part 1
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 1.5 kg object at the origin,
a 3.4 kg object at (0 m,1.8 m), and a 4.5 kg
object at (4.2 m,0 m).
THANK YOU!!
Find the resultant gravitational force exerted on the object at the origin by the other
two objects. The universal gravitational constant is 6.672 × 10
−11
N · m2
/kg
2
.
Answer in units of N
Part 2
At what angle does it act with respect to
the positive x axis? Let counterclockwise be
positive, within the limits of −180
◦
to 180
◦
.
Answer in units of
◦
F =G•m1•m2/R²
the gravitational constant G =6.67•10^-11 N•m²/kg²,
F1= G•1.5•3.4/1.8²
F1= G•1.5•4.5/4.2²
F=sqrt(F1²+F2²)
α=arctan(F2/F1)
The angle with respect to the positive x axis is
180°+ α
Part 1:
To find the resultant gravitational force exerted on the object at the origin, we need to calculate the gravitational force between each pair of objects and then add them up.
The formula for gravitational force between two objects is given by:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the universal gravitational constant (6.672 × 10^-11 N · m^2 / kg^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
Let's calculate the gravitational forces between the object at the origin and the other two objects.
For the object at (0 m, 1.8 m):
m1 = 1.5 kg (mass at the origin)
m2 = 3.4 kg (mass at (0 m, 1.8 m))
r = distance between the two objects = sqrt[(0 - 0)^2 + (1.8 - 0)^2] = sqrt(1.8^2) = 1.8 m
F1 = (6.672 × 10^-11 N · m^2 / kg^2) * (1.5 kg * 3.4 kg) / (1.8 m)^2
Similarly, for the object at (4.2 m, 0 m):
m1 = 1.5 kg (mass at the origin)
m2 = 4.5 kg (mass at (4.2 m, 0 m))
r = distance between the two objects = sqrt[(4.2 - 0)^2 + (0 - 0)^2] = sqrt(4.2^2) = 4.2 m
F2 = (6.672 × 10^-11 N · m^2 / kg^2) * (1.5 kg * 4.5 kg) / (4.2 m)^2
Now, we can add up these two forces to get the resultant force:
Resultant force = F1 + F2
Part 2:
To find the angle at which the resultant force acts with respect to the positive x-axis, we can use trigonometry. The angle can be determined as:
angle = tan^(-1)(Fy / Fx)
Where:
Fy is the vertical component of the resultant force
Fx is the horizontal component of the resultant force
We can calculate Fy and Fx using the magnitudes of the resultant force and the trigonometric relationships:
Fy = |Resultant force| * sin(angle)
Fx = |Resultant force| * cos(angle)
Substituting these values into the formula for the angle, we can find the angle at which the resultant force acts.
To find the resultant gravitational force exerted on the object at the origin by the other two objects, we can use Newton's law of universal gravitation.
Part 1:
1. Calculate the gravitational force between the first object (1.5 kg) at the origin and the second object (3.4 kg) located at (0 m, 1.8 m).
- Calculate the distance between the two objects using the Pythagorean theorem:
- distance = sqrt((0 m)^2 + (1.8 m)^2)
- Calculate the gravitational force:
- force1 = (6.672 × 10^(-11) N · m^2/kg^2) * (1.5 kg) * (3.4 kg) / (distance)^2
2. Calculate the gravitational force between the first object (1.5 kg) at the origin and the third object (4.5 kg) located at (4.2 m, 0 m).
- Calculate the distance between the two objects:
- distance = sqrt((4.2 m)^2 + (0 m)^2)
- Calculate the gravitational force:
- force2 = (6.672 × 10^(-11) N · m^2/kg^2) * (1.5 kg) * (4.5 kg) / (distance)^2
3. Add the gravitational forces calculated in steps 1 and 2 to find the resultant gravitational force:
- resultant force = force1 + force2
Part 2:
To find the angle at which the resultant gravitational force acts with respect to the positive x-axis, we can use trigonometry.
4. Calculate the horizontal component of the resultant force by multiplying the magnitude of the resultant force by the cosine of the angle:
- horizontal component = resultant force * cos(theta)
5. Calculate the vertical component of the resultant force by multiplying the magnitude of the resultant force by the sine of the angle:
- vertical component = resultant force * sin(theta)
6. Calculate the angle with respect to the positive x-axis using the inverse tangent of the ratio of the vertical component to the horizontal component:
- angle = atan(vertical component / horizontal component)
Make sure to substitute the values of the distances and masses given in the problem into the calculations to find the final answers for both the resultant gravitational force and the angle.