Question

Two objects, A and B, are connected by hinges to a rigid rod that has a length L. The objects slide along perpendicular guide rails as shown in the figure below. Assume object A slides to the left with a constant speed v.

(a) Find the velocity vB of object B as a function of the angle θ. (Use any variable or symbol stated above as necessary.)
vB =

(b) Describe vB relative to v. Is vB always smaller than v, larger than v, or the same as v, or does it have some other relationship?

Answer 1

Interesting question. Note that the position of B as a function of A is
\[ y = L\sin(\theta) \]
Or, because A and B trace out a right triangle,
\[y = \sqrt{L^2-x^2}\]
We can find the speed of B by differentiating with respect to time. Via the chain rule, we know that
\[\frac{d}{dt} = \frac{dx}{dt} \frac{d}{dx} = -v \frac{d}{dx}\]
where -v is the velocity of object A. so,
\[ v_B = \frac{d}{dt} y = -v \frac{d}{dx} \sqrt{L^2-x^2} = v\frac{x}{\sqrt{L^2-x^2}}\]
Finally, notice that
\[\frac{x}{\sqrt{L^2-x^2}} = \cot(\theta)\]
so finally
\[v_B = v\cot(\theta) \]

Answer 2

for help answering the second part, refer to this graph: http://www.wolframalpha.com/input/?i=cot%28x%29+from+zero+to+pi%2F2

Answer 3

Again, nicely done Jem.

Answer 4

Thanks, James