A big olive (m=0.50 kg) lies at the origin and a big Brazil nut (M=1.5 kg) lies at the point (

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A big olive (m=0.50 kg) lies at the origin and a big Brazil...

A big olive $(m=0.50 \mathrm{~kg})$ lies at the origin and a big Brazil nut $(M=1.5 \mathrm{~kg})$ lies at the point $(1.0,2.0) \mathrm{m}$ in an $x y$ plane. At $t_{1}=0$, a force $\vec{F}_{o}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(3.0 \mathrm{~N}) \hat{\mathrm{j}}$ begins to act on the olive, and a force $\vec{F}_{n}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}+(-2.0 \mathrm{~N}) \hat{\mathrm{j}}$ begins to act on the nut. In unit-vector notation, what is the displacement of the center of mass of the olive-nut system at $t_{2}=4.0 \mathrm{~s}$, with respect to its position at $t_{1}=0$ ?

Key Concepts

Kinematics of Constant Acceleration

When acceleration is constant, the displacement of an object (or the center of mass of a system) over a given time interval can be calculated using the kinematic equation: displacement equals initial velocity multiplied by time plus one-half the acceleration multiplied by the square of time. This equation provides a straightforward method for predicting the position change over time.

Newton's Second Law for Systems

Newton's second law applied to a system of particles states that the acceleration of the center of mass is equal to the net external force divided by the total mass of the system. This principle allows one to treat the system’s motion as if all the mass were concentrated at a single point, thereby simplifying the analysis of complex multi-body interactions.

Center of Mass

The center of mass of a system of particles is the point where the total mass of the system can be considered to be concentrated. It is calculated as the mass-weighted average position of all components. This concept simplifies the analysis of motion because the translational motion of the entire system, under external forces, can be analyzed as if all the mass were located at the center of mass.

Net External Force

The net external force acting on a system is the vector sum of all the external forces acting on each constituent of the system. According to Newton's second law, this net force determines the acceleration of the center of mass. Understanding the net force is crucial for predicting the overall motion of the system under various forces.

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