First slide

Work

Question

An object of mass 4 kg is moving along x-axis such that its position x- varies with time ‘t’ as $x^{2} = 16 t^{3}$ , with x in metre and t in sec. Work done as a function of time can be shown as

Moderate

A
B
C
D

Solution

$g i v e n p o s i t i o n x v a r y i n g w i t h t i m e t x^{2} = 16 t^{3} t a k e s q u a r e r o o t b o t h s i d e s$
$x = 4 t^{\frac{3}{2}}$
$d i f f e r e n t i a t e x w i t h r e s p e c t t o t i m e t \frac{d x}{d t} = 4 \times \frac{3}{2} t^{1 / 2} \frac{d x}{d t} = 6 t^{1 / 2} - - - (1) d x = 6 t^{1 / 2} d t - - - (2) a g a i n d i f f e r e n t i a t e e q n (1) w i t h r e s p e c t t o t i m e t$
$\Rightarrow \frac{d^{2} x}{d t^{2}} = \frac{3}{\sqrt{t}} =acceleration---(3) w o r k d o n e d W = F . d x h e r e f o r c e F = m a \Rightarrow d W = m a . d x s u b s t i t u t e e q u a t i o n s (2), (3) a n d m = 4 k g i n t h e a b o v e e q u a t i o n \Rightarrow d W = 4 (\frac{3}{\sqrt{t}}) 6 t^{1 / 2} d t \Rightarrow d W = 72 d t i n t e g r a t i n g \int d W = \int 72 d t \Rightarrow W = 72 t \Rightarrow W o r k α t i m e$

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