Friday Feb. 1, 2008

No real announcements at the start of class other than to say that it is often hard for students to make it to the 8am T Th section of NATS because it is too cold at that time of day.  It is equally difficult for the 2 pm MWF students because by then it is too nice outside.
Newton's Law of Universal Gravitation is an equation that allows you to calculate the gravitational attraction between two objects.  We really didn't work through the following figures in class (except for filling in the boxes at the bottom of p. 28 with the English and Metric units for mass and weight).  The reason that they have been included on the online notes is that with a little thought you can appreciate and understand why certain variables appear in Newton's Law and why they appear in either the numerator (direct proportionality) or in the denominator (inverse proportionality).


The gravitational attraction between two objects (M and m in the figures) depends first of all on the distance separating the objects.  The  gravitational force becomes weaker the further away the two objects are from each other.  In the bottom picture above and the top figure below we see that the attractive force also depends on the masses of the two objects.


The complete formula is shown in the middle of the page above.  G is a constant.  On the surface of the earth G, M, and R don't change.  The gravitational acceleration, g, is just the quantity  [G times Mearth divided by ( Rearth )2 ].  To determine the weight (on the earth's surface) of an object with mass m you simply multiply m x g. 

Down at the bottom of the page are the Metric and English units of mass and weight.  You have probably heard of pounds, grams, and kilograms.  You might not have heard of dynes, Newtons, and especially slugs.
Here's another page from the photocopied Class Notes that we didn't cover in class.  The weight of a person on the earth and the moon is calculated in English and metric units.

The mass of a person would be the same on the earth and on the moon. The weight of a person depends on the person's mass and on the strength of gravity (the acceleration of gravity term, the g variable below).


(1) The course instructor weighs about 160 pounds.   In (2) we see that the gravitational acceleration is 32 ft/sec2 in English units (on the earth).  The meaning of this value is shown in (3).  If you drop an object it will start to fall and will speed up as it continues to fall.  Gravity will cause a falling object to fall 32 ft/sec faster every second.  Dividing the instructor's weight by the gravitation acceleration in (4) we obtain the instructor's mass, 5 slugs, in English units.

In metric units, the instructor has a mass of 73 kilograms (5).  The gravitation acceleration is 9.8 m/sec (6).  Multiplying these two values, in (7), we find that the instructor weighs 715 Newtons.

On the moon, the mass stays the same.  Gravity is weaker, so the value of g is smaller.  The instructor would weigh quite a bit less (117 Newtons or 26 pounds) on the moon compared to the earth.
Mercury barometers are used to measure atmospheric pressure.  A mercury barometer is really just a balance that can be used to weigh the atmosphere.  A basic understanding of how a mercury barometer works is something that every college graduate should know.  You'll find most of what follows on p. 29 in the photocopied Class Notes.

The instrument above ( a u-shaped glass tube filled with a liquid of some kind) is a manometer and can be used to measure pressure difference.  The two ends of the tube are open so that air can get inside and air pressure can press on the liquid.  Given that the liquid levels on the two sides of  the manometer are equal, what could you about PL and PR?

The liquid can slosh back and forth just like the pans on a balance can move up and down.  A manometer really behaves just like a pan balance.


PL and PR are equal (note you don't really know what either pressure is just that they are equal).

Now the situation is a little different, the liquid levels are no longer equal.  You probably realize that the air pressure on the left, PL, is a little higher than the air pressure on the right, PR.  PL is now being balanced by PR + P acting together.  P is the pressure produced by the weight of the extra fluid on the right hand side of the manometer (the fluid that lies above the dotted line).  The height of the column of extra liquid provides a measure of the difference between PL and PR.

 Next we will go an extreme and close off the right hand side of the manometer.

 Air pressure can't get into the right tube any more.  Now at the level of the dotted line the balance is between Pair and P (pressure by the extra liquid on the right).  If Pair changes, the height of the right column, h,  will change.  You now have a barometer, an instrument that can measure and monitor the atmospheric pressure.

 Barometers like this are usually filled with mercury.  Mercury is a liquid.  You need a liquid that can slosh back and forth in response to changes in air pressure.  Mercury is also dense which means the barometer won't need to be as tall as if you used something like water.  A water barometer would need to be over 30 feet tall.  With mercury you will need only a 30 inch tall column to balance the weight of the atmosphere at sea level under normal conditions (remember the 30 inches of mercury pressure units mentioned earlier).  Mercury also has a low rate of evaporation so you don't have much mercury gas at the top of the right tube.

Finally here is a more conventional barometer design.  The bowl of mercury is usually covered in such a way that it can sense changes in pressure but not evaporate and fill the room with poisonous mercury vapor.

The figure above (p. 30 in the photocopied Class Notes) first shows average sea level pressure values (1000 mb or 30 inches of mercury are close enough in this class).

Sea level pressures usually fall between 950 mb and 1050 mb. 

Record high sea level pressure values occur during cold weather. 

Record low pressure values have all been set by intense hurricanes (the record setting low pressure is the reason these storms were so intense).  Hurricane Wilma in 2005 set a new record low sea level pressure reading for the Atlantic.  Hurricane Katrina had a pressure of 902 mb.  You'll find a list of the most intense, destructive, and deadly hurricanes on p. 146a.

Air pressure is a force that pushes downward, upward, and sideways.  If you fill a balloon with air and then push downward on it, you can feel the air in the balloon pushing back (pushing upward).  You'd see the air in the balloon pushing sideways as well.

The bottom person in the people pyramid above must push upward with enough force to support the other people.  The air in a layer at the bottom of the atmosphere must do the same thing.  It pushes upward with enough force to support the weight of all the air overhead. 

The air pressure in the four tires on your automobile pushes down on the road (that's something you would feel if the car ran over your foot) and pushes upward with enough force to keep the 1000 or 2000 pound vehicle off the road.

We used a stack of brick to understand air pressure and why it decreases with increasing altitude.  Now we will imagine a stack of matresses to understand why air density decreases with increasing altitude.



Mattresses are compressible.  The mattress at the bottom of the pile is compressed the most by the weight of all the mattresses above.  The mattresses higher up aren't squished as much because their is less weight remaining above.

In the case of the atmosphere you can think of layers of air instead of mattresses.


1. You can first notice and remember that pressure decreases with increasing altitude.

Each layer of air contain the same amount (mass) of air.  You can tell because the pressure decrease as you move upward through each layer is the same (100 mb). 

2. The densest air is found in the bottom layer because the air is squeezed into a smaller volume than the other layers. 

3. You again notice something that we covered earlier: the most rapid rate of pressure decrease with increasing altitude is in the densest air in the bottom air layer.
We finished class with a demonstration of the upward force caused by air pressure.
The demonstration is summarized on p. 35a in the photocopied Classnotes.
We'll come back to this demonstration briefly in class next Monday and look at the actual forces being exerted on the water and the glass
First the case of a water balloon.

The figure at left shows air pressure (red arrows) pushing on all the sides of the balloon.  Because pressure decreases with increasing altitude, the pressure pushing downward on the top of the balloon is a little weaker (strength=14) than the pressure pushing upward at the bottom of the balloon (strength=15).  The two sideways forces cancel each other out.  The total effect of the pressure is a weak upward force (shown on the right figure, you might have heard this called a bouyant force).  Gravity exerts a downward force on the water balloon.  In the figure at right you can see that the gravity force (strength=10) is stronger than the upward pressure difference force (strength=1).  The balloon falls as a result. 

In the demonstration a wine glass is filled with water.  A small plastic lid is used to cover the wine glass.  You can then turn the glass upside down without the water falling out. 


All the same forces are shown again in the left most figure.  In the right two figures we separate this into two parts.  First the water inside the glass isn't feeling the downward and sideways pressure forces (because they're pushing on the glass).  Gravity still pulls downward on the water but the upward pressure force is able to overcome the downward pull of gravity.  The upward pointing pressure force is used to overcome gravity not to cancel out the downward pointing pressure force.

The demonstration was repeated using a 4 Liter flash (more than a gallon of water, more than 8 pounds of water).  The upward pressure force was still able to keep the water in the flask (much of the weight of the water is pushing against the sides of the flask which the instructor was supporting with his arms).