First slide

Questions using constraints

Question

A block is dragged on a smooth plane with the help of a rope which moves with a velocity v as shown in the figure. The horizontal velocity of the block is

Moderate

A

$\frac{v}{\sin θ}$
B

$v sinθ$
C

$\frac{v}{\cos θ}$
D

$v cosθ$

Solution

Let at any instant of time, the length AB be l, here angle $θ$ and length l vary with time, then using Pythagoras theorem in $∆ A B C$ ,
$x^{2} + y^{2} = t^{2}$
On differentiating both sides w.r.t t, we get
or $2 x \frac{d x}{d t} + 2 y \frac{d y}{d t} = 2 l \frac{d l}{d t}$
As, there is no vertical motion of the block, so
$\frac{d y}{d t} = 0, \frac{d x}{d t} = v_{x} and \frac{d l}{d t} = v$
$∴ 2 {xv}_{x} = 2 lv$
$or v_{x} = \frac{l}{x} v or v_{x} = \frac{v}{(\frac{x}{l})} = \frac{v}{\sin θ}$

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