Newton's Law of Gravitation problem

In summary, a square of edge length 22.0 cm is formed by four spheres with masses of 7.00 g, 3.50 g, 1.50 g, and 7.00 g. The net gravitational force on a central sphere with mass 2.30 g can be found using unit-vector notation. It is recommended to define a unit vector r in the direction from m3 to m5, and then use the formula F = Gm5(m2-m3)/r^2 to find the force. The root 2 is necessary to account for the diagonal length of the square. Alternatively, the forces can be calculated separately and expressed in vector form. Using unit vectors is a useful
  • #1
Shadow Cloud
13
0
In Figure 13-34, a square of edge length 22.0 cm is formed by four spheres of masses m1 = 7.00 g, m2 = 3.50 g, m3 = 1.50 g, and m4 = 7.00 g. In unit-vector notation, what is the net gravitational force from them on a central sphere with mass m5 = 2.30 g?

hrw7_13-34.gif


Okay this is what I did so far.
Since m4 and m1 are equal, they can be neglected. This leaves...

F = Gm5(m2-m3) / r^2

I then tried to multiply it by cos 45 degrees to get the i component and sin 45 degrees to get the y component, but this isn't working. Why is that?
 
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  • #2
try defining your own unit vector r in the direction from m3 straight to m5.
then R= rr. Finding r is easy. The force will also be along r, so now the problem is really easy because you won't need trig functions.
If you were explicitly told to use x, y unit vectors then you'll have to change it back.
 
Last edited:
  • #3
What is rr in the equation of R = rr that you gave?
 
  • #4
R is the vector from m3 to m5. -R would be the vetor from m2 to m5. r is the unit vector in the direction of R.
r is the magnitude of R and is given by √(x²+y²).
r is found the same way, using the magnitudes of the unit vectors.
just be careful where you put the √2 in the unit vectors.
you want to have the r unit vector. remember what it means to be a unit vector.
 
  • #5
I'm kind of following this, but why is the root 2 necessary and where would I put it?
 
  • #6
Shadow Cloud said:
I'm kind of following this, but why is the root 2 necessary and where would I put it?

a simpler way seen as m1 and m4 cancel may be to work out the 2 forces sepetately using the formula you have already put at the top of the page

seen as they are opposite simply subtract one magnitude from the other (larger - smaller to stop any sign problems) then from that you will have the force and the direction of the line of action of the force, then express this in vector form

sorry if this isn't the way you have been taught but first time i have tried to post a solution on this forum :)
 
  • #7
Oh nevermind, I finally figure out the necessity of the root 2. It's pertains to the length of a diagonal of a square.
 
  • #8
that's right, it's from the diagonal of the square.

and it's the same thing you'd get if you did trig, but this is, in my opinion, easier and makes more sense.
not only that the ability to perform transformations by means of unit vectors is a very important thing.
/s
 

1. What is Newton's Law of Gravitation?

Newton's Law of Gravitation is a fundamental law of physics that states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This law explains the force of gravity between two objects, such as the Earth and the moon.

2. How do you calculate the force of gravity between two objects?

The force of gravity between two objects can be calculated using the formula F = G * (m1 * m2)/r^2, where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

3. Does Newton's Law of Gravitation only apply to objects on Earth?

No, Newton's Law of Gravitation applies to all objects in the universe. It is a universal law that governs the force of gravity between any two objects, regardless of their location.

4. What is the difference between mass and weight in relation to Newton's Law of Gravitation?

Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. Newton's Law of Gravitation relates to mass, as it determines the force of gravity between two objects based on their masses.

5. How did Newton's Law of Gravitation change our understanding of the universe?

Newton's Law of Gravitation revolutionized our understanding of the universe by providing a mathematical explanation for the force of gravity and its effects on objects. This law allowed scientists to make accurate predictions about the motion of celestial bodies and led to further discoveries and advancements in the field of physics.

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