Four 1,000,000- kg masses are arranged in a square, 150 meters on a side. To the nearest ten thousandth of a Newton, what is the magnitude of the force the mass in the lower left hand corner?
Can someone please explain this to me in a detailed manner? I know this somehow relates to gravitation/circular motion, but just would like to know a formula/method of going about solving this. I read up somewhere that since the 4 masses are in a square the opposite corners cancel out but then the diagonal needs to be taken into consideration. I also know that the degree measure of the bottom left corner's force is 45 degrees. Additional help would be greatly appreciated. Thanks.
draw the figure. figure the horizontal distance.
So the net force is the sum of the lleft-uleft;lleft-uright; lleft-lright
So it is adding three vectors. If you use arguements of symettry, you will see the ll-ul will add with the ll-lr to form a vector along the diagonal , of magnitude GMM/150^2 * cos45*2 the two at the last is because you are adding both. Add that to the force along the diagonal, and you have it.
the way you explained it does not make sense to me.
I need to see an equation. I have no idea what the "that" means in 'Add "that" to the force (what force?) along the diagonal...
The last thing you need to see is an equation. You are adding three vectors. Compute the vectors, and add them with vector math, the result will be along the direction of the diagonal. I suspect you are having breaking vectors into components, and need to understand that.
So, first calculate the force between two boxes which is on the right hand side
so apply formula F= (GMm)r^2
we know the value of G = 6.67* 10 ^(-11)
value r is 150
put the value for both masses nd calculte for F, after that box which is on left side up on the vertical line, calculate same for it, it is also the same force as above we get
so now calculate the force for the box which is remain (diagonally) . This time remember , you have to find "r" by pythagorean theorem , than do all same thing as we did above . than add all of them , you will get the force...
To find the magnitude of the force on the mass in the lower left-hand corner, we need to consider the gravitational attraction between the masses.
The formula we can use to calculate the gravitational force between two masses is the universal law of gravitation:
F = (G * m1 * m2) / r^2
where F is the force in Newtons, G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses in kilograms, and r is the distance between the masses in meters.
In this case, since we have a square arrangement of masses, we need to calculate the force exerted on the mass in the lower left corner by each of the other three masses. The magnitude of the net force on the lower left corner mass will be the sum of these individual forces.
Let's go through the steps to calculate the magnitude of the force:
Step 1: Calculate the force of each individual mass on the lower left corner mass.
- The distance between the masses is given as the side length of the square, which is 150 meters.
- The mass of each object is 1,000,000 kg (as given in the question).
- We can use the formula above to calculate the force exerted by each of the three masses on the lower left corner mass.
Step 2: Find the components of each force.
- Since the forces act diagonally, we need to find the components of these forces in the horizontal and vertical directions.
- The angle between the diagonal forces and the horizontal direction is 45 degrees.
- We can split the forces into their horizontal and vertical components using trigonometry.
Step 3: Combine the forces and components.
- We add the horizontal and vertical components of the three individual forces to find the net horizontal and vertical forces acting on the lower left corner mass.
- Finally, we can use the Pythagorean theorem to calculate the magnitude of the net force.
Although we know the angle between the diagonal forces and the horizontal direction is 45 degrees, we shouldn't directly consider this angle while calculating the magnitude of the force. This is because the angle phi in the formula for the gravitational force is the angle between the line connecting the two masses and the horizontal direction, not the angle between the forces themselves.
Hope that helps! Let me know if you have any further questions.