Gravitational Potential Energy - Revision history

Dsaxena8: /* Examples */

2024-04-24T20:34:55Z

Examples

← Older revision		Revision as of 16:34, 24 April 2024
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	Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.		Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.

			== Guide on Solving Problems ==

			The Law of Conservation of energy states that energy is neither created nor destroyed. This means that a system's energy is always the same at any point in time as long as it is not acted upon by an external influence.

			<math> E_i = E_f </math>

			<math> U_{g_{init}} + K_{init} = U_{g_{final}} + K_{final} </math>

			Upon seeing any problem, you need to calculate the energy it would have at the final/initial state (depending on what information is given). Isolate the unknown and solve.

	== Examples ==		== Examples ==
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	Which gives us our final answer of:		Which gives us our final answer of:

	<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>		<math>m_{asteroid} = 2.04648 \times 10^{13}kg</math>





	== Connectedness ==		== Connectedness ==

Dsaxena8: /* History */

2024-04-24T13:16:18Z

History

← Older revision		Revision as of 09:16, 24 April 2024
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	== History ==		== History ==

	Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.		Galileo Galilei and Isaac Newton discovered how forces are related to acceleration. Newton summed up this information with his Laws of Motion. Mechanical energy was discovered by analyzing the equations derived from Newton's Laws. It was later realized that when two objects interact they exert forces on each other and that work (<math>W = \vec F \cdot \vec d</math> ) must be replaced with potential energy in a system of interacting objects within a gravitational field.


			The idea of gravitational potential energy specifically was introduced by Johann Bernoulli in 1738. The concept was later taken up by several people in several different ways. For example Euler, Lagrange, Laplace and Legendre looked at gravity as it revolved around something while Newton focused on gravitational attraction if the object is a sphere. Euler and Co later confirmed the shape does not matter for the object.

	== See also ==		== See also ==

Dsaxena8: /* Computational Model */

2024-04-24T12:59:05Z

Computational Model

← Older revision		Revision as of 08:59, 24 April 2024
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			[[File:EnergyAndPositionGraphs.png\|frame\|none\|alt=\|caption ]]
			From https://www.glowscript.org/#/user/Dhruv3745/folder/MyPrograms/program/lab2
	[[File:Totalenergy.gif\|frame\|none\|alt=\|caption ]]		[[File:Totalenergy.gif\|frame\|none\|alt=\|caption ]]

Dsaxena8: /* Main Idea */

2024-04-24T12:10:30Z

Main Idea

← Older revision		Revision as of 08:10, 24 April 2024
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	Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>\|U_{grav}\|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>\|U_{grav}\|</math> will be greater.		Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>\|U_{grav}\|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>\|U_{grav}\|</math> will be greater.


	Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.		Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.
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	An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.		An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.


	An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.		An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.


	We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.		We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.

Dsaxena8: /* Gravitational Potential Energy Relation to Gravitational Force: */

2024-04-24T11:51:04Z

Gravitational Potential Energy Relation to Gravitational Force:

← Older revision		Revision as of 07:51, 24 April 2024
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	<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>		<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

			----

	=== Computational Model ===		=== Computational Model ===

Dsaxena8: /* Mathematical Model */

2024-04-24T11:50:31Z

Mathematical Model

← Older revision		Revision as of 07:50, 24 April 2024
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	-----		-----

			===== Gravitational Potential Energy Relation to Gravitational Force: =====

			* Potential energy is always the integral of the force over the distance the force is applied.
			* We can observe how the gravitational potential energy comes about by looking at the gravitational force

			<math>U_{grav} = \int_{-\infty}^{r} F_g dr</math>

			<math>Fg = -G\frac{m_1 m_2}{r^2}</math>

			<math>U_{grav} = \int_{-\infty}^{r} -G\frac{m_1 m_2}{r^2} dr = -G\frac{m_1 m_2}{r}</math>

	=== Computational Model ===		=== Computational Model ===

Dsaxena8: /* Main Idea */

2024-04-24T11:41:45Z

Main Idea

← Older revision		Revision as of 07:41, 24 April 2024
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	Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>\|U_{grav}\|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>\|U_{grav}\|</math> will be greater.		Gravitational Potential Energy (<math>U_{grav}</math>) is the energy stored in an object as a result of its position in a gravitational field and attractive forces from surrounding objects. The general case of gravitational potential energy, when two objects with mass are attracted to each other, depends on the distance between the two and masses of each. The total gravitational energy <math>U_{grav}</math> is inversely proportional to the distance between the two objects with mass. In other words, the closer together together two things are, the greater the magnitude of gravitational potential energy <math>\|U_{grav}\|</math> will be. Additionally, the total gravitational potential energy <math>U_{grav}</math> is directly proportional to the masses of the objects. When the objects are more massive, the magnitude <math>\|U_{grav}\|</math> will be greater.

			Gravitational Potential Energy, as implied by the name, is a form of potential energy. Potential energy is energy that is stored in an object due to its position or chemical properties that can be translated into kinetic or thermal energy. In these terms it clarifies how an object in space has gravitational potential energy and as it loses this potential energy while falling to the ground, it gains speed, or kinetic energy.


	An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.		An important note is also that gravitational potential energy is negative. This results due to the integration determining gravitational potential energy but an intuitive way to understand this is that the negative sign indicates that when masses approach each other in a gravitational field gravity is doing positive work. If that didn't make sense, remember the idea that kinetic energy is always positive so when <math>K</math> increases if we want to keep <math>E_{total}</math> the same, <math>U_{grav}</math> must be negative.

			An important aspect to acknowledge about Gravitational potential energy is that it must be looked at in the difference between two points. An example that showcases this is the difference in gravitational potential energy between a rock on the surface of the earth and a rock in space. While the rock in space has a smaller magnitude of potential energy in comparison to the rock on the surface, the change in energy from space to the ground is greater than the energy the rock on the ground has in respect to ground level. This is why the rock in space will gain speed and energy as it falls through the air towards the ground.

	We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.		We commonly use gravitational potential energy when we're trying to determine the work required to life an object with mass <math>m</math> to a height a <math>h</math>. Since the gravitational potential energy for objects near the surface of the Earth (As we will derive in the Mathematical Model section) can be represented as <math>U_{grav} = mgh</math> we can intuitively see that the force due to gravity on an object of mass <math>m</math> would be <math>mg</math> and the height we'd be lifting it to would be <math>h</math>. Since work <math>W = \vec F \cdot \vec d</math> we see a parallel idea between <math>U_{grav}</math> and <math>W</math>.
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	Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.		Click [https://trinket.io/embed/glowscript/370440404c?outputOnly=true here] to see it in action if you can't see it below.







	== Examples ==		== Examples ==

Dsaxena8: Claiming

2024-04-24T03:49:39Z

Claiming

← Older revision		Revision as of 23:49, 23 April 2024
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	'''Claimed by: ~~Aidan Handa~~ (~~Fall 2019~~)'''		'''Claimed by: Dhruv Saxena (Spring 2023)'''

	== Main Idea ==		== Main Idea ==

Ahanda32: Removed all copyright images from the page as well as added further explanation to all the questions plus hints plus step by step walk through.

2019-11-22T15:33:36Z

Removed all copyright images from the page as well as added further explanation to all the questions plus hints plus step by step walk through.

Show changes

Ahanda32: Updated and fleshed out the main idea. Added Mathematical Model and Computational Model. Added A VPython link + various examples

2019-11-18T05:39:22Z

Updated and fleshed out the main idea. Added Mathematical Model and Computational Model. Added A VPython link + various examples

Show changes