The position vector of a particle $\vec{R}$ as a function of time $t$ is given by: $\vec{R} = 4\sin(2\pi t)\ha

Question

The position vector of a particle $\vec{R}$ as a function of time $t$ is given by: $\vec{R} = 4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}$, where $R$ is in meters, $t$ is in seconds and $\hat{i}, \hat{j}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is **wrong** for the motion of the particle?

Accepted Answer

1. **Analyze the Path:** The position coordinates are $x = 4\sin(2\pi t)$ and $y = 4\cos(2\pi t)$. Squaring and adding gives $x^2 + y^2 = 16(\sin^2(2\pi t) + \cos^2(2\pi t)) = 16$. This represents a circle of radius $R = 4$ m. (Statement 4 is correct).
2. **Calculate Velocity:** Velocity is the time derivative of the position vector: $\vec{v} = \frac{d\vec{R}}{dt}$.
   $$ \vec{v} = \frac{d}{dt}[4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}] $$
   $$ \vec{v} = 4(2\pi)\cos(2\pi t)\hat{i} - 4(2\pi)\sin(2\pi t)\hat{j} = 8\pi\cos(2\pi t)\hat{i} - 8\pi\sin(2\pi t)\hat{j} $$
3. **Magnitude of Velocity (Speed):**
   $$ |\vec{v}| = \sqrt{(8\pi)^2 (\cos^2(2\pi t) + \sin^2(2\pi t))} = 8\pi \approx 25.1 	ext{ m/s} $$
   The statement claiming the speed is 8 m/s is **incorrect**. (Statement 3 is wrong).
4. **Calculate Acceleration:** Acceleration is the time derivative of velocity: $\vec{a} = \frac{d\vec{v}}{dt}$.
   $$ \vec{a} = \frac{d}{dt}[8\pi\cos(2\pi t)\hat{i} - 8\pi\sin(2\pi t)\hat{j}] $$
   $$ \vec{a} = -16\pi^2\sin(2\pi t)\hat{i} - 16\pi^2\cos(2\pi t)\hat{j} $$
   $$ \vec{a} = -(2\pi)^2 [4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}] = -(2\pi)^2\vec{R} $$
   The acceleration is directed along $-\vec{R}$ (towards the center). (Statement 1 is correct).
5. **Centripetal Acceleration Magnitude:** In uniform circular motion, acceleration magnitude is $a_c = \frac{v^2}{R}$ . Let's check: $\frac{v^2}{R} = \frac{(8\pi)^2}{4} = \frac{64\pi^2}{4} = 16\pi^2$. The magnitude of $\vec{a}$ calculated above is also $|-(2\pi)^2\vec{R}| = 4\pi^2(4) = 16\pi^2$. They match. (Statement 2 is correct).

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