Question 46 of 210

The machine as shown has $2$ rods of length $1m$ connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a $2kg$ weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :

ASpeed which is $\dfrac{3}{4}th$ of that of the roller when the weight is $0.4m$ above the groundBConstant speedCDecreasing speedDIncreasing speed

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Answer: (C) Decreasing speed

${v}_{A}=\dfrac{dx}{dt}=\dfrac{d\left[2l\cos \left(\theta \right)\right]}{dt}=-2l\sin \left(\theta \right)\dfrac{d\theta }{dt}$ and
Vertical velocity of $B$, ${v}_{B}=\dfrac{dy}{dt}=\dfrac{d\left[l\sin \left(\theta \right)\right]}{dt}=l\cos \left(\theta \right)\dfrac{d\theta }{dt}$
$\dfrac{{v}_{A}}{{v}_{B}}=-2\tan \left(\theta \right) \Rightarrow {v}_{B}=-\dfrac{{v}_{A}\cot \left(\theta \right)}{2}$

$\begin{matrix}\theta \rightarrow \end{matrix}$increase
$\cot \theta \rightarrow$decrease
Hence, $\left|{v}_{B}\right|\rightarrow$decrease.