Classical Mechanics Resource Packet (comments to: moloney@nextwork.rose-hulman.edu)

Suggestions for 'Active' Classroom Learning

When a light ball and a heavy ball are dropped together with the light ball just above the heavy ball, the lighter ball can rebound much higher than its initial height. Suppose that two balls of mass m and M are dropped with a spring (force constant k) between them. Will the maximum rebound occur when m and M are both touching the spring? What sort of qualitative condition is needed for the lighter mass (on top) to rebound to a maximum height above where it was dropped. [Hint: How long does it take the large mass to compress and release?]

(Good classroom problem after mass-spring behavior is discussed, including energy.)

A proposed automobile design has the bumper effectively
attached to the body of an 1100-kg car by a spring of constant 150 N/m.
The body structure is capable of withstanding a 5.5g's without suffering
damage due to crumpling. The car must pass a 'crash test' at 5 mph with
a stationary obstacle.

a) Is this a satisfactory design? State the reasons for your conclusion.
[No, g forces are very low, but bumper length is much too great.]

b) If the design is not satisfactory, improve the design
to meet the crash test specs without serious objections to your modified
design. [The spring constant can be greatly increased and still
meet the spec, with a much shorter length of bumper.]

**A mass
M hangs from a fixed support by a thin thread.An identical thread is connected
to the bottom of mass M. By grasping the bottom thread, a person can make
either the top thread or the bottom thread break. Discuss how this would
be done.** [1-2 min, Think-pair-share.Yank sharply on the lower thread
to break it first.]

**Drop a mass from height H onto a dashpot.
Design the dashpot spring constant and damping constant so that the time
to equilibrium is as short as possible. What are the qualitative considerations?
[Examine coefficients of the increasing and decreasing exponentials in
overdamped motion.]**

Why must a realistic potential energy function be continuous? (What would happen if there were a jump in the potential function?

Here are the components of a supposedly conservative force

Fx = x y^3, Fy = x^2 z^3, Fz=3/2 x^2 y z^2.

a) Convert Fx, Fy, and Fz into spherical polar coordinates using

x = r sin theta cos phi, y = r sin theta sin phi, and z = r cos theta.

b) For integrating in a circle at constant z, both r and theta must
be constant.

Show that an element of length dl along a path at constant r and theta
will be

**dl **= [ - **i **sin theta sin phi + **j** sin theta cos
phi ] r d phi

This must be done by means of a sketch, showing dl, and the angles theta
and phi . c) Integrate the dot product of the force **F** with the length
element **dl** around a path at constant z, holding theta and r constant,
showing that the value of the integral will be zero for all values of r
and theta. [The integral around any closed path should be zero for a conservative
force.]

Possible teaching strategies.

Might ask students to show that the potential function U is given by -x^2 y z^3/2 .

Instead of having a circular path at constant z centered on (0,0,0), one could ask for a circular path at constant z centered at (x0,0,0). This reinforces the idea of simple coordinate translations. U(x,y,z) would then be -(x-x0)^2 y z^3/2.

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

A 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 .**

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);**