91Ó°ÊÓ

The net force acting on an object or system is the sum of all the forces acting on it. It is important because it determines the resulting motion according to Newton's Second Law. In our exercise, two forces are acting on the olive and the Brazil nut: \

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• For the olive: \( \vec{F}_{o} = (2.0 \hat{\mathrm{i}} + 3.0 \hat{\mathrm{j}}) \mathrm{N} \).
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• For the nut: \( \vec{F}_{n} = (-3.0 \hat{\mathrm{i}} - 2.0 \hat{\mathrm{j}}) \mathrm{N} \).
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\To find the net force on the system, simply add these two forces together: \( \vec{F}_{\text{net}} = \vec{F}_{o} + \vec{F}_{n} = (-1.0 \hat{\mathrm{i}} + 1.0 \hat{\mathrm{j}}) \mathrm{N} \). The net force indicates the direction and magnitude of the total force acting on both objects together.

Acceleration is the rate of change of velocity of an object. It is directly related to the net force, as described by Newton's Second Law: \( F = ma \). This relationship tells us that the acceleration of an object is directly proportional to the net force and inversely proportional to its mass. \To calculate acceleration for the olive-nut system, use the formula: \\[ \vec{a}_{cm} = \frac{\vec{F}_{\text{net}}}{m+M} = \frac{(-1.0 \hat{\mathrm{i}} + 1.0 \hat{\mathrm{j}})}{0.5 + 1.5} = (-0.5 \hat{\mathrm{i}} + 0.5 \hat{\mathrm{j}}) \mathrm{m/s^2}. \] \This shows that the center of mass of the system accelerates in the direction dictated by the net force, which influences how the system moves over time.

Kinematics deals with the motion of objects without considering the forces that cause this motion. It provides equations to calculate different aspects of an object's trajectory, such as displacement, velocity, and acceleration over time. \In this exercise, we're interested in finding the displacement of the center of mass using the kinematic equation: \\[ \vec{s} = \vec{v}_0 \cdot t + \frac{1}{2} \cdot \vec{a} \cdot t^2. \] \Given that the initial velocity \( \vec{v}_0 \) is zero, the equation simplifies to: \\[ \vec{s}_{cm} = \frac{1}{2} \cdot (-0.5 \hat{\mathrm{i}} + 0.5 \hat{\mathrm{j}}) \cdot (4.0)^2 = (-4.0 \hat{\mathrm{i}} + 4.0 \hat{\mathrm{j}}) \mathrm{m}. \] \This displacement vector tells us how far and in what direction the center of mass moves from its initial position.

Vector notation is a way of representing quantities that have both magnitude and direction, crucial in physics for understanding forces, velocities, and displacements. In this problem, all forces, accelerations, and displacements are given in vector form using the unit vectors \( \hat{\mathrm{i}} \) and \( \hat{\mathrm{j}} \): \

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• \( \hat{\mathrm{i}} \) denotes the x-direction.
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• \( \hat{\mathrm{j}} \) denotes the y-direction.
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\In vector form, each physical quantity can be easily broken down into components along these perpendicular axes. This breakdown allows for straightforward calculations of combined effects, like the net force, through simple arithmetic operations component-wise. By expressing physical quantities in vector notation, we gain a clear and structured way to address motion problems involving multiple directions.