Essay questions for dynamics: Laws of motion and gravitation

by Quirino Sugon Jr.

I.  Physical Laws.  State the following laws in one sentence.

1. Law of universal gravitation
2. Gauss’s law for gravitation
3. Newton’s first law of motion
4. Newton’s second law of motion
5. Newton’s third law of motion

II.  Application.  Answer the following questions in at most two sentences.

1. Why are there high tides and low tides?
2. If you drill a hole from the North Pole to the South Pole and you drop the stone on the hole, what will happen to the stone?  Describe its motion.
3. If you are in a supermarket and you wish to buy meat, how do you estimate the force constant of the spring in the weighing scale?
4. Are the geostationary satellites not moving?
5.  You are standing on  top of a weighing scale while in an elevator.  If the elevator’s cables snap and the elevator falls freely, what happens to your weight and mass?  Why?
6. How do you measure the mass of the earth?
7. How did Cavendish measure the gravitational constant?
8. What is dark matter and why do scientists believe such matter exist?

Scientific method as drama: setting, characters, conflict, climax, and denouement.

by Quirino Sugon Jr.

Scientific method is a method for knowing the relationship between different physically measurable concepts.  There is no one way to do science, but its essential parts can be gleaned from the example of falling bodies.  You will notice that it has all the ingredients of a drama: setting, characters, conflict, climax, and denouement.

A.  Setting and Characters: Observation

For example, if you throw a stone upwards and video the result, you can measure both the height of the rock  and also the time it took for the rock to reach a particular height.  From this two data, you can plot the position vs time function and obtain a parabola.  Then you can find the instantaneous rate of change of position with respect to time to obtain velocity.  You can do likewise for velocity and obtain acceleration.  And you obtain a constant acceleration equal to .

The main characters of the story are time and position.  The supporting casts are velocity and acceleration.  The setting is free fall motion.

B.  Conflict:  The Search for Universals

Then you ask: is this value of acceleration constant for all objects on the surface of the earth?  Is the constant gravitational acceleration a universal law that governs the relationship between position and time?

And you find out that it is not true: a feather falls slower than a stone.  So you ask yourself why.

If you are Aristotle, you shall propose the following universal law in two statements:

1. All matter can be broken down into four components: earth, water, air, and fire.
2. The heavier elements goes toward the center of the earth; the lighter elements goes away from the center of the earth.
3. The speed of the body is proportional to its weight.

People tend to scoff at Aristotelian physics, but there are some truths in it.  Replace earth, water, and air by our concepts of solid, liquid, and gas.  Replace fire by temperature.   Plotting the pressure vs temperature of a substance leads to the classification of different phases of solid, liquid, and gas.

Aristotle’s third statement is also true, provided we assume that the speed is what we call as the terminal speed.  Raindrops fall faster and faster then slower and slower until they travels at a constant speed, due to friction with the air.  This constant speed is the terminal speed.

But what happens if the speed is not the terminal speed?  Then Galileo showed that Aristotelian physics collapses.  Galileo dropped a small and large cannonball from the top of the Tower of Pisa and showed that they arrived at the ground at the same time.  We need a new physics.

C.  Climax: The Scientific Laws

The endless struggle by scientists to find universals in the physical world resulted to the many laws of Physics.  But we must always remember that these laws have a certain range of validity.  Newton’s law of Universal Gravitation may be valid for the falling apple, for the moon around the earth, for the earth around the sun, and for other planets around the sun.  But when we go to the level of galaxies, things go awry.  Newton’s theory predicts that stars near the center of a galaxy travel faster than those far from the center.  But it turns out that in many galaxies, the stars near the center and away from the center have the same orbital speeds.

To resolve this problem, scientists propose a new kind of matter–not anymore earth, water, air, and fire–but something else, something dark and mysterious because we can’t observe it.  So they simply call this thing as “dark matter”, which permeates the voids between the stars.  Other scientists don’t like this idea.  They propose instead that the equation for Newton’s law of Gravitation should be modified.

D.  Denouement: Application of Universal Laws

Once the scientific law has been found, scientists apply them to different problems, instead of a smooth cannonball, we can have the whole London bridge falling down or an airplane flying up.  To do this, we cannot anymore rely on one universal.  We have to use others.

We first make a survey of what has been done before, define the concepts needed, determine the known values or imposed parameters,  write down the universal equations that relate these concepts and their values at certain conditions, describe the method for measuring the known and knowable parameters, present your results, write your conclusions, and list your references.  These are the parts of a scientific paper, which is published so that other scientists can duplicate what you did, learn from your mistakes, and use your results in proposing a new universal law.