Word Problems

lil_cookie · #1 $Old$ May 21st, 2009, 05:24 PM

Hey Guys, I'm having a lot of problems trying to answer a word problem and could use some help thanks.
....its a long one so sorry

A point on the outside of the axle of a truck has a circular motion that can be modelled by a since curve.
If you measure distance from the axle's centre to the bottom of the truck, that distance remains constant as long as the truck is on a level road. If the truck goes over a bump, the springs absorb the shock but the truck bounces for a while.
A truck is moving so that the axle, which is 6cm in diameter, rotates at 1 rotation per second. As the truck hits a bump, the spring depresses by 20cm and then continues to depress by 80% of the previous bounce as it bounces every half-second.
If the middle of the axle is 30cm from the bottom of the truck, construct a graph and equation illustrating the distance from a point on the outside of the axle to the bottom of the truck.

The question is only worth 2 marks but I would still like to try to get it....you don't need to make the graph but helping with the equation would be nice....

Thanks

Media_Man · #2 $Old$ May 22nd, 2009, 11:21 AM

First, the equation: $h(t)=30+20(.80)^{2t}\cos(4\pi t)+3\sin(2\pi t)$

Now, the explanation, term by term.

"30" represents the "natural" height of the center of the axle above the ground.
"20" represents the initial amplitude of the bump.
" $(.80)^{2t}$ " means that every half second, the amplitude of the bump decreases by 80%
" $\cos()$ " is used because when t=0, the truck has just hit the bump and is 20cm above its natural height, beginning its decline, hence the entire term $20(.80)^{2t}\cos(4\pi t)$ =+20 at t=0.
" $4\pi t$ " is used because the period of the bounce is half a second and the argument should be $\frac{2\pi t}{P}$ for period P.
"3" represents the amplitude of the point spinning around the axle. (half the diameter)
"sin" is used arbitrarily, as the problem does not state the orientation of this point at the exact moment the truck hits the bump.
" $2\pi$ " is the argument used because the axle has a period of 1 second.

Graph these three terms separately and it is easy to see. The straight line at h=30 represents the trucks "natural" height above the ground, the equilibrium of the system. The cosine term is a half second period bounce steadily decreasing in amplitude, and the sine term represents the point on the axle happily spinning round.

Lastly, we appeal to the Law of Superposition, which despite it's scary name, simply means that by adding all the separate terms governing this motion together, we get the end result of this motion, albeit a complicated-looking graph.

lil_cookie · #3 $Old$ May 22nd, 2009, 04:25 PM

Thank you so much!
There is no way I would have gotten this without you!

hebraica · #4 $Old$ November 6th, 2009, 02:01 PM

Hi,

I would like to know why does 2t in 20(0.8)^2t means "half second".

Thank you