A big olive (m=0.50 kg) lies at the origin of an x y...

A big olive $(m=0.50 \mathrm{kg})$ lies at the origin of an $x y$ coordinate system, and a big Brazil nut $(M=1.5 \mathrm{kg})$ lies at the point $(1.0,2.0) \mathrm{m} . \mathrm{At} t=0,$ a force $\vec{F}_{o}=(2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N}$ begins to act on the olive, and a force $\vec{F}_{n}=(-3.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}) \mathrm{N}$ 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 $t=4.0 \mathrm{s}$ , with respect to its position at $t=0 ?$

A big olive $(m=0.50 \mathrm{kg})$ lies at the origin of an $x y$ coordinate system, and a big Brazil nut $(M=1.5 \mathrm{kg})$ lies at the
point $(1.0,2.0) \mathrm{m} . \mathrm{At} t=0,$ a force $\vec{F}_{o}=(2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N}$ begins to
act on the olive, and a force $\vec{F}_{n}=(-3.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}) \mathrm{N}$ 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 $t=4.0 \mathrm{s}$ , with respect to
its position at $t=0 ?$

Key Concepts

Center-of-Mass Motion

The center-of-mass motion concept states that the motion of the center of mass of a system is determined solely by the net external force acting on the entire system, regardless of the internal forces. This principle allows us to treat a system of objects as a single particle whose acceleration is given by the total external force divided by the total mass of the system.

Newton's Second Law for Systems

Newton's Second Law, when applied to a system of particles, relates the net external force acting on the system to the total mass and the acceleration of its center of mass. This law is crucial for determining the acceleration of the center of mass by summing all external forces and dividing by the total mass of the objects involved.

Kinematics Under Constant Acceleration

The kinematics under constant acceleration provides the tools to calculate displacement when an object (or the center of mass) is subject to a constant acceleration. The standard equation, displacement = initial velocity × time + ½ × acceleration × time squared, is used to determine how far and in what direction the center of mass moves during a given time interval.

Vector Algebra and Unit-Vector Notation

Vector algebra is essential in physics for representing quantities that have both magnitude and direction, such as forces and displacements. Unit-vector notation breaks these vectors into components along standard axes, simplifying calculations involving the addition of vectors and helping to determine the resulting direction of motion in problems that span a coordinate system.

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