A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a 5.0 kg object at the origin of the coordinate system, a 12.0 kg object at (0, 2.0), and a 16.0 kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin.

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To find the resultant gravitational force exerted by the other two objects on the object at the origin, we can use Newton's Law of Universal Gravitation. The formula for the gravitational force between two objects is given by:

F = G * (m1 * m2) / r^2

where F is the gravitational force, G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the two objects.

In this case, we have three objects, so we need to calculate the gravitational force from each object and then find the resultant force.

Let's begin by calculating the gravitational force from the 12.0 kg object at (0, 2.0). The distance between the object at the origin and this object is the distance between the points (0, 0) and (0, 2.0), which is simply 2.0 meters.

F1 = G * (m1 * m2) / r^2
= (6.67 x 10^-11 N*m^2/kg^2) * ((5.0 kg) * (12.0 kg)) / (2.0 m)^2

Solving this equation will give us the gravitational force exerted by the 12.0 kg object on the object at the origin.

Next, let's calculate the gravitational force from the 16.0 kg object at (4.0, 0). The distance between the object at the origin and this object is the distance between the points (0, 0) and (4.0, 0), which is simply 4.0 meters.

F2 = G * (m1 * m2) / r^2
= (6.67 x 10^-11 N*m^2/kg^2) * ((5.0 kg) * (16.0 kg)) / (4.0 m)^2

Now we have the gravitational force exerted by the 16.0 kg object on the object at the origin.

To find the resultant force, we need to add the vector components of the forces. Since the forces are in opposite directions (one along the y-axis and the other along the x-axis), we can treat them as a vector sum.

Let's denote the force from the 12.0 kg object as F1 and the force from the 16.0 kg object as F2. The resultant force F_res can be calculated using the Pythagorean theorem:

F_res = sqrt(F1^2 + F2^2)

Substitute the values of F1 and F2 into the equation and solve to find the resultant gravitational force exerted by the two objects on the object at the origin.

Note: Ensure that the units are consistent throughout the calculation (e.g., mass in kilograms, distance in meters, and gravitational force in newtons).

Compute the forces due to the two other masses and add them vectorially. This procedure has been explained to you already. Show your work for further assistance.

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