Class Notes _070907

Physics I 070907

Summary of the course to date:

Definition of average rate; midpoint property of straight-line graphs

If the graph of quantity y vs. quantity t is a straight line, then over any interval of the graph, the average values of y and t occur at the midpoint of the interval.

The average rate of change of A with respect to B is

average rate = change in A / change in B.

Definition of average velocity and average acceleration

The average velocity of an object over an interval is the average rate of change of its position with respect to clock time over that interval.

On an interval an object has an initial velocity, a final velocity and an average velocity.

At an instant an object has an instantaneous velocity.

On an interval an object's change in velocity is the difference between its initial and final velocity.

The average acceleration of an object over an interval is the average rate of change of its velocity with respect to clock time over that interval.

An object's change in velocity of an interval corresponds to the 'rise' of its velocity vs. clock time graph over that interval, and the change in clock time corresponds to the 'run'. 

• So the slope between two points of a v vs. t graph, which is rise / run, corresponds to change in velocity / change in clock time, which is average rate of change of velocity with respect to clock time, or acceleration.

(For reasons we will see later, the above statements are well-defined and valid to the extent that what we refer to above as 'an object' behaves as 'a point mass'.  More about this later.)

Net force; constant and non-constant net forces

The net force on an object is the sum of all the forces acting on it.

In some situations the net force on an object is constant.  In those cases the graph of velocity vs. clock time is a straight line and we can say the following:

• If the v vs. t graph is a straight line, then the motion over any interval can be analyzed by considering a v vs. t trapezoid with 'altitudes' v0 and vf and width `dt.
• Complete analysis of this trapezoid defines the relationships among the quantities v0, vf, `dt, a, `ds, vAve and `dv, considered over the interval.
• The motion over the interval can be completely characterized in terms of the quantities v0, vf, `dt, a and `ds and the relationships among these quantities.
• The specific relationships are completely defined by four equations, call the equations of uniformly accelerated motion:

equation 1:  `ds = (vf + v0) / 2 * `dt

equation 2: vf = v0 + a `dt

equation 3: `ds = v0 `dt + 1/2 a `dt^2

equation 4: vf^2 = v0^2 + 2 a `ds.

• For any uniformly accelerated motion (i.e., for any motion which corresponds to constant net force, i.e., straight-line v vs. t graph), if we know three of the quantities v0, vf, `dt, a and `ds we can use the equations to find the other two.
• To develop and understanding of uniformly accelerated motion, we need to rely on these equations as little as possible and on direct reasoning as much as possible.  The equations involve only v0, vf, `ds, `dt and a; direct reasoning and understanding of uniform acceleration also requires consideration of the quantities `dv and vAve.
• Direct reasoning can be depicted by 'flow diagrams' with a triangular structure.
• There are ten possible combinations of three of the five variables v0, vf, a, `ds and `dt.  Any of these combinations can be easily analyzed, without understanding anything, using the equations of uniformly accelerated motion.  Eight of these combinations can be analyzed by direct reasoning, which leads to an understanding of uniformly accelerated motion.  Two of the ten combinations cannot easily be solved by direct reasoning and require the use of the equations of uniformly accelerated motion.
• To use the equations of uniformly accelerated motion, just list the three quantities you know. 

Two of the four equations will contain these three quantities. 

Each of these two equations will contain exactly one unknown quantity.

Solve each equation for its unknown quantity, substitute the values of your known quantities, and you will have the values of all five quantities.

One exception to this rule:  Equation 3 is a quadratic equation in `dt.  It can be confusing to solve the quadratic equation and interpret the resulting two solutions.  So you should probably avoid using equation 3 to solve for `dt.  If `dt is an unknown quantity and equation 3 arises, you will still have another equation to use.  Use that one, then use the information you have to solve for `dt without using equation 3.

One warning:  If you use equation 4 to find vf or v0, remember that the equation x^2 = c has two solutions, x = sqrt(c) and x = -sqrt(c), and both solutions must be considered.

All the above applies if the net force is constant.  Our typical example of a situation with constant net force is a ball rolling down a straight incline, with gravitational force acting straight down, the normal elastic or compressive force exerted by the incline directed perpendicular to the incline and balancing one component of the gravitational force, and the unbalanced component of the gravitational force accelerating the object down the incline.

Almost all the straight-line motion you analyze in this course will be uniformly accelerated motion.

In other situations the net force is not constant.  In those cases the graph of velocity vs. clock time is not a straight line and none of the above is valid.

• Almost any situation involving non-constant net forces must be based on calculus.
• We can accept and use certain results from calculus without knowing calculus.
• The ideas of energy conservation and conservation of momentum arise naturally in cases of uniform acceleration from the equations of uniformly accelerated motion.
• These ideas related to energy and momentum can be applied to situations in which acceleration does not have to be uniform.  To prove that it is so we need calculus, but we can use the ideas without using the calculus.
• Other types of motion, such as circular motion (things moving in circular paths) and simple harmonic motion (for example the motion of a pendulum or a mass on a spring) require calculus for proof, but can be analyzed using only the geometry of the circle (actually there's just a little trigonometry in there too, but the trigonometry used in this course is very easy).

Vectors

As you have already seen, vector quantities are quantities with magnitude and direction.

Vector quantities can be represented by arrows pointing in the direction of the quantitiy and having lengths which represent the magnitudes of the quantities.

Any vector drawn on an x-y coordinate system has the following:

• an angle which is measured counterclockwise from the positive x axis
• a magnitude which is represented by the length of the vector
• components parallel to the x and y axes which can be easily depicted using projection lines

the action of the vector is equivalent to the combined action of its components

A good vector picture of a situation is often the main thing you need in order to understand the situation.

You don't need to know it just yet, but here's a summary of the trigonometry you will use in this course.  You might know it already, and if not you can learn this much trigonometry in an hour or less.

• The x component of a vector of magnitude R at angle theta is R_x = R cos(theta).
• The y component of a vector of magnitude R at angle theta is R_y = R sin(theta).
• If a vector has x component R_x and y component R_y its magnitude is R = sqrt( R_x^2 + R_y^2).  This is just the pythagorean theorem.
• If a vector has x component R_x and y component R_y then its angle is theta = arcTan(R_y / R_x), plus 180 degrees if R_x is negative.
• cos, sin and arcTan are easy to calculate using a couple of buttons on your calculator, just like the square root button.  You can find the sin and cos buttons; arcTan typically uses the Tan button and the 2d function button.  It takes most people 5 minutes or less to learn to use these buttons.  You do have to be careful about degree mode vs. radian mode, but that's not hard to get used to either.