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 7.0-kg object at the origin of the coordinate system, a 8.0-kg object at (0, 2.0), and a 17.0-kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin.

magnitude N
direction °

To find the resultant gravitational force exerted by the other two objects on the object at the origin, you need to calculate the gravitational force between each pair of objects and then find the vector sum of these forces.

The gravitational force between two objects can be calculated using Newton's law of universal gravitation formula:

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

Where:
F is the gravitational force between the two objects,
G is the gravitational constant (6.67430 × 10^-11 m^3/kg/s^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the two objects.

Let's denote the 7.0-kg object as m1, the 8.0-kg object as m2, and the 17.0-kg object as m3.

The distance (r) between the object at the origin and the object at (0, 2.0) is given by the square root of ((0 - 0)^2 + (2.0 - 0)^2), which simplifies to 2.0 meters.

Now we can calculate the gravitational force between the object at the origin and the one at (0, 2.0):

F1 = G * ((m1 * m2) / r^2)

Substituting the values, we get:

F1 = 6.67430 × 10^-11 * ((7.0 * 8.0) / (2.0^2))

Similarly, calculate the gravitational force between the object at the origin and the one at (4.0, 0):

The distance (r) between the object at the origin and the object at (4.0, 0) is given by the square root of ((4.0 - 0)^2 + (0 - 0)^2), which simplifies to 4.0 meters.

F2 = G * ((m1 * m3) / r^2)

Substituting the values, we get:

F2 = 6.67430 × 10^-11 * ((7.0 * 17.0) / (4.0^2))

Now, we need to find the vector sum of these two forces to get the resultant gravitational force.

Using the Pythagorean theorem, the magnitude of the resultant force can be calculated as:

|FR| = sqrt(F1^2 + F2^2)

The direction of the resultant force can be found using trigonometry. Take the inverse tangent of (F2 / F1) to find the angle:

θ = atan(F2/F1)

Finally, substitute the values into the equations to calculate the magnitude and direction of the resultant gravitational force exerted by the other two objects on the object at the origin.

To find the resultant gravitational force exerted by the other two objects on the object at the origin, we need to calculate the individual gravitational forces and then find the vector sum of these forces.

1. Calculate the gravitational force between the 7.0-kg object at the origin and the 8.0-kg object at (0, 2.0):

The formula to calculate gravitational force is given by:

F = G * (m1 * m2) / r²

where F is the gravitational force, G is the gravitational constant (6.674 * 10⁻¹¹ N m²/kg²), m1 and m2 are the masses of the two objects, and r is the distance between them.

For the first object at the origin (7.0 kg) and the second object at (0, 2.0), the distance is the vertical distance between them, which is 2.0 m.

Plugging the values into the formula:

F1 = (6.674 * 10⁻¹¹ N m²/kg²) * (7.0 kg * 8.0 kg) / (2.0 m)²

Calculate the value of F1.

2. Next, calculate the gravitational force between the 7.0-kg object at the origin and the 17.0-kg object at (4.0, 0):

For the first object at the origin (7.0 kg) and the third object at (4.0, 0), the distance is the horizontal distance between them, which is 4.0 m.

Plugging the values into the formula:

F2 = (6.674 * 10⁻¹¹ N m²/kg²) * (7.0 kg * 17.0 kg) / (4.0 m)²

Calculate the value of F2.

3. Calculate the magnitude of the resultant gravitational force:

The resultant gravitational force can be calculated by finding the vector sum of F1 and F2. This can be done using vector addition.

F_resultant = sqrt(F1² + F2² + 2 * F1 * F2 * cosθ)

where θ is the angle between the two forces.

To find the angle, we can use trigonometry:

θ = atan(F2 * sinθ / (F1 + F2 * cosθ))

Calculate the magnitude of the resultant gravitational force, F_resultant.

4. Calculate the direction of the resultant gravitational force:

To find the direction, we can calculate the angle θ using the arctan function:

θ = atan(F2 * sinθ / (F1 + F2 * cosθ))

Calculate the value of θ in degrees.

The magnitude of the resultant gravitational force is F_resultant and the direction is θ degrees.