Newtons Laws EX 37 - Revision history

imported>Patrick: Created page with 'The two masses are placed on the x-axis as shown in the diagram below:

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Now, we want to place the small particle of mass
2011-05-27T03:28:15Z

Created page with 'The two masses are placed on the x-axis as shown in the diagram below:<br><br> TOP<br><br> Now, we want to place the small particle of mass <math…'

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The two masses are placed on the x-axis as shown in the diagram below:<br><br>
[[image:N_APP_EX_37_SOLNa.png|TOP]]<br><br>

Now, we want to place the small particle of mass <math>m</math> so that it is in equilibrium. There are only two forces acting on this particle: the gravitational force exerted by the mass <math>4M</math>, <math>F_{G_4}</math>, and the gravitational force exerted by the mass <math>M</math>, <math>F_G</math>. Both of these forces are attractive. In other words, <math>F_{G_4}</math> points towards the mass <math>4M</math> and <math>F_G</math> points towards the mass <math>M</math>. Since we want <math>m</math> to be in equilibrium we are looking for a place where these two forces have equal magnitudes and opposite directions.

Let's think where we could place the particle <math>m</math> so that the forces have equal magnitudes and opposite directions. We can think of three regions in the space around the masses <math>4M</math> and <math>M</math> (see the diagram below): x-axis, the gray region above and greenish region below the x-axis.<br><br>
[[image:N_APP_EX_37_SOLNb.png|TOP]]<br><br>

As you can see from the sketches above, the placement of <math>m</math> in either the blue or greenish region results in forces which do not have opposite directions. Therefore we should look for a placement along the x-axis. Try to visualize which way the two forces will act if the particle <math>m</math> is '''(i)''' to the left of <math>4M</math>, '''(ii)''' in between the two masses, and '''(iii)''' to the right of mass <math>M</math>. I think that you will reach the conclusion that the particle <math>m</math> should be located <u>between</u> <math>4M</math> and <math>M</math> at some distance <math>x</math> from <math>4M</math>. Its distance from <math>M</math> will then be (1 - x).<br><br>
[[image:N_APP_EX_37_SOLNc.png|TOP]]<br><br>

Now we can use the Newton Law of gravitation and write the equation stating that the magnitudes of the two forces are equal:

<math>F_{G_4} = F_G</math>

<math>{G (4M) m \over x^2} = {G (M) m \over (1 - x)^2}</math>
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Cancelling <math>G</math>, <math>M</math> and <math>m</math>, we get:

<math>{4 \over x^2} = {1 \over (1 - x)^2}</math>
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Take (positive) square roots:

<math>{2 \over x} = {1 \over 1 - x}</math>
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Cross multiply and solve for <math>x</math> to find <math>x = 2/3</math> m.