Force on a massive ring

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In summary, the problem asks for the calculation of the force F on a test mass m at different positions on the z-axis, given a massive ring with a radius of a and a mass of M on the xy-plane. Assuming a = 1 and GMm/a^2 = 1 in a specific system of units, the force on m at z = 0 is 0 and at z = a is 0.354. To find the maximum force on m, the equation F=(2GmMx)/(x^2+a^2)^3/2 is derived and then differentiated with respect to x. The resulting equation is set equal to zero and solved for x, which can then be plugged back into the original
  • #1
kraigandrews
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Homework Statement


A massive ring (radius = a and mass = M) lies on the xy plane. Calculate the force F on a test mass m at position z on the z axis.
Now assume a = 1 and GMm/a2 = 1 in some system of units.

(a) What is the force on m at z = 0?
I got the answer for this its 0
(b) What is the force on m at z = a?
the answer for this is 0.354
(c) What is the maximum force on m ?
I am stuck on this one.


Homework Equations



F=(GMm)/r^2

The Attempt at a Solution



after some work I get F=(2GmMx)/(x^2+a^2)^3/2
i know this is right because i got part b from it, however i don't Know how to find the maximum, my gut feeling is take the derivative with respect to x and set it equal to zero and then solve for x and plug back into F
 
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  • #2
kraigandrews said:
after some work I get F=(2GmMx)/(x^2+a^2)^3/2
i know this is right because i got part b from it, however i don't Know how to find the maximum, my gut feeling is take the derivative with respect to x and set it equal to zero and then solve for x and plug back into F

I'm not sure where that "2" comes from in your force expression. Better check that derivation.

And yes, you need to differentiate and solve as you've outlined.
 
Last edited:

1. What is the formula for calculating the force on a massive ring?

The formula for calculating the force on a massive ring is F = (G x m1 x m2)/r^2, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers of mass.

2. What is the direction of the force on a massive ring?

The direction of the force on a massive ring is towards the center of the ring, perpendicular to the plane of the ring.

3. How does the mass of the ring affect the force on it?

The mass of the ring does not affect the force on it. The force only depends on the masses of the objects and the distance between them.

4. What is the difference between force on a massive ring and force on a point mass?

The main difference is that force on a massive ring is exerted on the entire ring, while force on a point mass is exerted on a single point. The formula for calculating the force is also different, as force on a point mass is given by F = m x a, where m is the mass of the object and a is its acceleration.

5. Can the force on a massive ring be negative?

Yes, the force on a massive ring can be negative if the objects are of opposite masses and the distance between them is large enough. This means that the force is attractive, pulling the objects towards each other.

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