 A 4,000 pound-force car is resting on a 4-foot piston sitting in a cylinder as shown in the figure to the right. Attached to that piston is a large tube connected to a secondary piston which features a 1-foot diameter piston. The tube and cylinders are filled with liquid water. A. How much force needs to be applied to piston 1 to be able to support the weight of the car? And B. What if the diameter of the piston 1 were decreased to 6 inches? How would that affect the answer to part A? I'm going to begin this problem by recognizing that because of Pascal's law, which states that the pressure increased by a force on a constrained fluid is going to increase everywhere. This constrained fluid, therefore, will have the same pressure here as it does over here, which means that I can write P1 as being equal to P2. I don't actually care that much about the pressure. What I care about are the forces, so I'm going to recognize that a pressure defined as a force per unit area. Therefore, I can write F1 over A1 is equal to F2 over A2. And then I'll recognize that the area of effect here is the area of a circle because our pistons are inside of a cylinder, which means that they are going to be circular, which means that their area is going to be pi over 4 times diameter squared. Again, I'm writing a diameter because I happen to have diameters instead of radii and I don't want to do any extra math. So therefore, I can say F1 over pi over 4 times diameter 1 squared is equal to F2 over pi over 4 times diameter 2 squared. In this problem statement, I was told F2, I was also told D1 and D2. D1 is 1, D2 is 4, F2 is 4,000 pound force. Pi over 4 is going to cancel. So I can write F1 is equal to F2 times diameter 1 squared over diameter 2 squared. And then I'm going to simplify that square by writing this as F2 times diameter 1 over diameter 2 quantity squared. F2 is 4,000 pounds of force times 1 foot over 4 foot be cancel squared. 1 quarter squared is going to be 1 sixteenth. 1 sixteenth of 4,000 is 250. I'll confirm that with the calculator. 4,000 times. Yes, delete everything. That's what I want to calculate. Come on. Oh, why? Why must you keep doing that? Don't let my dreams be dreams calculated divided by 4. There you go. Look at you. Squared, 250 pounds. The reason I wrote it this way is because this quantity right here is our mechanical advantage. That's the power of hydraulics. Pascal's law, meaning that the pressure increase is going to increase everywhere, means that we have the same pressure applying a different force because the area is different. That gives us a lot of flexibility in regard to different forces applied at different points within the same system. And all we have to do is ensure that we are supplying the pressure required to accommodate those forces. In part B, the diameter of the piston at state 1 is decreased to 6 inches instead of 12 inches. How does that affect our answer? Would it have it or something else? Well, all I have to do now is change my 1 to a 0.5 and I get 62.5. Why is having the diameter not having the force? Because having the diameter does not have the area. Half the diameter means one quarter of the area, which means that there must be one quarter of the force to accommodate the same pressure or to hit the same pressure. So part A and part B.