This section of the workshop was concerned Newton's three Laws of Motion, the basis for almost all of classical mechanics. Calling them Newton's Laws is somewhat of a misnomer, since the first one was not developed by Sir Isaac, but by Galileo Galilei. It states:
A body continues in a state of rest, or motion with a constant velocity, unless compelled to change by an unbalanced force.
This law is considered by many to be the result of the first successful application of the scientific method. Before Galileo, most scientists felt that the natural state of an object was at rest. This, in fact, is still believed by many people today since our everyday experiences point to this. For example, if you take your foot off the gas pedal (stop applying a net force), the car slows. If you slide a book across a table, it stops. It is only when we realize that a net force (friction in these examples) is acting on the objects that we see that the law of inertia is true.
Newton's second law quantifies the relationship between the observed motion and the net force. Very simply, it states:
The acceleration of an object is directly proportional to the net force acting upon it and
inversely proportional to its mass.
Mathematically, this means F = ma, where F = net force, m = mass of the object, and a = the acceleration of the object = the time rate of change of the velocity. It should be noted that the direction of the acceleration is in the direction of the net force. Deceleration is the net force is in a direction that is opposite of velocity, which, of course, means that it is a form of acceleration.
Newton's third law is commonly known as the law of action-reaction. It states:
For every action force, there is an equal and opposite reaction force.
A good example of this law is two skaters at an ice rink. If one pushes on the other, both move, one due to the action force and the other due to the reaction force. While we do not notice it, this same thing happens every time we jump in the air. We push away from the Earth, and due to the reaction force, the Earth is pushed away from us. However, since the Earth is so massive in comparison to us, it does not accelerate very greatly.
The following topics can be found in this section
We will first investigate the concept of inertia. To do this, we will need the following materials: a smooth based glass, a cup, 5-10 equal sized coins, 3"x5" card, a piece of paper or small section of smooth cloth (handkerchief), 3' string, large weight (1 pound or more) with hooks on opposite sides, 8" dowel.
Demonstration 1: Fill the glass partially with water and place on top of paper or cloth. Slowly pull on the paper or cloth and observe the motion of the glass. Now jerk the paper or cloth very quickly and observe. What is the difference between the two situations?
Demonstration 2: Cut the string in two and tie one piece to the top of a ringstand. Tie the weight to the other end of string and then tie the other piece of string to the opposite end of the weight. Lastly, tie the bottom string to the middle of the dowel. Very carefully, pull slowly downward on the dowel (Make sure the weight does not hit your head when the spring breaks). Where does the string break? Reassemble the demonstration. Now quickly jerk down on the dowel. Where does the string break?
Experiment 1: Place the card on top of the cup. Take one coin and place it on the center of the card. Without lifting the card, try to get the coin into the cup. What is the best procedure?
Experiment 2: Stack several coins (4 or more) on a smooth table top. Place one coin a short distance from this stack. Flick the coin very quickly at the stack. What happens? Vary the speed of your flicking and observe the results.
Evaluation: Demonstration 1 works quite well with a medium sized beaker (500-100 mL). Be sure to have it at least half-filled with water and to jerk the paper or cloth very quickly straight down. Demonstration 2 was not done during the workshop, owing to a lack of an appropriate weight with hooks. Experiment 1 was a success, giving the participants an opportunity to be creative in their solutions. The best procedure was found to be to flick the card out from under the penny very quickly with your finger. Experiment 2 gave results that seemed somewhat puzzling at first glance. It was found that flicking the coins very hard resulted in the whole pile being upset. This seemed to go against the idea of inertia. However, we found that flicking to hard caused the penny to become slightly airborne and hit more than one penny.
NOTE: This is a demo that has been declared dangerous by a good deal of the physics community. It is being shown in this workshop as an example of the kind of demonstration that should not be done.
For this experiment, we will need a set of steel washers, string or thread, paper clips, and an elevated circular platform marked with a full 360 degrees (This can be made quite easily with thick cardboard or masonite on the end of a ringstand). Find the mass of an individual washer. This mass will become our unit of mass. Cut four equal-length pieces of string that are long enough to reach from the middle of the platform to halfway down to the table. With the strings all held together, tie one end, knotting all of the strings together. On the other end of each string, tie a paper clip. Place the knot at the middle of the platform, dangling the ends of the string with the paper clips over the edge.
We are now ready to begin applying forces. When washers are placed on a paper clip, gravity will act on them, pulling the entire system in that direction. Unless an equal force is applied in the other direction, the entire string system will fall over the edge of the platform. For this experiment, we will set up weights on three of the paper clips at various angles. The purpose of the experiment is to find how much weight at some other angle will exactly balance all of the other weight.
Evaluation: This experiment worked well on several levels. For middle grades students who have not had trigonometry, this experiment would provide an excellent hands-on experience in problem solving and critical thinking. For high school students who have had trig, this is a good mathematical problem solving exercise.
Newton's Third Law of equal and opposite forces is very important when considering collisions between objects or propulsion. In these situations, all of the forces are internal, which means that for every force, there will be an equally sized force in the opposite direction. An example is two skaters out on the ice. If one skater pushes on the other, not only does the skater that is pushed move, but also the skater that does the pushing. The speed with which they move is determined by their relative size. If they are the same mass, they move away from each other with the same velocity. If one is large than the other, then the smaller one moves away with a much larger velocity, whether they did the pushing or not.
The reason for this is that momentum (p=mv, where m is the mass and v is the velocity) is conserved. One way to show this is by having two different sized people sit on carts and push on each other. Another way is the following. Drop a tennis ball and a ping pong ball on the ground separately from a height of about one meter. You should note that both balls come back to about the same height, which is less than the height from which they were dropped. Now, with the ping pong ball on top of the tennis ball (they should be touching), drop the two simultaneously. While the tennis ball comes back to about the same height, the ping pong ball is shot up into the air over twice the height from which it is dropped.
The reason for this is the conservation of momentum. Both balls rebound off the floor with about the same velocity. However, the tennis ball is about 20 times more massive than the ping pong ball. Thus, while the balls are exchanging equal amounts of momentum, the difference in masses means that the ping pong balls velocity is much greater than that of the tennis ball. With this velocity coming off the tennis ball, the ping pong ball is able to go much higher in the air than what it would have had it only landed on the floor.
| To view a Quicktime movie displaying this experiment, click here. This movie shows a ping pong ball on top of a tennis ball being dropped from a height of 20 cm (each horizontal line in the movie corresponds to 10 cm). Note the height to which the ping pong ball rebounds. It is over 3.5 times as high as its original position. |
Evaluation: The two carts worked well. Using different sized people shows conservation of momentum quite well. However, you need to be sure to get wheels with very good bearings or else the carts stop very soon after pushing. It is probably best to get skateboard wheels. The superball projector also shows conservation of momentum well. However, it is very difficult to drop the balls so that the top ball lands directly on top of the other ball. It took several tries to get this to work for the participants. You might want to build some sort of ball dropper.
A paddle boat offers another example of Newton's Third Law. For this activity, the following supplies are needed: a 10 cm x 10 cm piece of cardboard or thin balsa wood, a rubber band, a pair of scissors or an exacto knife, a long tank of water (at least 5 cm deep), food coloring, a stopwatch, and a meter stick. First, cut the cardboard or wood as shown in the following picture. The slot in the boat should be about 4 cm x 4 cm, with the paddle being about .5 cm smaller in each direction. Loop the rubber band across the back end of the boat so that it crosses the slot. Insert the paddle between the rubber band and wind it such that it goes forward to back when viewed from above. Place the boat on the water and observe the motion. After placing the meter stick across the container, rewind the boat and release, this time measuring the distance and the time of travel to estimate the velocity. To see what the water is doing, place a drop of food coloring about 1 cm behind the boat as it is moving. Observe the motion of the drop.
Evaluation: This experiment did not work that well (we used cardboard). The rubber bands were too tight between the boat and paddle and often got stuck. After a couple of tries, the boats were so wet that they became quite flaccid. It would be better to try balsa wood and loose rubber bands.
Another demonstration of Newton's Third Law is rocket propulsion. Rockets are propelled by "pushing off" of their exhaust, i.e. the rocket pushes the exhaust in one direction and it is pushed in the other. Thus, even in outer space where there is nothing from which to push, a rocket can move. An interesting historical note is that one of the early pioneers, Robert Goddard, in rocketry was ridiculed for stating that a rocket would operate in outer space. A quotation from a New York Times editorial in 1921 stated:
That Professor Goddard with his "chair" at Clark College and the countenance of the Smithsonian Institution does not know the relation between action and reaction, and the need to have something better than a vacuum against which to react - to say that would be absurd. Of course, he only seems to lack the knowledge ladled out daily in high schools.(Quoted from College Physics, Paul Tipler, p. 167)
For our rockets, we will need the following materials: balloons (preferably long ones), straws, long string (at least 10 feet), tape, two liter plastic soda bottles, a cork or number 3 stopper, aluminum foil, baking soda, and vinegar.
Experiment 1: Attach one end of the string to a wall. Inflate the balloon and tape the straw or tubing to it. Place the other end of the string through the straw and hold the string taut. Release the balloon and observe the result. Modify the design to achieve maximum propulsion. Using a stopwatch and meter stick, estimate the velocity of your rocket.
Evaluation: This experiment worked quite well. You should definitely use fishing line and plastic straws. The balloon shape is not as important. However, the lines should be placed near the floor and the areas blocked off. We found that there was a potential danger of catching a line in the troat if these precautions were not taken. You also might want to use a colored fishing line.
Experiment 2: As the previous experiment shows, compressed gases make excellent propellants. Another source of these is to mix acetic acid (vinegar) and sodium hydrogen carbonate (baking soda) in an enclosed volume, creating compressed carbon dioxide. Pour about 1 cup of vinegar (250 mL) into a 2 liter soda bottle. Fold a small piece of foil into a canoe-shaped trough, small enough to fit through the bottle opening, but large enough to hold about 3 tablespoons of baking soda. Lay the bottle on its side in an outdoor location (preferably a concrete slab that can be rinsed off). With the stopper in one hand, slide the foil boat with baking soda into the bottle and quickly plug the bottle. Shake the bottle to mix the soda and vinegar. STAND BACK!! When the pressure becomes great enough, the stopper will be ejected in one direction and the bottle will go in the other. Bottles and stoppers traveling 30 meters are not unrealistic, so make sure that no one is standing in front of or behind the "rocket". Note the contents of the bottle after propulsion. How much does the bottle weigh before and after propulsion? Notice in patterns on the concrete slab.
To get an idea of how such factors as "fuel" amount and the tightness of the stopper in the hole affect the rocket, you might want to perform the following experiment: Set the bottle at a predetermined angle with respect to the ground. Follow the procedure as above for firing the rocket, except adjust the amount of baking soda or the tightness with which the cork is put into the bottle. Measure how far the stopper is thrown THROUGH the air. Using the formula
Distance traveled = (v2 sin 2)/g where g = 9.8 m/s2 and = angle above horizontal
calculate the velocity of the stopper. For the range of velocities that you get, is the force of the stopper on the bottle able to account for all the motion of the bottle (Note: Look at the conservation of momentum)?
Evaluation: Although I had run this experiment many times successfully, the rockets did not work for many of the participants. We later found out the reason why. The surface that we were laying the bottles on was slightly angled downward. Because of this, the initial pressure release was not causeing a jolt forward and throwing the fluid toward the back were it is shot out as propellant. This meant that the rockets simply stayed where they were and shot out carbon dioxide. The cannon idea did work, though. In order to increase repeatability, it was mentioned that a mark should be made on the stopper so that it would be pressed into the bottle the same distance each time.
A discussion of Newton's Laws of Motion would not be complete without discussing the center of mass. As it turns out, we can consider the net force to be acting through the center of mass of any object. As an example, we will find the center of mass of an irregularly shaped object using gravity as our force. For this activity, we will need: an irregularly shaped piece of cardboard, a piece of string, a ruler, and a pen or marker. First, poke a small hole near the edge of the cardboard and tie the string through it. Hold the string by the other end until the object has stopped swinging. Since gravity will act on the cardboard through the center of mass, we know that the center of mass will lie somewhere below the hole on line with the string (If it was not, then gravity would cause the object to swing to bring the center of mass to its lowest point.). Put the ruler along the string and draw a line with the marker on the object. Untie the string, make another hole near the edge of the object (at least a quarter of a turn from the first point), and tie the string there. Again, draw a line along the line on the object. The two lines should meet at the center of mass. Check this by repeating the procedure at a third point. After doing so, place the object on top of your finger at the intersection point. Does the object balance there?
Evaluation: The participants found this a good weigh to show center of gravity. Need at least two people to do demonstration (one to hold the string, the other to draw the lines).
When two are more objects are joined, the center of mass of the combined objects is found with the following formula