Force of gravity on one object caused by four objects

[drdude24]

1. A square of edge length s is formed by four spheres of masses, m1, m2, m3, and m4. What is the x-component and the y-component of the net gravitational force from them on a central sphere of mass m5. State your answers in terms of the given variables. (Use any variable or symbol stated above along with the following as necessary: and G for the gravitational constant.)

In the problem y is given positive direction up, x is positive right.
m1 - +y,-x (top left corner); m2 - +y,+x (top right); m3 - -y,-x (bot left); m4 - -y,+x (bot right)



2. The force of gravity between two objects equation: Fgrav = (G*M*m)/r^2



3. I would assume due to it being a square, the radius between the center and the corner squared (r^2) would equal (s^2)/2 and the angle we're using is 45 degrees. That being said it should be the Fgrav * sin(45) for y component and Fgrav * cos(45) for x comp. so it should look like:
Fx = (2Gm5cos(45))/(s^2) * (m2+m4-m1-m3)
Fy = (2Gm5sin(45))/(s^2) * (m1+m2-m3-m4)

I've worked it and reworked it thinking perhaps I missed something in the simplification process but no good. WebAssign (the computer program) says that it is wrong. Please help, my two roommates, one neighbor, and I all worked independently and got the same answer. The only other thing I thought of was maybe the radius of the spheres but that information is not given.

 

[haruspex]

Your answer looks correct. How smart is WebAssign at figuring out whether two algebraic expressions are equivalent? Are you actually entering "sin(45)"? Might it expect radians? Would putting √2 instead of 2 sin(45) help?

 

[Simon Bridge]

1. A square of edge length s is formed by four spheres of masses, m1, m2, m3, and m4. What is the x-component and the y-component of the net gravitational force from them on a central sphere of mass m5. State your answers in terms of the given variables. (Use any variable or symbol stated above along with the following as necessary: and G for the gravitational constant.)

In the problem y is given positive direction up, x is positive right.
m1 - +y,-x (top left corner); m2 - +y,+x (top right); m3 - -y,-x (bot left); m4 - -y,+x (bot right)

Those coordinates don't make any sense.

You mean that m₁ is at coordinate (x,y)=(-1,1)s/2 etc so that the square has sides length s and is centered on the origin?

2. The force of gravity between two objects equation: Fgrav = (G*M*m)/r^2

3. I would assume due to it being a square, the radius between the center and the corner squared (r^2) would equal (s^2)/2 and the angle we're using is 45 degrees.

Why "assume"? Didn't you work it out? i.e. using a sketch of the situation ... the distance from center to any corner forms the hypotenuse of a 1-1-√2 triangle where the other sides have length s/2 ... therefore the distance is s/√2 and r²=s²/2 ... no "assume"ing needed.

That being said it should be the Fgrav * sin(45) for y component and Fgrav * cos(45) for x comp. so it should look like:
Fx = (2Gm5cos(45))/(s^2) * (m2+m4-m1-m3)
Fy = (2Gm5sin(45))/(s^2) * (m1+m2-m3-m4)

I've worked it and reworked it thinking perhaps I missed something in the simplification process but no good. WebAssign (the computer program) says that it is wrong. Please help, my two roommates, one neighbor, and I all worked independently and got the same answer. The only other thing I thought of was maybe the radius of the spheres but that information is not given.

OK - so your trouble is with directions.

You don't need to resolve the forces into x and y components -
If you wanted to "brute force" it so the numbers come out automagically without much thought, then use: $$\vec{F}=\frac{Gm_1m_2}{|\vec{r}|^3}\vec{r}$$

But since the geometry is simple you know how it will come out...
just write out the magnitude of the force multiplied by a unit vector pointing in the direction of the force.

note: cos45 = sin45 = 1/√2
the unit vector pointing to the mass at position (1,1)s/2 is $\frac{1}{\sqrt{2}}(\hat{\imath}+\hat{\jmath})$

you have another shortcut in that two of the forces are opposite direction to the other two.

 

[drdude24]

Resolved it

Turns out that it was the sin(45) and cos(45). The program I'm guessing didn't register it as 45 degrees.

By substituting 2sin(45) and 2cos(45) for √2 everything worked out.

Thanks for the advice guys!

 

[Nickie]

I would first start by acknowledging that this is a complex problem and it's great to see that you and your peers have put in the effort to solve it independently. However, I would also suggest considering the fact that this is a hypothetical scenario and the variables given may not accurately represent a real-world situation. Therefore, it's possible that the answer you have calculated is correct, but it may not match the answer expected by the computer program.

In terms of your calculations, I would recommend double-checking your use of the gravitational constant, G. It should be multiplied by the product of the masses (m1, m2, m3, m4, m5), not divided. Additionally, I would suggest using the Pythagorean theorem to calculate the distance between the central sphere and each of the corner spheres, instead of assuming a radius of (s^2)/2. This may yield a more accurate result.

Lastly, I would advise communicating with your instructor or the person in charge of the computer program to clarify the expected answer and any specific assumptions or calculations that should be used. Science is about collaboration and seeking help when needed, so don't hesitate to reach out for assistance. Keep up the good work!