2021-04-18T00:39:50Z

Mkelly74:

'''claimed by MaKenna Kelly Spring 2021'''

This page defines and describes inertia.

==The Main Idea==

Inertia is the tendency of matter to resist change in [[Velocity]]. It is an inherent property of matter; the inertia of an object is directly proportional to its [[Mass]] and can in fact be used to define mass. [[Newton's First Law of Motion]] states that the velocity of an object does not change unless there is an unbalanced force acting on it. This is a consequence of the object's inertia. When a net external force acts on an object, the object will accelerate, meaning its velocity will change over time. For a given force, the rate of change of velocity is inversely proportional to the mass of an object; more massive objects have more inertia and therefore experience slower changes in velocity for a given force.

"Inertia" is not to be confused with "moment of inertia", a related but different topic. The moment of inertia of an object is the tendency of an object to resist change in angular velocity. It is the rotational analogue for inertia. For more information about moments of inertia, see [[The Moments of Inertia]].

Inertia is responsible fictitious forces, which are forces that appear to act on objects in accelerating [[Frame of Reference|frames of reference]].

===A Mathematical Model===

According to [[Newton's Second Law: the Momentum Principle]], <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. The more massive an object is, the less its velocity needs to change in order to achieve the same change in momentum in a given time interval.

The other form of Newton's Second Law is <math>\vec{F}_{net} = m \vec{a}</math>. Solving for acceleration yields <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>. This shows the inverse relationship between mass and acceleration for a given net force.

===A Computational Model===

Here is a link to a computational model that can aid in better understanding inertia. [https://demonstrations.wolfram.com/Inertia/] The situation shown is very similar to that described in the "Industry Applications" portion of this page.

Here are snapshots from a similar model:

[[File:Actual_inertia_1.PNG]]

[[File:Inertia_2.PNG]]

[[File:Inertia_3.PNG]]

==Inertial Reference Frames==

In physics, properties such as objects' positions, velocities, and accelerations depend on the [[Frame of Reference]] used. Different reference frames can have different origins, and can even move and accelerate with respect to one another. According to Einstein's idea of relativity, one cannot tell the position or velocity of one's reference frame by performing experiments. For example, a person standing in an elevator with no windows cannot tell if it is moving up or down by, say, dropping a ball from shoulder height and measuring how long it takes to hit the ground. The answer will be the same regardless of the velocity of the elevator, in part because when the ball is released from shoulder height, it is travelling at the same velocity as the elevator. However, one can tell the acceleration of one's reference frame by performing experiments. For example, if a person standing in an elevator with no windows were to drop a ball from shoulder height, if the elevator were accelerating upwards, it would take less time for the ball to hit the ground than if the elevator were moving at a constant velocity. This is because the ground accelerates towards the ball. From the point of view of the person in the elevator, the ball would appear to violate Newton's second law; it would appear to accelerate towards the ground faster than accounted for by the force of gravity. This is because Newton's second law was written for inertial reference frames. An inertial reference frame is a reference frame travelling at a constant velocity. The physics in this course is true only in inertial reference frames, and problems in this course should be solved using inertial reference frames. The surface of the earth is usually considered an inertial reference frame; its rotations about its axis and revolutions around the sun are technically forms of acceleration, but both are so small that they have negligible effects on most aspects of motion. The definition of an inertial reference frame is one in which matter exhibits inertia; in accelerating reference frames, objects may accelerate despite the absence of external forces acting on them.

==Examples==

===Simple===

Find the mass of an object whose acceleration is 4 m/s^2 and whose translational inertia is 20 kg.

<math>T = ma</math>

Where T is translational inertia, m is mass, and a is acceleration.

<math>m = \frac{T}{a}</math>

<math>m = \frac{20 kg}{4 m/s^2}</math>

<math>m = 5 kg</math>

===Middling===

A motor capable of producing a constant torque of 100 Nm and a maximum rotation speed of 150 rad/s is connected to a flywheel with rotational inertia 0.1 kgm^2.

a) What angular acceleration will the flywheel experience as the motor is switched on?

<math>\tau = I\alpha</math>

Where \tau is torque, I is rotational inertia, and \alpha is angular acceleration.

<math>\alpha = \frac{\tau}{I}<\math>

<math>\alpha = \frac{100 Nm}{0.1 kgm^2}<\math>

<math>\alpha = 1000 rad/s^2<\math>

b) How long will the flywheel take to reach a steady speed if starting from rest?

Using rotational kinematics,

<math>\omega = \omega_0 + \alphat</math>

Since we know the maximum rotational velocity of the motor, we can solve to find the time taken to accelerate up to that rotational
velocity.

<math>t = \frac{\omega_max}{α}</math>

<math>t = \frac{150 rad/s}{1000 rad/s^2}</math>

<math>t = 0.15 s</math>

===Difficult===

[[File:Inertia_diificult.PNG]]

What is the rotational inertia of the object shown above?

<math>I = m_1r_1^2 + m_2r_2^2 + ... = \sum m_ir_i^2</math>

Where I is rotational inertia, m is mass, and r is distance from the axis of rotation.

<math>I = (1 kg \times 1^2 m^2) + (1 kg \times 1.5^2 m^2) + (1 kg \times 0.75^2 m^2) + (2 kg \times 0.75^2 m^2)</math>

<math>I = 4.9375 kgm^2</math>

==Connectedness==
===Interesting Applications===

Scenario: Tablecloth Party Trick

A classic demonstration of inertia is a party trick in which a tablecloth is yanked out from underneath an assortment of dinnerware, which barely moves and remains on the table. The tablecloth accelerates because a strong external force- a person's arm- acts on it, but the only force acting on the dinnerware is kinetic friction with the sliding tablecloth. This force is significantly weaker, and if the tablecloth is pulled quickly enough, does not have enough time to impart a significant impulse on the dinnerware. This trick demonstrates the inertia of the dinnerware, which has an initial velocity of 0 and resists change in velocity.

Scenario: Turning car

You have probably experienced a situation in which you were driving or riding in a car when the driver takes a sharp turn. As a result, you were pressed against the side of the car to the outside of the turn. This is the result of your inertia; when the car changed direction, your body's natural tendency to continue moving in a straight line caused it to collide with the side of the car (or your seatbelt), which then applied enough normal force to cause your body to turn along with the car.

===Connection to Mathematics===

Inertia is studied using Newton's Second Law, which states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. The <math>\frac{d\vec{p}}{dt}</math> in this equation stands for the derivative of momentum.

Derivatives are core concept of a branch of Mathematics called Calculus, which studies constant motion. In this way, understanding inertia gives one a deeper understanding of Calculus, which is an integral member of the Mathematics family.

===Industry Application===

Using a seat belt is one of the best methods for preserving one's life in the event of a car crash. For this reason, they are a big deal in the vehicular safety industry.

Passengers in a moving car have the same motion as the car they are in. When that car stops abruptly, as cars often do in the event of a car crash, its passengers do not stop. Instead, they continue to have the same motion that the car did before it stopped because of inertia. This motion can often propel passengers of out the car, which increases the likelihood of death.

Seat belts are designed to fight against inertia and hold passengers in their seats, often saving their lives in the process.

==History==

Before Newton's Laws of Motion came to prominence, models of motion were based on of the observation that objects on Earth always ended up in a resting state regardless of their mass and initial velocity. Today we know this belief to be the result of [[Friction]]. Since friction is present for all macroscopic motion on earth, it was difficult for academics of the time to imagine that motion could exist without it, so it was not separated from the general motion of objects. However, this posed a problem: the perpetual motion of planets and other celestial bodies. Galileo Galilei (1564-1642) was the first to propose that perpetual motion was actually the natural state of objects, and that forces such as friction were necessary to bring them to rest or otherwise change their velocities.

Galileo performed an experiment with two ramps and a bronze ball. The two ramps were set up at the same angle of incline, facing each other. Galileo observed that if a ball was released on one of the ramps from a certain height, it would roll down that ramp and up the other and reach that same height. He then experimented with altering the angle of the second ramp. He observed that even when the second ramp was less steep than the first, the ball would reach the same height it was dropped from. (Today, this is known to be the result of conservation of energy.) Galileo reasoned that if the second ramp were removed entirely, and the ball rolled down the first ramp and onto a flat surface, it would never be able to reach the height it was dropped from, and would therefore never stop moving if conditions were ideal. This led to his idea of inertia.

== See also ==

[[Mass]]

[[Newton's First Law of Motion]]

[[Newton's Second Law: the Momentum Principle]]

[[The Moments of Inertia]]

[[Acceleration]]

[[Velocity]]

[[Galileo Galilei]]

==References==

*http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Laws-of-Motion-Real-life-applications.html
*http://education.seattlepi.com/galileos-experiments-theory-rolling-balls-down-inclined-planes-4831.html
*http://www.physicsclassroom.com/class/newtlaws/Lesson-1/Inertia-and-Mass
*http://etext.library.adelaide.edu.au/a/aristotle/a8ph/
*https://rachelpetrucciano3100.weebly.com/newtons-first-law.html#:~:text=If%20you%20were%20wearing%20a,you%20from%20being%20in%20motion.&text=Inertia%20is%20the%20property%20of,resist%20a%20change%20in%20motion.&text=Because%2C%20according%20to%20Newton's%20first,unbalanced%20force%20acts%20on%20it.
*https://www.physicsclassroom.com/mmedia/newtlaws/cci.cfm
*https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/a/rotational-inertia

[[Category:Properties of Matter]]

2021-03-26T04:44:03Z

Mkelly74:

File:Inertia 2.PNG

2021-03-26T04:42:56Z

Mkelly74:

2021-03-26T03:51:52Z

Mkelly74:

2021-03-23T08:39:00Z

Mkelly74:

Edited by Nallammai Kannan (Fall 2017)
'''Claimed by MaKenna Kelly (Spring 2021)'''
==The Main Idea==

Ductility is a solids ability to deform under tensile stress. It is similar to [[malleability]], which characterizes a materials ability to deform under an applied stress. Both of these are plastic properties of materials. While they are often similar, sometimes a materials ductility is independent from its malleability[1]. Ductility is the percentage of plastic deformation right before fracture, where plastic deformation means permanent deformation or change in shape of a solid body without fracture under the action of a sustained force[10]. Materials with low ductility are defined as brittle. Materials with metallic bonds have much higher ductility due to the mobile electrons that tend to deform, rather than fracture. Therefore, the most common ductile materials are steel, copper, gold and aluminum. Ductility is an important property in material science and metal-working industries, where solids are deformed and molded with outside forces. Ductile materials can absorb a large amount of energy before they start to show signs of deformation, whereas brittle materials tend to show deformations and cracks relatively easily.[8]
[[File:Cast iron tensile test.JPG|thumb|Fig. 1- Highly brittle fracture]]
[[File:Al tensile test.jpg|thumb| Fig. 2- Semi-ductile fracture]]

Environmental factors can also affect the ductility of a material. A temperature increase causes a material to stretch, and thus increases ductility. A temperature decreases leads to brittle and fragile behavior of the material and as such decreases ductility. Generally, low temperatures adversely affect the tensile toughness of many metals. Similarly, pressure can be used to control ductile-brittle effects. Sufficiently large superimposed pressure can convert a generally brittle material into a ductile material.[11]

Metals like aluminum, gold, silver, and copper have a face-centered cubic crystal lattice structure, and most do not experience a shift from ductile to brittle behavior. Other metals, like iron, chromium, and tungsten, have a body-centered cubic crystal structure and experience a sharp shift in ductility. [7]

The Ductile - Brittle Transition Temperature (DBTT) is the temperature at which the fracture energy passes below a predetermined value (typically 40 J) or the point at which the material absorbs 15 ft*lb of impact energy during fracture[7]. The Ductile - Brittle Transition Temperature is an important consideration when determining which material to select, when said material is subjected to mechanical stresses (as shown at https://www.doitpoms.ac.uk/tlplib/BD6/images/graph0.gif). A low DBTT is integral for designs which will need to function in low temperatures [7].

The Ductile to Brittle Transition can also occur when dislocation motion occurs. Dislocation is defined as areas where the atoms are out of position in the crystal structure[12]. The stress required to move a dislocation depends on the atomic bonding, crystal structure, and obstacles. If the stress required to move the dislocation is too high, the metal will fail instead and form cracks or other deformations instead and the failure will be brittle[6].

===A Mathematical Model===

Mathematically, ductility can be defined as the fracture strain, or the tensile strain along one axis that causes a fracture to occur. Fractures range from brittle fractures (Fig. 1) to fully ductile fractures (Fig. 2), resulting in very different physical appearances associated with the different types. This can be modeled on a stress/strain curve (https://www.nde-ed.org/EducationResources/CommunityCollege/Materials/Graphics/Mechanical/Brittle-Ductile.gif) showing where fracture occurs along the graph.

Quantitatively being able to measure ductility is important with regards to comparing ductility between different materials. Ductility can be measured through two main methods: percent elongation and percent reduction of area[5]. The formulas can be found below: (http://www.engineersedge.com/material_science/ductility.htm)

Percent Elongation = (Final Gage Length - Initial Gage Length) / Initial Gage Length
= ((Lf - Lo) / Lo) * 100

Percent Reduction of Area = (Area of Original Cross Section - Minimum Final Area) / Area of Original Cross Section
= (Decrease in Area / Original Area)
[2]
==Connectedness==
Stress-Strain Testing Background

When a material experiences forces such as stress and strain, understanding its performance under such conditions is critical to performance and application.

Stress = Force / Area
Strain = Delta L / L

Hooke’s Law:  stress = Modulus of Elasticity * Strain

Modulus of elasticity is a property of the material itself, and is a material’s resistance to elastic deformation (non-permanent). From these equations, the behavior of a material undergoing deformation may be determined.

Based on a material’s modulus of elasticity, a material can be categorized as either “brittle” or “ductile”. A brittle material does not , and fractures comparatively quickly. Ductile materials

[[File:Fracture.png]]
[14]
The common points that a ductile material experiences during deformation are identified below. One important characteristic to note is that fracture strength is typically lower than the Ultimate strength of the material. A ductile material is able to withstand additional elongation (strain) even after ultimate strength has been achieved.

[[File:Strain.jpeg ]]

[15]
This knowledge provides a background to how stress-strain testing is conducted.
 Different materials are placed inside a stress-strain testing rig. A load is continuously applied to the material until material fracture. A force transducer, combined with a position sensor, allow for the determination of key characteristics. This data allows for further examination using the equations and principles explained above.

As an engineering major, determining the correct material for components can be high risk. Knowing different materials ranges of ductility, can be integral in choosing the best option. This is especially important in materials that have a high applied tensile strength. Significant brittle fractures can cause a lot of damage. This was seen in Liberty Ships in World War 2, where ships had hull cracks and other major defects due to the cold temperatures of the water. which caused the ship's materials to be brittle. Eventually, a few ships sank and were lost to these brittle fractures.[13]

==History==
Percy Williams Bridgman's findings on tensile strength and material properties led to much of what is known about ductility, including that it is highly influenced by temperature and pressure. These findings led him to win the 1946 Nobel Prize in physics.[4]

== See also ==

*[[Malleability]]
*[[Temperature]]
*[[Pressure]]

===Further reading===

https://en.wikipedia.org/wiki/Ductility

===External links===

[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]

==References==

[1]https://en.wikipedia.org/wiki/Ductility
[2]https://en.wikibooks.org/wiki/Advanced_Structural_Analysis/Part_I_-_Theory/Materials/Properties/Ductility
[3]https://en.wikipedia.org/wiki/Ductility#/media/File:Ductility.svg
[4]https://en.wikipedia.org/wiki/Percy_Williams_Bridgman
[5]http://www.engineersedge.com/material_science/ductility.htm
[6]https://www.doitpoms.ac.uk/tlplib/BD6/ductile-to-brittle.php
[7]http://www.spartaengineering.com/effects-of-low-temperature-on-performance-of-steel-equipment/
[8]http://people.clarkson.edu/~isuni/Chap-7.pdf
[9]http://www.etomica.org/app/modules/sites/MaterialFracture/Background1.html
[10]https://www.merriam-webster.com/dictionary/plastic%20deformation
[11]http://www.failurecriteria.com/theductile-britt.html
[12]https://www.nde-ed.org/EducationResources/CommunityCollege/Materials/Structure/linear_defects.htm
[13]https://en.wikipedia.org/wiki/Liberty_ship
[14]https://www.researchgate.net/figure/Engineering-Stress-Strain-curve-for-both-Brittle-and-Ductile-material-Source_fig4_326753159
[15]https://www.instructables.com/Steps-to-Analyzing-a-Materials-Properties-from-its/

[[Category: Properties of Matter ]]