Of course, a real block and a real ramp would have friction, which would tend
to retard (or even stop altogether) the motion of the block. In fact, we can
think of friction as being a force that tries to accelerate the block up
the ramp in opposition to gravity. The following function gives this
opposing acceleration:
friction[theta_] := 1/2 * 9.8 * 1.0 * Cos[theta]
(For those in the know about such things, the 1.0 that appears in the formula
is the coefficient of friction between the block and the ramp. The Greek
letter mu is often used to stand for the coefficient of friction.
Slippery things have small coefficients of friction, and rough things have high
coefficients of friction.)
Since gravity will tend to accelerate the block down the ramp, and friction
will tend to accelerate the block up the ramp, we should be able to calculate
the overall acceleration by subtracting the effects of friction from the
effects of gravity. We can redefine our distance function accordingly:
distance[t_, theta_] :=
1/2 * (gravity[theta]-friction[theta]) * t^2
This uses the same formula as before, only now we take both gravity and
friction into account. But there is something completely wrong with this
function. If you can't see right away what it is, plot it just as we did
before and the problem should become obvious.
Click here for the answer
The graph should reveal that the block does not move at all if the ramp is
inclined at Pi/4 radians (45 degrees). At this angle, friction exactly cancels
out gravity. (In case you're interested, this is a consequence of choosing 1.0
as the coefficient of friction.) For angles greater than Pi/4 radians, the
block moves down the ramp, and the steeper the ramp, the further it moves in
five seconds. But for angles less than Pi/4 radians, the graph shows that the
block starts moving backwards, up the ramp! Your everyday experience, of
course, tells you that this is not what will happen.