Question
The position of an object having mass 0.1 kg as a function of time $t$ is given as $\overrightarrow{r} = \left( 10t^{2}\widehat{i} + 5t^{3}\widehat{j} \right)m$. At $t = 1\text{ }s$, which of the following statements are correct?
A. The linear momentum $\overrightarrow{p} = (2\widehat{i} + 1.5\widehat{j})kg \cdot m/s$.
B. The force acting on the object $\overrightarrow{F} = (2\widehat{i} + 3\widehat{j})N$.
C. The angular momentum of the object about its origin $\overrightarrow{L} = 15\widehat{k}Js$.
D. The torque acting on the object about its origin $\overrightarrow{\tau} = 20\widehat{k}Nm$.
Choose the correct answer from the option given below.
Options
Solution
Sol. [Key Concept: Compute $p = mv$, $F = ma$, $L = r \times \ p$, $\tau = r \times \ F$ at $t = 1$ and test each statement.]
Given $\overrightarrow{r} = 10t^{2}\widehat{i} + 5t^{3}\widehat{j} \Rightarrow {\overrightarrow{r}}_{t = 1} = 10\widehat{i} + 5\widehat{j}$
$$\begin{matrix} & \overrightarrow{v} = \frac{d\overrightarrow{r}}{dt} = 20t\widehat{i} + 15t^{2}\widehat{j} \Rightarrow {\overrightarrow{v}}_{t = 1sec} = 20\widehat{i} + 15\widehat{j} \\ & \overrightarrow{a} = \frac{d^{2}\overrightarrow{r}}{{dt}^{2}} = 20\widehat{i} + 30\widehat{j} \Rightarrow {\overrightarrow{a}}_{t = 1sec} = 20\widehat{i} + 30\widehat{j} \\ & \ \therefore\overrightarrow{P} = m\overrightarrow{v} = 2\widehat{i} + 1.5\widehat{j} \\ & \overrightarrow{\text{ }F} = m\overrightarrow{a} = 2\widehat{i} + 3\widehat{j} \\ & \overrightarrow{\tau} = \overrightarrow{r} \times \overrightarrow{F} = 20\widehat{k} \\ & \overrightarrow{\text{ }L} = \overrightarrow{r} \times \overrightarrow{P} = 5\widehat{k} \end{matrix}$$
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