class 050826

1.  Write down the definition of the average rate of change of A with respect to B.

Average rate of change of A with respect to B is (change in A) / (change in B).

• If A is position and B is clock time, then by the definition, what is the average rate of change of position with respect to clock time?

Average rate of change of A with respect to B is (change in A) / (change in B).

If A is position and B is clock time then we just substitute 'position' for A and 'clock time' for B to get the statement

• average rate of change of position with respect to clock time is (change in position) / (change in clock time).

Note also that average velocity is average rate of change of position with respect to clock time.

• If A is velocity and B is clock time, then by the definition, what is the rate of change of velocity with respect to clock time?

Average rate of change of A with respect to B is (change in A) / (change in B).  Substituting 'velocity' for A and 'clock time' for B we get

• Average rate of change of velocity with respect to clock time is (change in velocity) / (change in clock time).

2.  Answer the following:

• If position changes from 10 cm to 30 cm while clock time changes from 3 seconds to 7 seconds, then what is the average rate of change of position with respect to clock time?

average rate of change of position with respect to clock time is (change in position) / (change in clock time).

For the given information, change in position is 30 cm - 10 cm = 20 cm and change in clock time is 7 sec - 3 sec = 4 sec.

So ave rate of change of position with respect to clock time is

• vAve = `ds / `dt = 20 cm / (4 sec) = 5 cm/s.

• If velocity changes from 50 cm / second to 30 cm / second while clock time changes from 42 seconds to 47 seconds, then what is the average rate of change of position with respect to clock time?

Average rate of change of velocity with respect to clock time is (change in velocity) / (change in clock time).

For the given information change in velocity is 30 cm/s - 50 cm/s = -20 cm/s.

Change in clock time is 47 sec - 42 sec = 5 sec.

So average rate of change of velocity with respect to clock time is (-20 cm/s) / (5 s) = -4 ( cm/s) / s = -4 (cm / s) ( 1 / s) = -4 cm /s^2.

Note that average rate of change of velocity with respect to clock time is called average acceleration.

3. The period of a pendulum of length L is T = .2 L^.5, where T is in sec and L is in cm.

If a bracket pendulum of length 6 cm 'hits' 5 times between the release of a ball from rest on an incline to the time the ball reaches the end of the incline 50 cm away, then

• How much time did the ball spend on the incline?

6 cm length implies T = .2 * 6^.5 sec = .48 sec.

5 'hits' will require 1/4 cycle + 4 * 1/2 cycle = 2.25 cycles.

2.25 cycles of a .48 sec pendulum will require 2.25 * .48 sec = 1.08 sec.

• What was the average velocity of the ball on the incline?

ave. vel is ave rate of change of position with respect to clock time = `ds / `dt = 50 cm / (1.08 sec) = 46 cm/s, approx..

• Sketch a graph of the ball's velocity vs. clock time, assuming that this graph is a straight line. 

The graph will be a straight line from the origin to the t = 1.08 sec point, sloping upward.

We know that the average velocity is 46 cm/s.

On a straight-line graph the average velocity occurs at the midpoint of the corresponding segment.

So the midpoint of the graph is at (.54 sec, 46 cm/s).

Since the initial velocity is 0, the final velocity must therefore be double the average velocity, or 2 * 46 cm/s = 92 cm/s.

• Indicate on your graph the coordinates of the starting point, the coordinates of the midpoint and the coordinates of the ending point.  The t = 0 velocity is 0.
• What is the final velocity of the ball on the incline?

We got carried away with the previous analysis and already found that the final velocity is 92 cm/s.

• At what rate is velocity changing with respect to clock time?

Ave rate of change of vel with resp to clock time is (change in vel) / (change in clock time) = `dv / `dt = (92 cm/s - 0) / (1.08 s - 0) = 92 cm/s / (1.08 s) = 84 cm/s^2, approx..

Observe the metal strap rotating on top of the die.  Estimate how many degrees in how much time for three trials, one in which the strap rotated less than 360 deg, on in which it rotates through several revolutions, and one in which the number of revolutions is somewhere between.

• Angular position will be the total angle of rotation, counting an additional 360 deg for every complete revolution, as measured from some fixed reference line.
• Average angular velocity is the average rate of change of angular position with respect to clock time.
• Average angular acceleration is the average rate of change of angular velocity with respect to clock time.

For each of your trials determine the following:

• The total change in angular position.
• The average angular velocity.
• The angular acceleration (hint: start by sketching a graph of angular velocity vs. clock time; you may assume that a graph of angular velocity vs. clock time is a straight line).