First slide

Second law

Question

A particle of mass 'm' and initially at rest is acted by a force F = a t. N . Best representation of force displacement graph is (S assumes positive values only ) :

Moderate

A
B
C
D

Solution

F = at; $m \frac{dv}{dt} = at; \int_{0}^{v} mdv = \int_{0}^{t} at dt$
$mv = \frac{{at}^{2}}{2}; m \frac{ds}{dt} = \frac{{at}^{2}}{2}$
$\int_{0}^{v} mds = \int_{0}^{t} \frac{{at}^{2}}{2} dt$
ms = $\frac{{at}^{3}}{6} = \frac{a}{6} {(\frac{F}{a})}^{3};$
$∴ S \propto F^{3} or F \propto S^{\frac{1}{3}}$

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