A disc is centered at the origin and is free to rotate about its center. A rod is attached to the disc at point and to another rod that can only move horizontally along the -axis, at point . Both and are revolute joints. If , and the disc rotates at a constant angular velocity of , where is the counter clockwise angle that makes with the positive -axis, find the maximum speed of point , (speed = , and ) , in , where is the -coordinate of point A ( is negative).
Note: The answer is a positive number.
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O P = R A P = L
Coordinates of P are:
x p = R cos θ y p = R sin θ
Say the link AP makes an angle ϕ with the positive X-axis. Coordinates of point A are then:
x a = − L cos ϕ + R cos θ y a = 0
This gives rise to the constraint equation for the Y coordinate of point P which is:
y p = R sin θ = L sin ϕ
Eliminating ϕ from the equation of x a gives the following equation (Simplification left out):
x a = R cos θ − L 2 − R 2 sin 2 θ
Differentiating wrt. time gives:
v a = ( − R sin θ + L 2 − R 2 sin 2 θ R 2 sin θ cos θ ) ω
The above equation is v a as a function of θ which varies uniformly with time. This is a standard unconstrained maximisation problem which I solved using a script of code. I swept across the range 0 ≤ θ ≤ 2 π and stored the maximum value.
The problem can also be solved by differentiating wrt θ and equating the result to zero, solving, and doing a second derivative test. I have not done so, as programming is convenient here.
The result is:
v a , m a x ≈ 1 3 . 1 8 3