Kinematics 6 Question 7
7. In the arrangement shown in the figure, the ends $P$ and $Q$ of an unstretchable string move downwards with uniform speed $U$. Pulleys $A$ and $B$ are fixed. Mass $M$ moves upwards with a speed
(1982, 3M)
(a) $2 U \cos \theta$
(b) $\frac{U}{\cos \theta}$
(c) $\frac{2 U}{\cos \theta}$
(d) $U \cos \theta$
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Answer:
Correct Answer: 7. (b)
Solution:
- In the right angle $\triangle P Q R$
$$ l^{2}=c^{2}+y^{2} $$
$C=$ constant, $/$ and $y$ are variable
Differentiating this equation with respect to time, we get
$2 l \frac{d l}{d t}=0+2 y \frac{d y}{d t}$ or $-\frac{d y}{d t}=\frac{l}{y}-\frac{d l}{d t}$
Here, $\quad-\frac{d y}{d t}=v _M$
$\Rightarrow \quad \frac{l}{y}=\frac{1}{\cos \theta}$ and $\quad-d l / d t=U$
Hence,
$$ v _M=\frac{U}{\cos \theta} $$
