Q.29. The position of an object having mass kg as a function of time is given as m. At s, which

Q: What is the correct answer for JEE Main Physics Q29?

The correct answer is D. Sol. Given mass $m = 0.1$ kg. Position vector $\vec{r} = (10t^2\hat{i}+5t^2\hat{j})$ m. Velocity $\vec{v} = \frac{d\vec{r}}{dt} = (20t\hat{i}+10t\hat{j})$ m/s. At $t=1$ s, $\vec{v}_{t=1} = (20(1)\hat{i}+10(1)\hat{j}) = (20\hat{i}+10\hat{j})$ m/s. Acceleration $\vec{a} = \frac{d\vec{v}}{dt} = (20\hat{i}+10\hat{j})$ m/s$^2$. At $t=1$ s, $\vec{a}_{t=1} = (20\hat{i}+10\hat{j})$ m/s$^2$. Let's check

Q: Q.29. The position of an object having mass kg as a function of time is given as m. At s, which of the following statements are correct? A. The linear momentum kg.m/s. B. The force acting on the…

The correct answer is D. Sol. Given mass $m = 0.1$ kg. Position vector $\vec{r} = (10t^2\hat{i}+5t^2\hat{j})$ m. Velocity $\vec{v} = \frac{d\vec{r}}{dt} = (20t\hat{i}+10t\hat{j})$ m/s. At $t=1$ s, $\vec{v}_{t=1} = (20(1)\hat{i}+10(1)\hat{j}) = (20\hat{i}+10\hat{j})$ m/s. Acceleration $\vec{a} = \frac{d\vec{v}}{dt} = (20\hat{i}+10\hat{j})$ m/s$^2$. At $t=1$ s, $\vec{a}_{t=1} = (20\hat{i}+10\hat{j})$ m/s$^2$. Let's check

Q: What are the options for this JEE Main Physics question?

A. A, B and C only · B. B, C and D only · C. A, C and D only · D. A, B and D only

Question

Q.29. The position of an object having mass  kg as a function of time  is given as  m. At  s, which of the following statements are correct?
A. The linear momentum  kg.m/s.
B. The force acting on the object  N.
C. The angular momentum of the object about its origin  Js.
D. The torque acting on the object about its origin  Nm.
Choose the correct answer from the option given below.

FirstInTest · Accepted Answer

Sol. Given mass $m = 0.1$ kg.
Position vector $\vec{r} = (10t^2\hat{i}+5t^2\hat{j})$ m.

Velocity $\vec{v} = \frac{d\vec{r}}{dt} = (20t\hat{i}+10t\hat{j})$ m/s.
At $t=1$ s, $\vec{v}_{t=1} = (20(1)\hat{i}+10(1)\hat{j}) = (20\hat{i}+10\hat{j})$ m/s.

Acceleration $\vec{a} = \frac{d\vec{v}}{dt} = (20\hat{i}+10\hat{j})$ m/s$^2$.
At $t=1$ s, $\vec{a}_{t=1} = (20\hat{i}+10\hat{j})$ m/s$^2$.

Let's check each statement at $t=1$ s:

A. Linear momentum $\vec{p} = m\vec{v} = 0.1(20\hat{i}+10\hat{j}) = (2\hat{i}+1\hat{j})$ kg.m/s.
Statement A says $\vec{p} = (2\hat{i}+1.5\hat{j})$ kg.m/s. So A is incorrect.

B. Force acting on the object $\vec{F} = m\vec{a} = 0.1(20\hat{i}+10\hat{j}) = (2\hat{i}+1\hat{j})$ N.
Statement B says $\vec{F} = (2\hat{i}+3\hat{j})$ N. So B is incorrect.

C. Angular momentum about origin $\vec{L} = \vec{r} 	imes \vec{p}$.
At $t=1$ s, $\vec{r}_{t=1} = (10(1)^2\hat{i}+5(1)^2\hat{j}) = (10\hat{i}+5\hat{j})$ m.
$\vec{p}_{t=1} = (2\hat{i}+1\hat{j})$ kg.m/s.
$\vec{L} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+1\hat{j}) = (10 	imes 1 - 5 	imes 2)\hat{k} = (10-10)\hat{k} = 0$ Js.
Statement C says $\vec{L}=15\hat{k}$ Js. So C is incorrect.

D. Torque acting on the object about origin $\vec{	au} = \vec{r} 	imes \vec{F}$.
At $t=1$ s, $\vec{r}_{t=1} = (10\hat{i}+5\hat{j})$ m.
$\vec{F}_{t=1} = (2\hat{i}+1\hat{j})$ N.
$\vec{	au} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+1\hat{j}) = (10 	imes 1 - 5 	imes 2)\hat{k} = (10-10)\hat{k} = 0$ Nm.
Statement D says $\vec{	au} = 20\hat{k}$ Nm. So D is incorrect.

All statements A, B, C, D are incorrect based on my calculations. This implies there might be a mistake in my calculation or the problem/solution.

Let's re-check the solution's steps:
Sol. Given $\vec{r} = 10t^2\hat{i}+5t^2\hat{j}$.
$\vec{v} = \frac{d\vec{r}}{dt} = 20t\hat{i}+10t\hat{j}$. At $t=1$s, $\vec{v}_{t=1} = 20\hat{i}+10\hat{j}$. (Matches)
$\vec{a} = \frac{d\vec{v}}{dt} = 20\hat{i}+10\hat{j}$. At $t=1$s, $\vec{a}_{t=1} = 20\hat{i}+10\hat{j}$. (Matches)

Statement A: $\vec{p} = m\vec{v} = 0.1(20\hat{i}+10\hat{j}) = 2\hat{i}+1\hat{j}$. The statement says $2\hat{i}+1.5\hat{j}$. So A is incorrect. (Matches)

Statement B: $\vec{F} = m\vec{a} = 0.1(20\hat{i}+10\hat{j}) = 2\hat{i}+1\hat{j}$. The statement says $2\hat{i}+3\hat{j}$. So B is incorrect. (Matches)

Statement C: $\vec{L} = \vec{r} 	imes \vec{p}$. At $t=1$s, $\vec{r} = 10\hat{i}+5\hat{j}$, $\vec{p} = 2\hat{i}+1\hat{j}$.
$\vec{L} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+1\hat{j}) = (10 	imes 1 - 5 	imes 2)\hat{k} = (10-10)\hat{k} = 0$. The statement says $15\hat{k}$. So C is incorrect. (Matches)

Statement D: $\vec{	au} = \vec{r} 	imes \vec{F}$. At $t=1$s, $\vec{r} = 10\hat{i}+5\hat{j}$, $\vec{F} = 2\hat{i}+1\hat{j}$.
$\vec{	au} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+1\hat{j}) = (10 	imes 1 - 5 	imes 2)\hat{k} = (10-10)\hat{k} = 0$. The statement says $20\hat{k}$. So D is incorrect. (Matches)

All statements are incorrect. This means the question or the options or the provided answer (4) is wrong. If all statements are incorrect, then none of the options (1) A, B and C only, (2) B, C and D only, (3) A, C and D only, (4) A, B and D only can be correct. This is a severe issue.

Let's assume there's a typo in the problem statement for $\vec{r}$. If $\vec{r} = (10t^2\hat{i}+5t^3\hat{j})$:
$\vec{v} = (20t\hat{i}+15t^2\hat{j})$. At $t=1$s, $\vec{v} = 20\hat{i}+15\hat{j}$.
$\vec{a} = (20\hat{i}+30t\hat{j})$. At $t=1$s, $\vec{a} = 20\hat{i}+30\hat{j}$.

Let's check statements with this modified $\vec{r}$:
Mass $m=0.1$ kg.

A. $\vec{p} = m\vec{v} = 0.1(20\hat{i}+15\hat{j}) = 2\hat{i}+1.5\hat{j}$. This matches statement A. So A is correct.

B. $\vec{F} = m\vec{a} = 0.1(20\hat{i}+30\hat{j}) = 2\hat{i}+3\hat{j}$. This matches statement B. So B is correct.

C. $\vec{L} = \vec{r} 	imes \vec{p}$. At $t=1$s, $\vec{r} = 10\hat{i}+5\hat{j}$. $\vec{p} = 2\hat{i}+1.5\hat{j}$.
$\vec{L} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+1.5\hat{j}) = (10 	imes 1.5 - 5 	imes 2)\hat{k} = (15-10)\hat{k} = 5\hat{k}$ Js.
Statement C says $15\hat{k}$ Js. So C is incorrect.

D. $\vec{	au} = \vec{r} 	imes \vec{F}$. At $t=1$s, $\vec{r} = 10\hat{i}+5\hat{j}$. $\vec{F} = 2\hat{i}+3\hat{j}$.
$\vec{	au} = (10\hat{i}+5\hat{j}) 	imes (2\hat{i}+3\hat{j}) = (10 	imes 3 - 5 	imes 2)\hat{k} = (30-10)\hat{k} = 20\hat{k}$ Nm.
Statement D says $20\hat{k}$ Nm. So D is correct.

So, if $\vec{r} = (10t^2\hat{i}+5t^3\hat{j})$, then A, B, D are correct. This corresponds to option (4) "A, B and D only".
This is a very common type of typo in physics problems where $t^2$ is written instead of $t^3$ or vice-versa. Given that option (4) is the answer, I will assume this typo was intended to be corrected for the solution.

I will use the corrected $\vec{r} = (10t^2\hat{i}+5t^3\hat{j})$ for the explanation.

Answer

A, B and C only

Answer

B, C and D only

Answer

A, C and D only

Answer

A, B and D only

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