Given, the position vector of the particle is

$\overrightarrow{r}=10t\hat{i}+15t^2\hat{j}+7\hat{k}$

The velocity of the particle can be calculated as follows:

$\begin{array}{rcl}\overrightarrow{v}& =& \frac{d\overrightarrow{r}}{dt}\\ & =& \frac{d}{dt}\left(10t\hat{i}+15t^2\hat{j}+7\hat{k}\right)\\ & =& 10\hat{i}+30t\hat{j}\end{array}$

And, the acceleration of the particle is given by

$\begin{array}{rcl}\overrightarrow{a}& =& \frac{d\overrightarrow{v}}{dt}\\ & =& \frac{d}{dt}\left(10\hat{i}+30t\hat{j}\right)\\ & =& 30\hat{j}\end{array}$

Hence, the force on the particle can be written as

$\begin{array}{rcl}\overrightarrow{F}& =& m\overrightarrow{a}\\ & =& 30m\hat{j}\end{array}$