of aluminum and is two centimeters thick. When the gas temperature is increased to 100 °C, the piston is found to have moved up to a position 15 centimeters above the bottom of the cylinder. What is the new pressure of the helium gas in the cylinder? We're told to assume the density of aluminum is 2710 kg per cubic meter. And the cross sectional area of the cylinder is constant throughout our multiple choice answers in units of pascals are a 531 B 451 C 154 or D 135. So the key to this problem is recognizing the information that we need out of all of the information that was given to us and that is really tricky when they give us so much information. But in the end, this is essentially a free body diagram, Newton's Second law pro so the piston, if we treat it as a point has the force of the gas that helium gas acting on it, maybe we will call that the force of the helium in the upward direction. And then the weight of that piston in the negative y direction. So the force of the helium is given by the equation pressure multiplied by the area. And then we can recall that weight is simply mass multiplied by gravity. After the temperature is increased, the system is again an equilibrium. So the sum of the forces in the y direction is zero. So we can say that the pressure from the helium at 100 °C multiplied by the area after the piston has moved is equal to the mass of the piston multiplied by gravity. Now, we don't know the mass of the piston, but we do know the material and that material's density. And so we can recall that mass is equal to density multiplied by the volume and volume in turn is equal to the area multiplied by the thickness. And so when we combine those two equations, we get that pressure is equal to the density multiplied by the area multiplied by the thickness multiplied by gravity divided by the area. And so that area cancels and we are simply left with needing the density of the piston material, the thickness of the piston material and gravity, which is of course a constant. And so we can plug in those known values 2710 kg per cubic meter multiplied by two centimeters. Keeping everything in standard units that will be 0.02 m multiplied by gravity 9.8 m per second squared, which gives us 531 pascals. And so that's the final answer to this problem which aligns with answer choice. A that's all we have for this one. We'll see you in the next video.