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

• A. $\frac{v}{\sin \theta}$
• B. $v \sin \theta$
• C. $\frac{v}{\cos \theta}$
• D. $v \cos \theta$

Answer 1

Answer: (A)

Let at any instant of time, the length $A B$ be $l$.
Here angle $\theta$ and length $l$ vary with time, then using Pythagorus theorem in $\triangle A B C$

Differentiating both sides w.r.t. $t$, we get
$x^{2}+y^{2}=l^{2}$
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 no vertical motion of the block, so
$\frac{d y}{d t}=0, \frac{d x}{d t}=v_{x}$ and $\frac{d l}{d t}=v$
$\therefore 2 x v_{x}=2 l v$
or $v_{x}=\frac{l}{x} v$
or $v_{x}=\frac{v}{\left(\frac{x}{l}\right)}=\frac{v}{\sin \theta}$