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Recall our classical spring problem deals with modeling the position (dis- placement) of a spring as it vibrates/oscillates. We start with a spring hanging from some surface, and we hang an object of mass m on it, which causes it to elongate L units. This is what we call the equilibrium/original state or position. Letting y(t) be the function which represents the displace- ment from this equilibrium state at time t, we will use the convention that upwards movements/forces are negative quantities, while downward ones are positive. So:
y(t) > 0 _ spring is stretched down y(t) = 0 _ spring is at equilibrium (no displacement) y(t) < 0 _ spring is compressed up
We used Newton’s Law, F (t) = ma(t), to get a DE for y(t), since acceler- ation is just y′′(t). To do this, we added up all of the forces acting on the object at time t. These consisted of gravity (Fg ), resistance/drag (Fr (t)), and spring force (Fs(t)). That is:
my′′(t) = F (t) = Fg + Fr(t) + Fs(t)
my′′(t) = mg − γy′(t) − k(L + y(t))
my′′(t) = −γy′(t) − ky(t) (since mg = kL)
So the our spring problem is modeled by the homogeneous DE:
my′′(t) + γy′(t) + ky(t) = 0, y(0) = y 0 , y′(0) = y 0 ′
where m = mass of the object, γ is the damping/resistance constant, and k = mg L is the spring constant.
Now suppose that in addition to the forces above, there is an external/additional force Fe(t) which acts on the object. Fe(t) can be a function, like cosine, or just a constant. Modifying the above derivation, we’d get that this spring problem is modeled by:
my′′(t) + γy′(t) + ky(t) = Fe(t), y(0) = y 0 , y′(0) = y′ 0
which is now a non-homogeneous DE.
PROBLEMS
For both problems, we assume the following setup. Suppose there is a 3- meter spring hanging down from the ceiling of a room which is 5 meters high. A tile of mass 2 kg is hung on the end of the spring, causing it to elongate 12 meters after coming to rest. So, the equilibrium state is when the spring is hanging down 3.5 meters. It’s also known that when the spring has velocity 2 m/s, the drag/resistance force is -2. You should use a gravi- tational constant of 9.8.
Problem 1 (Homogeneous) Suppose you will set the spring into motion by compressing the spring upwards 1 meter, and giving it an initial downward velocity of 2 m/s.
a) Write down and solve a DE for y(t), which is the displacement from the equilibrium state (which is when the spring has total length 3.5 = 3 (original) + .5 (elongation))?
b) Graph your solution. Remember, our orientation is ”backwards” in the sense that positive y-values correspond to the spring moving down.
c) After t is past zero, how close does it come to touching the ceiling and the floor? That is, during the first rebound, how close is it to the floor, and during the second rebound, how close is it to the ceiling?
d) When does the spring first return to its equilibrium state (that is, y(t) = 0)?
e) Is it stretched down or compressed up at t = 2?
f) How long would a 1-meter tall child have to wait until it’s safe to walk under the spring?
g) What is the limit of your solution as t → ∞?
g) Suppose someone will throw this block of weight 20 down from the ceiling and onto the tile. This can be modeled as a falling body problem, where the weight is 20 (so the mass is (^920). 8 ) and let’s say the drag coefficient of the block is 0.25. With how much initial velocity must the person throw this block in order to guarantee that when it hits the tray, it will have a velocity twice the size of the y′ 0 that you calculated you needed for the spring in part e? That is, if you said y′ 0 needed to be 5 m/s, then find out how fast you have to throw this block down from the ceiling so that when it hits the tile, it has velocity 10 m/s. THE MATH GETS CRAZY ON THIS ONE...JUST TELL ME HOW YOU WOULD DO IT...
h) Please comment on the feasibility of this model in problem 2. That is, if you think it disregards something (like conservation of momen- tum, kinetic energy, etc), please feel free to tell me about these issues here. It’s my hope that you’ll believe the model isn’t totally crap and that it provides a reasonably accurate picture of what’s happening.
