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Math 142 Practice problems for final exam
1. Hinged pendulum
Consider the pendulum shown above. Sketch the potential energy curve and the phase plane diagram for this pendulum.
2. Earth oscillator Imagine we drill a hole through the center of the earth from one side to the other. The force of gravity inside the earth is given by
F ( r ) = −
GmM ( r ) r^2
where M ( r ) is the mass of the sphere of radius r within the earth. (Assume the density of the earth is uniform.)
If we drop a ball from the surface of the earth, how long does it take to return.
3. Mafia dynamics Suppose there are two mob families in the same area. Each member of family x recruits new members at a rate γ, and each member of family y recruits at a rate σ. Also, each member of family x kills off members of family y at a rate a , and vice versa at a rate b.
Can these two families coexist in equilibrium? If so, what is required for them to coexist?
4. Traffic flow diagram Consider the point marked on the traffic flow diagram;
q
Draw a line with slope equal to the car velocity at this density. Also draw a line with slope equal to the density wave velocity. If a light turns red, then the cars will come to a stop. Draw a line representing the shock velocity for such a situation.
5. Gravity We did not have to include gravity in our spring-mass model. Why?
a. Our model only applied in zero-gravity. b. Including gravity only changes the equilibrium displacement, not the motion with respect to equilib- rium. c. Using the small angle approximation, the effect of gravity became negligible. d. The effect of gravity is included in Hooke’s Law.
6. Roots of the characteristic equation A saddle point exits when the two roots of the characteristic equation are described by
a. r 1 < 0 and r 2 > 0 b. r 1 and r 2 complex c. r 1 and r 2 positive d. r 1 < 0 and r 2 complex
7. Characteristics and shocks Let the initial density of cars on a roadway be described by
ρ( x ) =
ρ 0 x < 0 0 x > 0
Draw the characteristics for this situation. Now let
ρ( x ) =
0 x < 0 ρ 0 x > 0
Draw the characteristics including the shock.
