Kinematics Ques 14
- The position vector $\mathbf{r}$ of particle of mass $m$ is given by the following equation $\mathbf{r}(t)=\alpha t^{3} \hat{\mathbf{i}}+\beta t^{2} \hat{\mathbf{j}} \quad$ where, $\alpha=\frac{10}{3} ms^{-3}, \quad \beta=5 ms^{-2}$ and $m=0.1 kg$. At $t=1 s$, which of the following statement(s) is (are) true about the particle?
(2016 Adv.)
(a) The velocity $\mathbf{v}$ is given by $\mathbf{v}=(10 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}) ms^{-1}$
(b) The angular momentum $\mathbf{L}$ with respect to the origin is given by $\mathbf{L}=(5 / 3) \hat{\mathbf{k}} Nms$
(c) The force $\mathbf{F}$ is given by $\mathbf{F}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}) N$
(d) The torque $\tau$ with respect to the origin is given by $\tau=-\frac{20}{3} \hat{\mathbf{k}} Nm$
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Answer:
Correct Answer: 14.(a, d)
Solution:
Formula:
- $\mathbf{r}=\alpha t^{3} \hat{\mathbf{i}}+\beta t^{2} \hat{\mathbf{j}}$
$\mathbf{v}=\frac{d \mathbf{r}}{d t}=3 \alpha t^{2} \hat{\mathbf{i}}+2 \beta t \hat{\mathbf{j}}$
$\mathbf{a}=\frac{d^{2} \mathbf{r}}{d t^{2}}=6 \alpha \hat{\mathbf{i}}+2 \beta \hat{\mathbf{j}}$ At $t=1 s$,
(a) $\mathbf{v}=3 \times \frac{10}{3} \times 1 \hat{\mathbf{i}}+2 \times 5 \times 1 \hat{\mathbf{j}}=(10 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}) m / s$
(b) $\mathbf{L}=\mathbf{r} \times \mathbf{p}=\frac{10}{3} \times 1 \hat{\mathbf{i}}+5 \times 1 \hat{\mathbf{j}} \times 0.1(10 \hat{\mathbf{i}}+10 \hat{\mathbf{j}})$
$=-\frac{5}{3} \hat{\mathbf{k}} N-ms$
(c) $\mathbf{F}=m \mathbf{a}=m \times 6 \times \frac{10}{3} \times 1 \hat{\mathbf{i}}+2 \times 5 \hat{\mathbf{j}}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}) N$
(d) $\tau=r \times \mathbf{F}=\frac{10}{3} \hat{\mathbf{i}}+5 \hat{\mathbf{j}} \times(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})$
$$ =+\frac{10}{3} \hat{\mathbf{k}}+10(-\hat{\mathbf{k}})=\frac{-20}{3} \hat{\mathbf{k}} \quad N-m $$
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