Hydraulic Arm

The diagram below shows an exploded drawing of the Hydraulic Arm assembly. Refer to this drawing as necessary as you proceed through the steps that follow.

1. Drill holes in the four pieces of wood—the post, arm, and both bases—as shown in the diagram below (click to enlarge). By making two bases, you’ll be able to flip the Hydraulic Arm and use it in two different positions.
   
2. Assemble the two bases and the post using the two wood screws. Countersink to make sure the heads of the screws don’t stick out from the top and bottom bases.
3. Attach the arm to the post using a machine screw, three washers, and a wing nut.
4. Screw the cup hook into the end of the arm.
5. Drill a small hole (smaller than the sheet-metal screws) in the body of each of the two 10-mL syringes, just below the top flange. Screw a sheet-metal screw into each of these holes and rotate the syringe plungers so the screws can go in as far as possible without hitting the plungers (see photo below). The screws should now act as stops to keep the plungers from being pulled completely out of the syringes.
   
6. Select one of the 10 mL syringes and drill a 1/8-inch (0.3 cm) hole through the shaft of its plunger, approximately 1/4 inch (0.5 cm) from the end of the plunger, as shown below. This “fixed syringe” will be attached to the apparatus; a nail will later be placed through this hole.
   
7. Place the cable tie near the bottom of the fixed syringe as shown in the photo below. Pull the cable tie as tight as possible so the syringe does not easily slip. If necessary, use pliers to pull the cable tie tighter after you have initially tightened it by hand. Use scissors to cut off all but about 1/4 inch (0.6 cm) of the excess tie.
   
8. Attach the fixed syringe to the post through the cable-tie loop using a machine screw, two washers, four hex nuts, and a wing nut (see photo below). Adjust the hex nuts to allow enough space between the last hex nut and the head of the machine screw so that the mounting head of the cable tie will allow the syringe to pivot freely, but not slide sideways excessively. When this adjustment has been made, make sure that the four hex nuts are tight against each other, and then tighten the wing nut.
   
9. Put the nail through the hole in the syringe plunger and force it into the hole in the wooden assembly arm until the head of the nail is almost up against the plunger, but not so far that the point of the nail protrudes significantly from the other side of the arm (see photo below). If necessary, use pliers to push the nail into the hole.
   
10. Attach the tubing firmly to the tip of the second 10 mL syringe. This will be the “free syringe.” Pull the plunger out until it is fully extended. Fill the syringe and tube completely with water. (If there are air bubbles, try flicking the syringe or tube with your finger to get them to rise, and then top off with more water as necessary. If you have difficulty, try detaching the hose and filling it separately, and then reattaching it to the full syringe.)
11. Push the plunger all the way into the fixed syringe (the one attached to the post and arm) so that there’s no air in the syringe. Attach the open end of the water-filled tube firmly to the tip of this syringe. Be sure there are no large air bubbles anywhere in the system.

Push on the plunger of the free syringe. What happens? Pull on the plunger. What happens now?

Use the arm to lift a small object. (If the arm tips, put a book or other heavy weight on the base to steady it, or find a lighter object to lift.) How does the force of your push on the free syringe’s plunger compare to the force that the fixed syringe’s plunger exerts on the arm? How do the distances that the two plungers move compare with each other?

Notice carefully how hard you have to push on the plunger to lift a particular object, and notice how far the arm can move the object.

Remove the object, then push water into the fixed syringe so that the arm is elevated as much as possible. Support the elevated arm so that it can’t fall. (You could have a friend hold the arm, or you could support it with a stack of books.) Then raise the free syringe until the end of the tube attached to it is well above the fixed syringe. Keeping the end of the tube raised (to prevent water from coming out when the syringe is removed), remove the 10-mL syringe and replace it with the 1-mL syringe.

Remove the support from the elevated arm. Pull the plunger on the 1-mL syringe until the syringe is full. Replace the object on the hook, and then push the plunger on the 1-mL syringe to lift the object. Notice the difference in how hard you have to push on the plunger to lift the object this time. Notice also how far the object is lifted.

Put the 10-mL syringe back in place at the end of the tube, using the same technique you used to replace it with the 1-mL syringe. Turn the whole device upside down and use the syringe to raise and lower the arm. Compared to the right-side-up position, what’s different about the process of elevating the arm?

When you push on the plunger of the movable syringe, the arm rises; when you pull on the plunger, the arm descends.

Pushing on the plunger applies pressure on the water in the movable syringe. Since the water is confined and incompressible, Pascal’s principle comes into play, telling us that the pressure is transmitted undiminished to all parts of the water and to the walls of its container. Since the plunger of the fixed syringe at the other end of the tube forms part of the “container” for the water, and is the only part of the container that can expand, the pressure causes the plunger in the fixed syringe to move.

Pascal’s principle and a little mathematics can be used to show that—if the syringes are identical—the force you apply to one plunger is transmitted in full to the other plunger (see Math Root, below). Additionally, as you can observe, each plunger moves the same distance.

With the 1-mL syringe, you need to push with less force than with the 10-mL syringe, but the arm is not lifted nearly as far. In accordance with Pascal’s principle, the pressure on the plunger of the 10-mL syringe is the same as the pressure on the plunger of the 1-mL syringe. However, since the area of the 10-mL plunger is far larger than the area of the 1-mL plunger, the force exerted on the 10-mL plunger is far larger than the force you push with (remember, $F = pA$ ). The good news is that you have obtained a force advantage, but the bad news is that you’re paying for it with a distance penalty. Mechanical work is the product of force times the distance the force moves through ($W = Fd$), and this product remains constant.

In the right-side-up position, the plunger pushes on the arm to raise it. But when you turn the whole assembly upside down, the syringe pulls on the arm to raise it, just like your muscles do with your own arms. The muscle that allows your forearm to lift things, called the biceps, is attached near your shoulder and just below your elbow. When the biceps contracts, it has the same effect on your arm as the syringe has on the hydraulic arm when the assembly is upside down. In both cases, a large force is exerted so that a small weight can be lifted, but the weight can be lifted a large distance compared to the distance the force moves (the distance the syringe plunger moves, or the distance your muscles contract).

Hydraulic systems are used in countless applications: brakes and steering on cars; hydraulic lifts and jacks for servicing cars; airplane wing flaps, stabilizer controls, and landing gear; mechanical arms on garbage trucks; blades on bulldozers; and so on.

Math Root

Pressure is defined as force per unit area \(p = \frac{F}{A}\). If you divide the force you push with by the area of the plunger that is in contact with the water, you can find the pressure exerted on the water. You can mathematically rearrange \(p = \frac{F}{A}\) to become $F = pA$. This tells you that if you multiply pressure (expressed in pounds per square inch) by area (expressed in square inches) the square inches cancel out, and you are left with force expressed in pounds.

\[
\text {pressure } x \text { area} = \left[\frac{\text {pounds}}{\text {square inch}}\right] \times \text { [square inch]} = [\text {pounds}]
\]

Since the pressure on both plungers is the same, and the areas of both plungers are identical, then the force on both plungers is the same. (In the SI system, force is expressed in newtons.)

Louis Bloomfield. How Things Work: The Physics of Everyday Life (New York: John Wiley & Sons, 1997). There is an excellent discussion of hydraulic elevators on pages 236–237.

John Cameron, James Skofronick, and Roderick Grant. Physics of the Body, 2nd ed. (Madison, WI: Medical Physics Publishing, 1999). Pages 41–50 have a good discussion of the biomechanics of the arm.