Inertia: Difference between revisions

Latest revision as of 22:13, 18 April 2021

Edited by MaKenna Kelly Spring 2021

This page defines and describes inertia.

All Khan Academy content is available for free at www.khanacademy.org

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 frames of reference.

A Mathematical Model

According to Newton's Second Law: the Momentum Principle, [math]\displaystyle{ \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]\displaystyle{ \vec{F}_{net} = m \vec{a} }[/math]. Solving for acceleration yields [math]\displaystyle{ \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. [1] The situation shown is very similar to that described in the "Industry Applications" portion of this page.

Here are snapshots from a similar model:

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]\displaystyle{ T = ma }[/math]

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

[math]\displaystyle{ m = \frac{T}{a} }[/math]

[math]\displaystyle{ m = \frac{20 kg}{4 m/s^2} }[/math]

[math]\displaystyle{ 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]\displaystyle{ \tau = I\alpha }[/math]

Where [math]\displaystyle{ \tau }[/math] is torque, I is rotational inertia, and [math]\displaystyle{ \alpha }[/math] is angular acceleration.

[math]\displaystyle{ \alpha = \frac{\tau}{I} }[/math]

[math]\displaystyle{ \alpha = \frac{100 Nm}{0.1 kgm^2} }[/math]

[math]\displaystyle{ \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]\displaystyle{ \omega = \omega_0 + \alpha \times t }[/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]\displaystyle{ t = \frac{{\omega}_{max}}{\alpha} }[/math]

[math]\displaystyle{ t = \frac{150 rad/s}{1000 rad/s^2} }[/math]

[math]\displaystyle{ t = 0.15 s }[/math]

Difficult

What is the rotational inertia of the object shown above?

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]. The [math]\displaystyle{ \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.

References

@@ Line 1: / Line 1: @@
+'''Edited by MaKenna Kelly Spring 2021'''
 This page defines and describes inertia.
+All Khan Academy content is available for free at www.khanacademy.org
 ==The Main Idea==
-Inertia is the tendency of matter to resist change in motion. It is an inherent property of matter; the inertia of an object is directly proportional to its mass and is in fact sometimes used to define mass. Newton's first law states that the velocity of an object does not change unless there is an unbalanced force acting on it. This means that all non-zero resultant forces on a cause ''acceleration'' (not ''velocity''). Forces on a body are easy to identify through a Free Body Diagram, through which you can determine the magnitude and direction of forces. If the sum of the forces in all directions are zero, then the object does not accelerate. It could, however, be moving with constant velocity. Inertia allows us to understand a key concept of kinematics which is that forces cause a change in velocity (acceleration). Furthermore, this acceleration is inversely proportional to the mass of the body.
+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]].
-==History==
+===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 <b>can</b> tell the <i>acceleration</i> 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?
-Before Newton's Laws of Motion came to prominence, academics worked based off of their observations that objects on Earth always ended up in a resting state no matter the initial velocity or mass. This observation was actually due to the presence of friction. Friction is a force that acts in the opposite direction of motion. Without friction, a ball rolling across a surface would continue at the same velocity until an outside force acted upon it. This was counter-intuitive for people at the time, however, they had no other way to explain the constant motion of heavenly bodies based on the Copernican model of a heliocentric solar system. For, the only force acting on bodies in space is that of gravity, which causes a radially inward acceleration and a constant orbital velocity. Since the force from gravity didn't accelerate or decelerate the planets along their orbits, and no other forces acted on them, it could be proven that objects do not tend to decelerate on their own.
+<math>\tau = I\alpha</math>
+Where <math>\tau</math> is torque, I is rotational  inertia, and <math>\alpha</math> is angular acceleration.
-Galileo performed an experiment with two ramps and a bronze ball.  To begin, the two were set up at the same angle. Galileo observed that if a ball was released at one height, it would roll to the same height at which the ball was released.  He then experimented with altering the angle of the second ramp. He concluded that even though it may take longer, when the angle is smaller, the ball will still roll up to the same height. Because the height was conserved, Galileo believed that if a ball was rolled from a ramp to a flat surface, it would stay in motion unless a force stopped it.
+<math>\alpha = \frac{\tau}{I}</math>
+<math>\alpha = \frac{100 Nm}{0.1 kgm^2}</math>
-[[File:newtonsexperiment.png|frameless|600px]]
+<math>\alpha = 1000 rad/s^2</math>
-==Calculations with Inertia==
+b) How long will the flywheel take to reach a steady speed if starting from rest?
-Mass denoted by ''m'' is more accurately referred to as inertial mass. That is because the true definition of mass is a body's resistance to motion. Oftentimes mass is incorrectly thought of with regard to weight, which is the force of gravity acting upon a body. Inertial mass is present in a majority of kinematics equations, such as F=ma, and p=mv.
+Using rotational kinematics,
+<math>\omega = \omega_0 + \alpha \times t</math>
-The formula for the standard moment of inertia, typically denoted as I, depends on what the particular statically determinate object is as well as its rotational axis.
+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.
-[[File:Inertia1.png|850px]]
+<math>t = \frac{{\omega}_{max}}{\alpha}</math>
-Calculating the polar moment of inertia, typically denoted as J, depends on what the deformable body is and also depends on the rotational axis; the cross sectional area, distance from the center of mass, and projected deformation are also key components. J comes from the relationship between the total internal torque at a cross section and the stress distribution at a cross section.
+<math>t = \frac{150 rad/s}{1000 rad/s^2}</math>
-[[File:Inertia2.png|1100px]]
+<math>t = 0.15 s</math>
-==Inertial Reference Frame==
+===Difficult===
-Reference frames are important for how velocity and acceleration are perceived. Imagine a person sitting in a car moving at a constant speed due north. To that persons left there is a truck moving at the same speed due north. To the persons right there is a pedestrian moving much slower but in the same direction. Think of the  reference frame as the point of view of the person in the car. From their perspective the truck would be standing still and the pedestrian would be moving backwards. This is because in this reference frame the person in the car is being treated as a fixed point. All motion is perceived relative to that fixed point. The distance between the car and the pedestrian is increasing which is why the pedestrian appears to be moving away.
+[[File:Inertia_diificult_.png]]
-Examples:
+What is the rotational inertia of the object shown above?
-If object A is moving at 1 mph and object B is moving behind object A it at 2 mph, how fast is object be moving relative to object A?
+<math>I = m_1r_1^2 + m_2r_2^2 + ... = \sum m_ir_i^2</math>
--In this example from the perspective of object A, object B is approaching object A at 1 mph
+Where I is rotational inertia, m is mass, and r is distance from the axis of rotation.
-What if object A is moving 1 mph towards object B?
+<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>
--In this example from the perspective of object A, object B is approaching object A at 3 mph
+<math>I = 4.9375 kgm^2</math>
-Important Concepts:
+==Connectedness==
+===Interesting Applications===
--The velocity of the reference frame needs to be taken into consideration when computing inertial reference frame problems.
+Scenario: Tablecloth Party Trick
-It's important to note that this concept also applies to acceleration. If someone were to drop a ball from an accelerating car the ball would appear to have a force accelerating it in the reverse direction. This force would not actually exist. From the perspective on someone standing on the side of the road the ball would appear to move in the direction of the car (inheriting the velocity of the car) and downwards ( due to gravity).
+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.
-Example:
+Scenario: Turning car
-Person A is in a car accelerating North at 15 m/s^2. Describe the motion of a ball dropped from the car after 5 seconds.
--The ball would appear to accelerate down at 9.81m/s^2 due to gravity, and would move 75 m/s north.
+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.
-Describe the motion of the ball in reference to the car.
+===Connection to Mathematics===
--The ball would appear to accelerate down at 9.81m/s^2 due to gravity, and would accelerate  15m/s^2 south.
--keep in mind the ball isn't truly undergoing that acceleration south. It just appears to from that reference frame.
-==Examples==
+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.
-'''Simple'''
+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.
-A simple example of inertia is the classic trick of pulling a table cloth off of a table while the plates and cups stay in place. The tablecloth is acted upon by a force applied through the persons arm, while the only horizontal force acting on the plates and cups is friction. The person performing this trick must keep that in mind because a tablecloth with too high of a friction coefficient will act on the plates and cups and cause them to accelerate off of the table. It is clear from this example, though, that the plates and cups remain still unless acted upon by an outside force due to their inertia.
+===Industry Application===
-'''Meddling'''
+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.
-Area Moment of InertiExample.jpg:
+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.
-[[File:AreaI.jpeg|1000px]]
+Seat belts are designed to fight against inertia and hold passengers in their seats, often saving their lives in the process.
-'''Difficult'''
+==History==
-Polar Moment of Inertia Example:
+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.
-[[File:PolarJ.jpeg|1000px]]
+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 ==
-[[Galileo Galilei]]
-[[Conservation of Momentum]]
+[[Mass]]
+[[Newton's First Law of Motion]]
-[[Newton's Laws and Linear Momentum]]
+[[Newton's Second Law: the Momentum Principle]]
 [[The Moments of Inertia]]
@@ Line 89: / Line 141: @@
 [[Velocity]]
+[[Galileo Galilei]]
+===Further Reading===
+Books:
+''Gravitation and Inertia'' by Ignazio Ciufolini and John Archibald Wheeler
+''Focus on Inertia'' by Joanne Mattern
+''Inertia and Gravitation'' by Herbert Pfister and Markus King
+===External Links===
+[https://www.physicsclassroom.com/class/newtlaws/Lesson-1/Inertia-and-Mass The Physics Classroom]
+[https://www.qrg.northwestern.edu/projects/vss/docs/propulsion/2-what-is-inertia.html Northwestern University]
+[https://www.thoughtco.com/inertia-2698982 ThoughtCo]
 ==References==
-[http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Laws-of-Motion-Real-life-applications.html]
+*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://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://www.physicsclassroom.com/class/newtlaws/Lesson-1/Inertia-and-Mass
-[http://etext.library.adelaide.edu.au/a/aristotle/a8ph/]
+*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]]