Three uniform spheres of masses m1 = 3.50 kg, m2 = 4.00 kg, and m3 = 6.50 kg are placed at the corners of a right triangle (see figure below). Calculate the resultant gravitational force on the object of mass m2, assuming the spheres are isolated from the rest of the Universe.
The figure depicts m2 at the origin with m3 going along the x-axis to <-4.00,0>m, and m1 going along the y-axis to <0,3.00>m. The force vectors are labelled as F32 and F12.
To calculate the resultant gravitational force on the object of mass m2, we need to find the individual forces F32 and F12 and then determine their vector sum.
1. Calculate the force between m3 and m2 (F32):
According to Newton's law of universal gravitation, the gravitational force between two objects is given by:
F32 = G * ((m3 * m2) / r32^2)
where G is the gravitational constant (6.67430 x 10^-11 N m^2 / kg^2), m3 is the mass of the first sphere (6.50 kg), m2 is the mass of the second sphere (4.00 kg), and r32 is the distance between the centers of the spheres.
2. Calculate the force between m1 and m2 (F12):
Similarly, we can calculate the gravitational force between m1 and m2:
F12 = G * ((m1 * m2) / r12^2)
where m1 is the mass of the third sphere (3.50 kg), and r12 is the distance between the centers of the spheres.
3. Calculate the vector sum of F32 and F12:
Since the spheres are placed at the corners of a right triangle, the forces F32 and F12 are perpendicular to each other. Therefore, we can calculate the vector sum of these forces using the Pythagorean theorem.
F_resultant = sqrt(F32^2 + F12^2)
Let's plug in the respective values and calculate the resultant gravitational force.
Given:
m1 = 3.50 kg
m2 = 4.00 kg
m3 = 6.50 kg
G = 6.67430 x 10^-11 N m^2 / kg^2
r12 = distance along y-axis (from m1 to m2) = 3.00 m
r32 = distance along x-axis (from m3 to m2) = 4.00 m
1. Calculate F32:
F32 = (6.67430 x 10^-11 N m^2 / kg^2) * ((6.50 kg * 4.00 kg) / (4.00 m)^2)
2. Calculate F12:
F12 = (6.67430 x 10^-11 N m^2 / kg^2) * ((3.50 kg * 4.00 kg) / (3.00 m)^2)
3. Calculate F_resultant:
F_resultant = sqrt(F32^2 + F12^2)
By plugging in the values and performing the calculations, you will obtain the resultant gravitational force on the object of mass m2.
To calculate the resultant gravitational force on the object of mass m2, we need to calculate the individual gravitational forces between m2 and m1, and between m2 and m3.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the objects, and r is the distance between them.
Let's calculate the individual forces:
1. Gravitational force between m2 and m1 (F12):
The distance between m2 and m1 can be found using the Pythagorean theorem since they are placed at the corners of a right triangle.
Distance = sqrt((3.00)^2 + (4.00)^2) = 5.00 m
F12 = G * (m1 * m2) / r^2
F12 = (6.67430 × 10^-11 N m^2 / kg^2) * (3.50 kg * 4.00 kg) / (5.00 m)^2
2. Gravitational force between m2 and m3 (F32):
The distance between m2 and m3 is simply the x-component of the position vector of m3.
Distance = 4.00 m
F32 = G * (m2 * m3) / r^2
F32 = (6.67430 × 10^-11 N m^2 / kg^2) * (4.00 kg * 6.50 kg) / (4.00 m)^2
Now, to calculate the resultant gravitational force, we need to find the vector sum of F12 and F32. Since F12 is acting in the positive y-direction and F32 is acting in the negative x-direction, we can use vector addition to find the resultant force.
First, we need to find the components of each force:
Fx12 = 0
Fy12 = F12
Fx32 = -F32
Fy32 = 0
Then, we can find the resultant force components:
Fx = Fx12 + Fx32
Fy = Fy12 + Fy32
Finally, we can calculate the magnitude and direction of the resultant force:
Resultant force = sqrt(Fx^2 + Fy^2)
Direction = arctan(Fy / Fx)
the gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²,
F₁₂ =G•m₁•m₂/y²= ...
F₃₂ =G•m₃•m₂/x²= ...
F=sqrt(F₃₂²+F₁₂²)=...
φ is the angle respectively the negative
direction of x-axis
tan φ=F₁₂/F₃₂= ...