Ball and Spring Model - Revision history

Jayanimannam: /* Conclusion */

2026-04-27T00:34:52Z

Conclusion

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Jayanimannam: /* Connectedness */

2026-04-27T00:33:06Z

Connectedness

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Jayanimannam:

2. Hooke's Law

2026-04-27T00:22:33Z

<h2>2. Hooke's Law</h2>

← Older revision		Revision as of 20:22, 26 April 2026
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	<hr>		<hr>

	~~==Examples==~~

	~~===Simple===~~		scene.append_to_caption("""
	~~[[File:simple ex~~.~~png\| 200px \|right]]~~

	'~~''Question'''~~		<h2>2. Hooke's Law</h2>

	~~Young’s modulus for brass is 8.96*10^11 Pa. A 100N weight is attached to a 5m length of brass wire. Find~~ the ~~increase in length of the wire. The diameter~~ is ~~1.5mm.~~		The main equation behind the model is:

	~~'''Solution'''~~		<br><br>
			<b>F_spring = -k Δs</b>
			<br><br>

	~~[[File~~:~~Simple exam.png]]~~		where:

	===~~Middling===~~		<ul>
			<li><b>F_spring</b> = restoring force</li>
			<li><b>k</b> = spring constant</li>
			<li><b>Δs</b> = displacement from equilibrium</li>
			</ul>

	~~'''Question'''~~		The negative sign means the force acts opposite the displacement.

	~~You stack 1000 pennies, face to face, and apply a force of 25~~,~~000N to~~ the ~~top penny~~. ~~While~~ the ~~force is applied~~, ~~you find the thickness of~~ the ~~stack decrease by 1mm. The diameter of a penny is 1.905e-2m. The thickness of a penny is 1.3*10^-3m)~~.		If the atom moves right, the spring pulls left.
			If the atom moves downward, the spring pulls upward.

	~~A) Calculate Young’s modulus of zinc.~~		For a vertical spring:

	B) ~~Calculate the interatomic spring stiffness for zinc. The diameter of a single zinc atom is 2.5 * 10^-10m.~~		<br><br>
			<b>F_spring = -k(L - L0)L_hat</b>
			<br><br>

	~~'''Solution~~		where:
			<ul>
			<li><b>L</b> = current spring length</li>
			<li><b>L0</b> = natural spring length</li>
			<li><b>L_hat</b> = unit vector along spring</li>
			</ul>

	~~A)[[File:Part 1.png]]~~		<hr>

	~~B)[[file:Part 2~~.~~png]]~~		<h2>3. Harmonic Approximation</h2>

	~~===Difficult===~~		The model works best for small displacements.

	~~'''Question'''~~		Real atomic forces are more complex, but near equilibrium the potential energy curve behaves like a parabola.

	~~A wire made of an unknown alloy hangs from~~ a support in the ceiling. You measure the relaxed length of the wire to be 1.8m long, and the radius of the wire to be 0.00025m. When you hang a 5kg mass from the wire, you measure that it stretches a distance of 4 x 10^-3m. the average bond length between atoms ~~is 1.3 x 10^-10m for this alloy~~.		That is why Hooke's Law gives a good approximation for vibrating atoms.

	~~A) If you treat the wire as a macroscopic spring, what is the overall spring stiffness of the wire?~~		<hr>
			""")
	~~B) What is the value of Young’s modulus for this alloy?~~

	~~C) What is the stiffness of a typical interatomic bond in the alloy?~~

	~~D) You cut the wire into four pieces of equal length and hang a 5kg mass from one of the wires. How much does this wire stretch?~~

	~~E) You bundle (side by side) each of the four pieces of wire together and hang a 5kg mass from the bundle. How much does this bundle of wires stretch?~~

	~~'''Solution'''~~

	~~A)[[File:Part A.png]]~~

	~~B)[[File: New2.png]]~~

	~~C)[[File: New3.png]]~~

	~~D)[[File: New4.png]]~~

	E)~~[[File: New5.png]]~~

	==Connectedness==		==Connectedness==

Jayanimannam: /* The Main Idea */

2026-04-27T00:18:30Z

The Main Idea

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Sydney: /* References */

2019-08-18T23:46:49Z

References

← Older revision		Revision as of 19:46, 18 August 2019
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	[[Category:~~Which Category did you place this in?~~]]		[[Category:Energy]]

Sydney: /* The Main Idea */

2019-08-18T23:45:52Z

The Main Idea

← Older revision		Revision as of 19:45, 18 August 2019
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	:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).		:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).

	:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File:~~Krypton hydride structure~~.~~png~~\| thumb \| ~~250px~~ \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).		:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File: Imageedit_3_2948105260.gif\| thumb \| 200px \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).

	'''Macroscopic Stretch'''		'''Macroscopic Stretch'''

Sydney: /* The Main Idea */

2019-08-18T23:44:11Z

The Main Idea

← Older revision		Revision as of 19:44, 18 August 2019
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	==The Main Idea==		==The Main Idea==

	:Everything that has mass (solids, liquids, gases, etc.) is made of matter, which is made up of atoms. Each atom is composed of a dense positively charged nucleus and a much less dense negatively charged electron cloud (1). [[File:~~atom~~.png\|200px\|right]] The charges of the atom affect how they interact with other atoms. Simply, the positive charge of one positively charged nucleus of one atom is attracted to the negatively charged electron cloud of the other atom, which bounds the two atoms together. This interaction is known as the electric force. However, this attraction only occurs up to a specific point. Once the atoms near too close to one another, the attractive forces of the nucleus and electron cloud are outweighed by the repulsive forces of the two atoms’ electron clouds (while opposite charges attract, alike charges repel). Therefore, if the atoms are pushed too close to one another, they will repel each other rather than attract. (1).		:Everything that has mass (solids, liquids, gases, etc.) is made of matter, which is made up of atoms. Each atom is composed of a dense positively charged nucleus and a much less dense negatively charged electron cloud (1). [[File:Atom diagram.png\|200px\|right]] The charges of the atom affect how they interact with other atoms. Simply, the positive charge of one positively charged nucleus of one atom is attracted to the negatively charged electron cloud of the other atom, which bounds the two atoms together. This interaction is known as the electric force. However, this attraction only occurs up to a specific point. Once the atoms near too close to one another, the attractive forces of the nucleus and electron cloud are outweighed by the repulsive forces of the two atoms’ electron clouds (while opposite charges attract, alike charges repel). Therefore, if the atoms are pushed too close to one another, they will repel each other rather than attract. (1).

	:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).		:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).

	:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File: ~~Ball Spring Trim 1~~.~~gif~~\| thumb \| 250px \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).		:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File:Krypton hydride structure.png\| thumb \| 250px \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).

	'''Macroscopic Stretch'''		'''Macroscopic Stretch'''

Sydney: /* Examples */

2019-08-18T22:40:52Z

Examples

← Older revision		Revision as of 18:40, 18 August 2019
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	===Simple===		===Simple===
	[[File:simple ex.png\| 200px \|right]]		[[File:simple ex.png\| 200px \|right]]

			'''Question'''

	Young’s modulus for brass is 8.96*10^11 Pa. A 100N weight is attached to a 5m length of brass wire. Find the increase in length of the wire. The diameter is 1.5mm.		Young’s modulus for brass is 8.96*10^11 Pa. A 100N weight is attached to a 5m length of brass wire. Find the increase in length of the wire. The diameter is 1.5mm.

			'''Solution'''

	[[File:Simple exam.png]]		[[File:Simple exam.png]]

	===Middling===		===Middling===

			'''Question'''

	You stack 1000 pennies, face to face, and apply a force of 25,000N to the top penny. While the force is applied, you find the thickness of the stack decrease by 1mm. The diameter of a penny is 1.905e-2m. The thickness of a penny is 1.3*10^-3m).		You stack 1000 pennies, face to face, and apply a force of 25,000N to the top penny. While the force is applied, you find the thickness of the stack decrease by 1mm. The diameter of a penny is 1.905e-2m. The thickness of a penny is 1.3*10^-3m).

	A) Calculate Young’s modulus of zinc.		A) Calculate Young’s modulus of zinc.

	~~[[File:Part 1~~.~~png]]~~		B) Calculate the interatomic spring stiffness for zinc. The diameter of a single zinc atom is 2.5 * 10^-10m.

	B) ~~Calculate the interatomic spring stiffness for zinc. The diameter of a single zinc atom is 2.5 * 10^-10m~~.		'''Solution

			A)[[File:Part 1.png]]

	[[file:Part 2.png]]		B)[[file:Part 2.png]]

	===Difficult===		===Difficult===

			'''Question'''

	A wire made of an unknown alloy hangs from a support in the ceiling. You measure the relaxed length of the wire to be 1.8m long, and the radius of the wire to be 0.00025m. When you hang a 5kg mass from the wire, you measure that it stretches a distance of 4 x 10^-3m. the average bond length between atoms is 1.3 x 10^-10m for this alloy.		A wire made of an unknown alloy hangs from a support in the ceiling. You measure the relaxed length of the wire to be 1.8m long, and the radius of the wire to be 0.00025m. When you hang a 5kg mass from the wire, you measure that it stretches a distance of 4 x 10^-3m. the average bond length between atoms is 1.3 x 10^-10m for this alloy.

	A) If you treat the wire as a macroscopic spring, what is the overall spring stiffness of the wire?		A) If you treat the wire as a macroscopic spring, what is the overall spring stiffness of the wire?

	~~[[File:Part A.png]]~~

	B) What is the value of Young’s modulus for this alloy?		B) What is the value of Young’s modulus for this alloy?

	~~[[File: New2.png]]~~		C) What is the stiffness of a typical interatomic bond in the alloy?

	C) ~~What is~~ the ~~stiffness~~ of a ~~typical interatomic bond in~~ the ~~alloy~~?		D) You cut the wire into four pieces of equal length and hang a 5kg mass from one of the wires. How much does this wire stretch?

			E) You bundle (side by side) each of the four pieces of wire together and hang a 5kg mass from the bundle. How much does this bundle of wires stretch?

			'''Solution'''

	[[File: ~~New3~~.png]]		A)[[File:Part A.png]]

	D) ~~You cut the wire into four pieces of equal length and hang a 5kg mass from one of the wires~~. ~~How much does this wire stretch?~~		B)[[File: New2.png]]

	[[File: ~~New4~~.png]]		C)[[File: New3.png]]

	E) ~~You bundle (side by side) each of the four pieces of wire together and hang a 5kg mass from the bundle~~. ~~How much does this bundle of wires stretch?~~		D)[[File: New4.png]]

	[[File: New5.png]]		E)[[File: New5.png]]

	==Connectedness==		==Connectedness==

Sydney: /* The Main Idea */

2019-08-18T22:36:54Z

The Main Idea

← Older revision		Revision as of 18:36, 18 August 2019
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	:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File: Ball Spring Trim 1.gif\| thumb \| 250px \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).		:This simple model of just two atoms connected by a spring can then be extended to represent an entire solid! If each atom was connected to six atoms (each at a 90 degree angle to the others), it would form a cube, or what’s known as a simple cubic lattice. A simple cubic lattice is the simplest model of a solid. A model of this can be seen below. [[File: Ball Spring Trim 1.gif\| thumb \| 250px \| right \|Example of Coded Simple Cubic Lattice]] If one of the atoms is moved in any direction, it has a ripple effect with the other atoms (causing some to move away and some to come closer). Because they are all connected by these springs (which again, represents the electric forces between the atoms through a chemical bond), the movement of one atom affects all the other atoms (4).

	~~===~~Macroscopic Stretch~~===~~		'''Macroscopic Stretch'''
	:That force exerted by a spring, mentioned earlier, in order to restore it back to its relaxed length after it has been compressed or stretched can be calculated. Every spring has a specific stiffness that is unique to that particular spring known as the spring constant (stiffness), k. It has units N/m. The stretch or compression of the spring is modeled by the difference in the final length minus the initial (relaxed length) and is denoted as ∆L = L – L0, where L is the final length and L0 is the relaxed length. Change in L is referred to as the macroscopic stretch. The force the spring exerts is a direct relationship between the spring stiffness and the macroscopic stretch, and is therefore:		:That force exerted by a spring, mentioned earlier, in order to restore it back to its relaxed length after it has been compressed or stretched can be calculated. Every spring has a specific stiffness that is unique to that particular spring known as the spring constant (stiffness), k. It has units N/m. The stretch or compression of the spring is modeled by the difference in the final length minus the initial (relaxed length) and is denoted as ∆L = L – L0, where L is the final length and L0 is the relaxed length. Change in L is referred to as the macroscopic stretch. The force the spring exerts is a direct relationship between the spring stiffness and the macroscopic stretch, and is therefore:
	:::Fspring = k*∆L		:::Fspring = k*∆L

	~~===~~Microscopic Stretch~~===~~		'''Microscopic Stretch'''
	:The ball-and-spring model of a solid consists of balls that represent atoms held together by springs representing chemical bonds as a result of electric forces. As with a spring, a relaxed length is needed as a reference point to determine stretch or compression; the relaxed length of the spring between the two atoms is the center-to-center distance between the atoms, d. This distance would simply be twice the radius since the atoms would fill the space up entirely, minimizing the space between atoms. Since these bonds are modeled as springs, they also maintain a spring stiffness, k, specific to the spring and how much it will stretch. This stretch is defined as s, and is referred to as the microscopic stretch.		:The ball-and-spring model of a solid consists of balls that represent atoms held together by springs representing chemical bonds as a result of electric forces. As with a spring, a relaxed length is needed as a reference point to determine stretch or compression; the relaxed length of the spring between the two atoms is the center-to-center distance between the atoms, d. This distance would simply be twice the radius since the atoms would fill the space up entirely, minimizing the space between atoms. Since these bonds are modeled as springs, they also maintain a spring stiffness, k, specific to the spring and how much it will stretch. This stretch is defined as s, and is referred to as the microscopic stretch.

	~~===~~Young’s modulus~~===~~		'''Young’s modulus'''
	:Recall that the stiffness of a wire depends on how long the wire is and the thickness of that wire only; it does not depend on the material of wire, so different wires from the same material could still have different stiffnesses.		:Recall that the stiffness of a wire depends on how long the wire is and the thickness of that wire only; it does not depend on the material of wire, so different wires from the same material could still have different stiffnesses.
	:Calculating Young’s modulus is a way to measure the “springiness” of a material and factor out the size and shape of the particular wire. Young’s modulus is the ratio of the force exerted per square meter of area (F/A) and the stretch of the wire (∆L/L). It is a way to measure the “springiness” of a material.		:Calculating Young’s modulus is a way to measure the “springiness” of a material and factor out the size and shape of the particular wire. Young’s modulus is the ratio of the force exerted per square meter of area (F/A) and the stretch of the wire (∆L/L). It is a way to measure the “springiness” of a material.

Sydney: /* Introduction */

2019-08-18T22:31:44Z

Introduction

← Older revision		Revision as of 18:31, 18 August 2019
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	The interactions of atoms can be modeled using balls to represent the atoms and a spring to represent the chemical bond between them.		The interactions of atoms can be modeled using balls to represent the atoms and a spring to represent the chemical bond between them.

	==~~Introduction~~==		==The Main Idea==

	:Everything that has mass (solids, liquids, gases, etc.) is made of matter, which is made up of atoms. Each atom is composed of a dense positively charged nucleus and a much less dense negatively charged electron cloud (1). [[File:atom.png\|200px\|right]] The charges of the atom affect how they interact with other atoms. Simply, the positive charge of one positively charged nucleus of one atom is attracted to the negatively charged electron cloud of the other atom, which bounds the two atoms together. This interaction is known as the electric force. However, this attraction only occurs up to a specific point. Once the atoms near too close to one another, the attractive forces of the nucleus and electron cloud are outweighed by the repulsive forces of the two atoms’ electron clouds (while opposite charges attract, alike charges repel). Therefore, if the atoms are pushed too close to one another, they will repel each other rather than attract. (1).		:Everything that has mass (solids, liquids, gases, etc.) is made of matter, which is made up of atoms. Each atom is composed of a dense positively charged nucleus and a much less dense negatively charged electron cloud (1). [[File:atom.png\|200px\|right]] The charges of the atom affect how they interact with other atoms. Simply, the positive charge of one positively charged nucleus of one atom is attracted to the negatively charged electron cloud of the other atom, which bounds the two atoms together. This interaction is known as the electric force. However, this attraction only occurs up to a specific point. Once the atoms near too close to one another, the attractive forces of the nucleus and electron cloud are outweighed by the repulsive forces of the two atoms’ electron clouds (while opposite charges attract, alike charges repel). Therefore, if the atoms are pushed too close to one another, they will repel each other rather than attract. (1).


	:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).		:These interactions can be modeled using a familiar object: a spring! Every spring has a relaxed length. Once the spring is stretched past this relaxed length, the spring exerts a force inward in order to restore it to its original, relaxed length. If the spring is compressed smaller than the relaxed length, then it exerts a force outward in order to restore it to its original, relaxed length. Atoms can be modeled in the same way. Suppose two balls represent two atoms. These two atoms are connected by a spring, which represents the chemical bond (a bond between atoms that is a result of electric forces that bind protons and electrons to each other) between the two atoms (5).