Question

A time varying force applied on a body of mass $m$ is $F=at-b{t}^2$ where $a$ and $b$ are arbitrary constants. The correct options is/are

• A. The force is maximum at $t=\frac{a}{2b}$
• B. Impulse imparted between $t=0$ and the instant at which force is maximum is $\frac{a^3}{12b^2}$
• C. Maximum force is $\frac{a^2}{4b}$
• D. Maximum force is $\frac{a^2}{2b}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A The force is maximum at $t=\frac{a}{2b}$
B Impulse imparted between $t=0$ and the instant at which force is maximum is $\frac{a^3}{12b^2}$
C Maximum force is $\frac{a^2}{4b}$

$F=at-b{t}^2$
For $F$ to be maximum $\Rightarrow \frac{dF}{dt}=0$
$\therefore a-2bt=0$ i.e., $t=\frac{a}{2b}$

$F_{t=a/2b}=\frac{a\times a}{2b}-b\times \frac{a^2}{4b^2}=\frac{a^2}{4b}$

$\text{Impulse}=\underset{0}{\overset{a/2b}{\int }}Fdt={(\frac{a{t}^2}{2}-\frac{b{t}^3}{3})}_0^{a/2b}$

$=\frac{a^3}{8b^2}-\frac{a^3}{24b^2}=\frac{a^3}{12b^2}$