Maple Lab in PH 314 Wed 9/22/99

We will not meet on Tuesday October 12, and possibly one other day to compensate for this meeting.

Energy diagrams, forces, and equilibrium (by J. W. Harrell, U of Alabama)

(To be collected as homework in a few days)

A 2-kg particle is subject to a conservative force for which the potential energy function U(x) is given by

U(x) = 11x^2 -7x^3 + x^4 .

Part A (10),

1. Use Maple to plot U(x) from x= -1 to x= +5 .

2. Examine the plot and estimate the following

• a) The values of x for which the force on the particle is zero
• b) The value of x for which the force on the particle is a maximum
• c) The value of x for which the force is a minimum
• d) The ranges of x for which the force is positive
• e) The ranges of x for which the force is negative

3. a) At what points would the particle be in stable equilibrium if placed there at rest?

   b) At what points would the particle be in unstable equilibrium if placed there at rest?

4. Now use Maple to calculate the force function F(x) = -dU(x)/dx.

5. On a single graph, plot both U(x) and F(x). Is your graph consistent with your answers to questions 2 and 3? If not, then reconcile your answers.

6. Now suppose the particle has an energy of 10 units.

On a single graph, plot both U(x) and total energy. (Don't use E for energy; Maple has claimed E for something else.)

7. If no other forces act on the particle, determine from your graph

• a) Where the speed of the partricle is greatest
• b) Where the speed is zero

Notes:

• to plot a single function use plot(f(x),x=a..b);
• to plot two functions on the same graph, use plot({f(x),g(x)},x=a..b);

to differentiate, use diff(f(x),x);

Part B (5)

Determine the oscillation frequency of a 1.5-kg mass near the potential minima between x=-1 and x=+2 m in the potential U(x) given above. ( Using an analytic technique.)

Part C (10)

Suppose the particle has an energy of +1 units.

• Locate the 'turning points' of the motion near x=0 (with evalf and solve commands)
• Write an integral which will give the time of travel between the turning points.
• Evaluate the integral using evalf
• Compare the time to half the period of the motion found by the analytic technique

Part D Osage orange problem

Bring up the osage orange problem and try different wind speeds until you find the correct value to make the orange drop 20 m while travelling laterally 12 m. Convert your answer of wind speed to miles per hour. osage.mws