91Ó°ÊÓ

Newton's law of universal gravitation is a fundamental principle that describes the attractive force between two masses. According to this law, every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between their centers.
This relationship is captured by the formula: \[ F = G \frac{m1 \cdot m2}{r^2} \]where:

• \( F \) is the gravitational force between the masses.
• \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \mathrm{Nm}^2/\mathrm{kg}^2 \).
• \( m1 \) and \( m2 \) are the masses of the objects.
• \( r \) is the distance between the centers of the two masses.

In our exercise, this principle is applied to calculate the forces exerted on the 2.0-kg object at the origin by the 3.0-kg and 4.0-kg objects, respectively. Each force is computed by determining the mass products and the distance squared, which then allows for solving the gravitational force using the above formula.
This calculation reveals how the forces act in different directions, guiding us to the next step: vector addition.

The coordinate system is crucial for solving problems involving multiple objects in physics. It allows us to clearly define the positions of objects within a structured framework, using axes for reference. In this problem, a coordinate system on a pool table defines the locations of masses with point coordinates.
The origin (0,0) holds the 2.0-kg object, the 3.0-kg object is at (0,2.0), and the 4.0-kg object is found at (4.0,0). Using this system, we can easily calculate the distances between different objects necessary for applying Newton's law of universal gravitation.
For instance:

• Distance between the 2.0-kg and 3.0-kg objects is 2 meters vertical (y-direction).
• Distance between the 2.0-kg and 4.0-kg objects is 4 meters horizontal (x-direction).

These distances, observed from the coordinate points, are critical in computing the forces correctly. Having a coordinate system not only simplifies the problem but also illustrates the direction of forces, clarifying the geometry involved.

After calculating individual forces, understanding how to add them helps determine the net gravitational effect on an object. Forces are vector quantities, meaning they have both magnitude and direction. Simply summing them requires consideration of their directional components.
The gravitational forces on the 2.0-kg object have specific directions:

• Force from the 3.0-kg is upward (negative y-direction).
• Force from the 4.0-kg is rightward (positive x-direction).

To find the resultant force, each force is broken into its component form. The force's x-component is horizontal, while the y-component is vertical. We add like components together: x-components with x-components and y-components with y-components.
This piecewise vector addition allows us to find the net vector, combining the magnitudes and directions into a single force vector acting on the object at the origin. This is exemplified in how one might merge the gravitational pull from different sides, leading into using the Pythagorean theorem for final calculations.

The Pythagorean theorem comes into play when determining the magnitude of the resultant vector, particularly after combining component forces in vector addition. This theorem is best known from geometry and states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of squares of the lengths of the other two sides. It can be expressed as: \[ c^2 = a^2 + b^2 \]where:

• \( c \) is the hypotenuse or resultant vector magnitude.
• \( a \) and \( b \) are the x and y components of the forces.

In this problem, each gravitational force contributes to the resultant force acting on the 2.0-kg object. By calculating the vector's magnitude using the Pythagorean theorem, we understand how different vectors add geometrically.
This technique distills all our prior work into a single clear answer. It's a succinct method to assert not just how strongly the object at the origin is attracted, but also in which direction the total force is acting, providing a comprehensive understanding of the forces in play.