A particle of mass 𝑀 = 0.2 kg is initially at rest in the xy-plane at a point (𝑥 = −𝑙, y =−ℎ), where 𝑙 = 10 m and ℎ = 1 m. The particle is accelerated at time 𝑡 = 0 with a constant acceleration 𝑎 = 10 m/s2 along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by 𝐿 ⃗ and 𝜏 , respectively. 𝑖, 𝑗 and 𝑘 ̂ are unit vectors along the positive x, y and z-directions, respectively. If 𝑘 ̂ = 𝑖 × 𝑗 then which of the following statement(s) is(are) correct? (A) The particle arrives at the point (𝑥 = 𝑙, 𝑦 = −ℎ) at time 𝑡 = 2 s (B) 𝜏 = 2 𝑘 ̂ when the particle passes through the point (𝑥 = 𝑙, 𝑦 = −ℎ) (C) 𝐿 ⃗ = 4 𝑘 ̂ when the particle passes through the point (𝑥 = 𝑙, 𝑦 = −ℎ) (D) 𝜏 = 𝑘 ̂ when the particle passes through the point (𝑥 = 0, 𝑦 = −ℎ)

 To determine which statements are correct, we need to analyze the motion of the particle and calculate its angular momentum and torque with respect to the origin at specific points in its trajectory.

1. Understanding the Problem

• Mass of the particle (M): 0.2 kg

• Initial position: $(x=-l,y=-h)$, where $l=10$ m and $h=1$ m

• Initial velocity: 0 m/s (particle is at rest)

• Acceleration (a): 10 m/s² along the positive x-direction

• Time of acceleration start: $t=0$ s

2. Calculating the Time to Reach $(x=l,y=-h)$

The particle moves along the x-axis with constant acceleration. The position as a function of time is given by:

$$x(t)=x_0+v_{0x}t+\frac{1}{2}a{t}^2$$

where:

• $x_0=-l=-10$ m

• $v_{0x}=0$ m/s

• $a=10$ m/s²

We want to find the time $t$ when $x(t)=l=10$ m:

$$10=-10+0\cdot t+\frac{1}{2}\cdot 10\cdot t^2$$

Simplifying:

$$10=-10+5t^2⟹20=5t^2⟹t^2=4⟹t=2\text{ s}$$

So, the particle arrives at $(x=l,y=-h)$ at $t=2$ s.

Statement (A) is correct.

3. Calculating Angular Momentum ($\vec{L}$) at $(x=l,y=-h)$

Angular momentum is given by:

$$\vec{L}=\vec{r}\times \vec{p}$$

where:

• $\vec{r}$ is the position vector,

• $\vec{p}$ is the linear momentum ($\vec{p}=M\vec{v}$).

At $t=2$ s:

• Position vector $\vec{r}=l\widehat{i}-h\widehat{j}=10\widehat{i}-1\widehat{j}$

• Velocity $\vec{v}=at\widehat{i}=10\times 2\widehat{i}=20\widehat{i}$ m/s

• Momentum $\vec{p}=M\vec{v}=0.2\times 20\widehat{i}=4\widehat{i}$ kg·m/s

Now, calculate the cross product:

$$\vec{L}=\vec{r}\times \vec{p}=(10\widehat{i}-1\widehat{j})\times (4\widehat{i})=10\times 4(\widehat{i}\times \widehat{i})-1\times 4(\widehat{j}\times \widehat{i})=0-4(-\widehat{k})=4\widehat{k}$$

So, $\vec{L}=4\widehat{k}$ kg·m²/s.

Statement (C) is correct.

4. Calculating Torque ($\vec{\tau }$) at $(x=l,y=-h)$

Torque is given by:

$$\vec{\tau }=\vec{r}\times \vec{F}$$

where:

• $\vec{F}$ is the force acting on the particle.

Since the particle is accelerating along the x-axis, the force is:

$$\vec{F}=M\vec{a}=0.2\times 10\widehat{i}=2\widehat{i}\text{ N}$$

Now, calculate the cross product:

$$\vec{\tau }=\vec{r}\times \vec{F}=(10\widehat{i}-1\widehat{j})\times (2\widehat{i})=10\times 2(\widehat{i}\times \widehat{i})-1\times 2(\widehat{j}\times \widehat{i})=0-2(-\widehat{k})=2\widehat{k}$$

So, $\vec{\tau }=2\widehat{k}$ N·m.

Statement (B) is correct.

5. Calculating Torque ($\vec{\tau }$) at $(x=0,y=-h)$

First, find the time when the particle passes through $x=0$:

$$0=-10+0\cdot t+\frac{1}{2}\cdot 10\cdot t^2⟹0=-10+5t^2⟹t^2=2⟹t=\sqrt{2}\text{ s}$$

At $t=\sqrt{2}$ s:

• Position vector $\vec{r}=0\widehat{i}-h\widehat{j}=-1\widehat{j}$

• Velocity $\vec{v}=at\widehat{i}=10\times \sqrt{2}\widehat{i}=10\sqrt{2}\widehat{i}$ m/s

• Force $\vec{F}=M\vec{a}=0.2\times 10\widehat{i}=2\widehat{i}$ N

Now, calculate the cross product:

$$\vec{\tau }=\vec{r}\times \vec{F}=(-1\widehat{j})\times (2\widehat{i})=-1\times 2(\widehat{j}\times \widehat{i})=-2(-\widehat{k})=2\widehat{k}$$

So, $\vec{\tau }=2\widehat{k}$ N·m.

Statement (D) is incorrect because it claims $\vec{\tau }=\widehat{k}$, but we found $\vec{\tau }=2\widehat{k}$.

6. Conclusion

After analyzing each statement:

• Statement (A): Correct.

• Statement (B): Correct.

• Statement (C): Correct.

• Statement (D): Incorrect.

Final Answer: Statements (A), (B), and (C) are correct.