Question

A force $\overrightarrow{F}=(kx-9x^2)\hat{i}$ acts on a particle as the particle moves along an x-axis, with $\overrightarrow{F}$ in newtons, $x$ in meters. At $x=9$, the particle's kinetic energy is $60J$, and at $x=0m$, it is $33J$, then determine the value of $k$.

• A. 55
• B. 27.5
• C. 86
• D. 110

Answer 1

The correct option is A 55

Given that
$\overrightarrow{F}=(kx-9x^2)\hat{i}$
As the body moves along x-axis from $x_i=0$ to $x_f=9m$, the work done by the force is
$w={\int }_{x_i}^{x_f}Fxdx$
$\Rightarrow w={\int }_{x_i}^{x_f}(kx-9x^2)dx$
$\Rightarrow w={(\frac{k{x}^2}{2}-\frac{9x^3}{3}+c)}_0^9$
$\Rightarrow w=\frac{k{9}^2}{2}-\frac{9\times 9^3}{3}$
$\Rightarrow w=\frac{81k}{2}-2187$
$\Rightarrow w=40.5k-2187$
From work energy theorem
$w=\Delta k=(60-33)=27$
Thus, $40.5k-2187=27$
$=k=\frac{27+2187}{40.5}$
$\Rightarrow k=54.66\approx 55$
Hence, option (a) is correct