Physics Solution: Newton’s Laws in Non-Inertial Frames
Core Principle: Pseudo Force
When observing motion from a non-inertial reference frame (accelerating with $\vec{a}_{frame}$), a pseudo force $\vec{F}_p = -m\vec{a}_{frame}$ must be applied to every particle of mass $m$. The “Centroidal Frame” is attached to the Center of Mass (COM), so $\vec{a}_{frame} = \vec{a}_{cm}$.
When observing motion from a non-inertial reference frame (accelerating with $\vec{a}_{frame}$), a pseudo force $\vec{F}_p = -m\vec{a}_{frame}$ must be applied to every particle of mass $m$. The “Centroidal Frame” is attached to the Center of Mass (COM), so $\vec{a}_{frame} = \vec{a}_{cm}$.
Step-by-Step Calculation
1. Identify Parameters
- Total mass of system: $M = 10 \text{ kg}$
- Mass of single particle: $m = 2 \text{ kg}$
- Pseudo force on the particle: $\vec{F}_p = (4\hat{i} – 2\hat{j}) \text{ N}$
2. Determine Acceleration of COM
Using the definition of pseudo force:
$$\vec{F}_p = -m \vec{a}_{cm}$$
$$4\hat{i} – 2\hat{j} = -2 \vec{a}_{cm}$$
$$\vec{a}_{cm} = \frac{4\hat{i} – 2\hat{j}}{-2} = -2\hat{i} + 1\hat{j} \text{ m/s}^2$$
3. Calculate Net External Force
Newton’s Second Law for the whole system states that the net external force determines the acceleration of the center of mass.
$$\vec{F}_{net} = M \vec{a}_{cm}$$
$$\vec{F}_{net} = 10 \times (-2\hat{i} + 1\hat{j})$$
$$\vec{F}_{net} = (-20\hat{i} + 10\hat{j}) \text{ N}$$
Final Answer: (d) $(-20\hat{i} + 10\hat{j})$ N