A coordinate system (in meters) is constructed on the surface of a pool table, and
three objects are placed on the coordinate system as follows: a 0.8 kg object at the origin,
a 2.8 kg object at (0 m,1.7 m), and a 4.4 kg
object at (3.8 m,0 m).
Find the resultant gravitational force exerted on the object at the origin by the other
two objects. The universal gravitational con-stant is 6.672 × 10-11N · m2/kg2
.
Answer in units of N
To find the resultant gravitational force exerted on the object at the origin by the other two objects, we need to find the gravitational force between each pair of objects and then vectorially sum them.
The gravitational force between two objects is given by Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force between the two objects,
G is the universal gravitational constant (G = 6.672 × 10^-11 N · m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
Let's start by finding the gravitational force between the object at the origin (m1 = 0.8 kg) and the object at (0 m, 1.7 m) (m2 = 2.8 kg).
First, calculate the distance between the two objects:
r1 = sqrt((0 - 0)^2 + (0 - 1.7)^2) = 1.7 m
Now, calculate the gravitational force between these two objects:
F1 = (G * m1 * m2) / r1^2 = (6.672 × 10^-11 N · m^2/kg^2 * 0.8 kg * 2.8 kg) / (1.7 m)^2
Simplifying the equation:
F1 = (18.7392 × 10^-11 N · m^2/kg^2) / 2.89 m^2
F1 ≈ 6.47 × 10^-11 N
Next, let's calculate the gravitational force between the object at the origin (m1 = 0.8 kg) and the object at (3.8 m, 0 m) (m2 = 4.4 kg).
The distance between the two objects is:
r2 = sqrt((0 - 3.8)^2 + (0 - 0)^2) = 3.8 m
Calculating the gravitational force:
F2 = (G * m1 * m2) / r2^2 = (6.672 × 10^-11 N · m^2/kg^2 * 0.8 kg * 4.4 kg) / (3.8 m)^2
Simplifying:
F2 = (23.3664 × 10^-11 N · m^2/kg^2) / 14.44 m^2
F2 ≈ 1.62 × 10^-11 N
Now, to find the resultant gravitational force, we need to sum the individual forces vectorially. Since they are acting in perpendicular directions, we can use the Pythagorean theorem:
F_total = sqrt(F1^2 + F2^2) = sqrt((6.47 × 10^-11 N)^2 + (1.62 × 10^-11 N)^2)
Calculating the result:
F_total ≈ 6.69 × 10^-11 N
Therefore, the resultant gravitational force exerted on the object at the origin by the other two objects is approximately 6.69 × 10^-11 N.