Discussion Forum

The position vector r of particle of mass m is given by the following equation $r(t)=\alpha t^{3}\hat{i}+\beta t^{2}\hat{j}$ where , $\alpha= \frac{10}{3}ms^{-3}$, $\beta=5 ms^{-2}$ and m=0.1 kg .At t = 1s ,which of the following statement is (are ) true about the particle?

Answer:

Explanation:

 $r=\alpha t^{3}\hat{i}+\beta t^{2}\hat{j}$

$v=\frac{\text{d}r}{\text{d}t}=3\alpha t^{2}\hat{i}+2\beta t\hat{j}$

$a=\frac{\text{d}^{2}}{\text{d}t^{2}}=6\alpha t\hat{i}+2\beta \hat{j}$

At t=1 s,

(a) $v=3\times \frac{10}{3}\times 1\hat{i}+2\times 5\times 1\hat{j}$

$=(10\hat{i}+10\hat{j})m/s$

(b) $\hat{L}=\hat{r}\times \hat{p}$

$\left(\frac{10}{3}\times 1\hat{i}+5\times 1\hat{j}\right)\times 0.1(10\hat{i}+10\hat{j})$

$=\left(-\frac{5}{3}\hat{k}\right)Nms$

(c) $F=ma$

$=m\times \left(6\times \frac{10}{3}\times 1\hat{i}+2\times 5\hat{j}\right)$

$=(2\hat{i}+\hat{j})N$