Gravitational Self Potential Energy

In summary, the question discusses the gravitational self potential energy of a solid ball of mass density ρ and radius R, denoted as E. The question then asks about the gravitational self potential energy of a ball with twice the radius, 2R, and the same mass density ρ. A possible solution is to use dimensional analysis and the equation PE = -GM1M2/R, where M1 and M2 represent the masses of the two balls. Another approach is to recognize that the mass of the larger ball will be 8 times the mass of the original ball, while the radius is doubled. This results in a gravitational potential energy that is 32 times larger than the original energy.
  • #1
Rmehtany
27
2
Note: I know this question has been asked before, but I wasn't allowed to ask my question on that thread

1. Homework Statement

The gravitational self potential energy of a solid ball of mass density ρ and radius R is E. What is the gravitational self potential energy of a ball of mass density ρ and radius 2R?

Homework Equations


$$PE = -\frac{GMM}{R}$$

The Attempt at a Solution


My attempt was dimensional analysis, because I had no other idea on how to approach this. The energy was somehow going to be related to G, radius, density, and other constants. PE has units $$kg \frac{m^2}{s^2}$$, density $$\frac{kg}{m^3}$$, and G in terms of $$\frac{m^3}{s^2}$$. I tried using E = $$k G^a \rho^b R^c$$, but I couldn't eliminate enough masses from the equation.
 
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  • #2
Rmehtany said:
G in terms of ##\frac{m^3}{s^2}##.
These are not the units for G.

Otherwise, your approach should work.

See if it works out once you have the correct units for G. If not, post the details of your attempt.
 
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  • #3
TSny said:
These are not the units for G.

Otherwise, your approach should work.

See if it works out once you have the correct units for G. If not, post the details of your attempt.
I will try that, but are there other ways to solve this question? I kinda think my method is a bit sloppy.
 
  • #4
Rmehtany said:
I will try that, but are there other ways to solve this question? I kinda think my method is a bit sloppy.
Yes. Look at the equation you wrote for the PE of two point masses. This equation tells you how to dimensionally get PE from G, M and R. Thus, you need to think about relating M to density and R so that you can see how PE is related to G, ρ, and R.
 
  • #5
TSny said:
Yes. Look at the equation you wrote for the PE of two point masses. This equation tells you how to dimensionally get PE from G, M and R. Thus, you need to think about relating M to density and R so that you can see how PE is related to G, ρ, and R.

Uh huh, so let me see if I understand you:

The mass will be 8 times the original mass of the ball due to unvarying density, and radius is doubled. Since $$PE = \frac{-GMM}{R}$$, this equals $$\frac{8^2}{2}$$ = 32 times? Is that correct?
 
  • #6
Yes, that's correct. The actual formula for the gravitational PE of a solid sphere will have some numerical factor out front, but it must be proportional to GM2/R.
 
  • #7
Thank you for helping
 

1. What is gravitational self potential energy?

Gravitational self potential energy is the potential energy stored in an object due to its position in a gravitational field. It is the energy that an object possesses by virtue of its mass and its distance from the center of gravity.

2. How is gravitational self potential energy calculated?

The formula for calculating gravitational self potential energy is U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height or distance from the center of gravity.

3. What factors affect gravitational self potential energy?

The two main factors that affect gravitational self potential energy are the mass of the object and its distance or height from the center of gravity. The greater the mass and the higher the height, the more potential energy an object will have.

4. What is the relationship between gravitational self potential energy and gravitational potential energy?

Gravitational self potential energy is a type of gravitational potential energy. Gravitational potential energy is the total potential energy of an object in a gravitational field, including the energy due to its own mass (gravitational self potential energy) and the energy due to its position relative to other objects in the gravitational field.

5. How is gravitational self potential energy related to work and kinetic energy?

Gravitational self potential energy can be converted into other forms of energy, such as work or kinetic energy. For example, if an object falls from a height, its gravitational self potential energy is converted into kinetic energy as it gains speed. Similarly, lifting an object against the force of gravity requires an input of work, which increases the object's gravitational self potential energy.

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