- #1
Poetria
- 267
- 42
Homework Statement
Consider a bumpy road in which each hump has the shape
##y=\cosh(a)-\cosh(x)## for ##-a\leq x \leq a##
where ##y'(a)=-1## so ##a=arcsinh(1)##
L=2
##(x_p(t), y_p(t)) = (t, \cosh(a)-\cosh(t))##
##(x_q(t), y_q(t)) = (t, \cosh(a))##
We place the square wheel onto the bumpy road so that the point C touches the point A when t=0. The wheel rolls along so that the point D lands on the point B. Hence, the size (the length of one side) of the square wheel is fixed at the value found previously.
B is where the 0th hump and the 1st hump meet.
Q is the center of the wheel.
C is the middle of the side of the square which touches A at t=0
P is the point of contact of the wheel and the road.
At time t=0, the points A, C, and P coincide.
At all time t, the point P lies directly below the point Q.
If the wheel rotates at constant, unit speed, then the angle ##\theta## between P, Q and C satisfies ##\frac {d\theta}{dt} = 1##
For ##0\leq\theta\leq \frac \pi 4## find the speed of Q as a function of t.
2. The attempt at a solution
I have taken the derivatives of x and y: ##(x_q(t), y_q(t)) = (t, \cosh(a))##
##\sqrt{1+\sinh(t)^2}##
but this is wrong. :(