Hello, Welcome to this new video on moment of inertia from "Feel Science!".
Enjoying riding in automobile!. Thanks to moment of inertia, it helps automobile engineers in designing automobile that provides jerk free super smooth rides.
Wondering how were engineers able to build world's tallest structure like Eiffel Tower?. Thanks to Moment of inertia, it helps engineers in design of structural elements with high strength.
Using google maps for getting direction to desired destination or seeing images of your home google earth earth or seeing images of distant planet, star or galaxy. All this images are taken from satellite. To take such a sharp image at such a high zoom, satellite must remain rock steady. Again thanks to Moment of inertia, it helps satellite to remain rock steady such that it can take super sharp images.
Wondering how these things are related to monent of inertia. Then watch this complete video where we are going to cover a topic of science called " Moment of Inertia". So let’s begin with the topic.
To understand and have a intuitive feeling of moment of Inertia we need to know all the 4 things
First: What is Moment of Inertia
Second: Origin of Moment of Inertia. What made physicist define such a term called moment of Inertia, what kind of problem it solved.
Third: How to have a feel of it. By this I mean I want to feel number and how does it correlate to other known scientific quantity that we know.
And finally, what is the application of Moment of Inertia in real world. How and where it is used.
So Let’s start with what it is. Let’s say that we have to find Moment of inertia of this rectangular mass rotation about axis as shown. Mathematically moment of inertia is integration of r square dm. In simple terms it is also summation of r square dm over all the particles. In this formulae , dm is mass of small element which is at a distance r from the axis of rotation.Now this is just the mathematical definition of moment of inertia. However this mathematical equation or mathematical definition doesn’t give us an insight into what moment of inertia is.
Now we know its definition, i mean mathematical definition but it is not sufficient to have a feel of it. This definition leaves us with lot of question like What is the origin of this formulation?. Why it is defined in such a way?. How to have a feel of moment of inertia in real world? What is the significance of the numerical value of Moment of Inertia? And finally the most important thing, where it is used and what are the application of moment of inertia? So let’s begin our journey about understanding what is moment of inertia.
First we will start with the origin and why it is defined in such a way. To understand this let us go back a little. Kinetic energy!. Origin of moment of inertia has a strong relation with kinetic energy. We know kinetic energy is defined as ½ m v square, where m is mass in kg and v is velocity in meter. Anyone who studied science would knows this formulae. But wait!, This formula is actually applicable only to point mass. Now If we need to find kinetic energy of a body with finite dimension having mass m and moving with velocity V along a straight line. How will we find it?. And most of the real word body that we are interested are not body with point mass. Then how to find kinetic energy of finite body?. Lets us start. We will use the above formula and try to find Kinetic energy of finite body. But this formula is only applicable for point mass. So we divide this body into large number of small section or particle say n and name part these particle or part as 1,2 ,3 and so on. Then kinetic energy of complete body is equal to sum of kinetic energy of all these small particle. Right !. So total Kinetic energy of body is equal to sum of ½ m i times v i square for all the particles, where m i is mass of ith particle and v i is velocity of ith particle.. Now we know that this body is moving in straight line, hence velocity of all the particle are same i.e. V1 = V2 = V3 = Vn= V. Now from above formulae when we can remove all constant outside summation sign. Then we have Kinetic energy is equal to ½ V square times sum of m i over all the particles of body. Now sum of all m i is equal m, mass of body. And finally we get Kinetic energy of body as ½ m V square. This formula is similar to Kinetic energy of particle.Right
Now let us come to origin of Moment of inertia and why it is defined in such a way. Long before there was a time when physicist wanted to find kinetic energy of mass rotating about an axis with constant angular velocity. Obviously they cannot use the above formula ½ m v square in this case as velocity at each point varies according to V = r omega, where r is the distance of point from axis of rotation and omega is angular velocity by which body is rotating. Moment of Inertia is solution to this problem.
Let us see how. To find Kinetic energy of body rotating about an axis we will go back to the basics. kinetic energy of particle or point mass is defined as ½ m v square. And Kinetic energy of whole body rotating about axis is equal to Sum of kinetic energy of all the particles of this body. We use the same formula. sum of ½ m i times v i square for all the particles 1, 2 and so on where m i is mass of ith particle and v i is velocity of ith particle. In this case, both m i and v i is not constant and hence we cannot take it outside of summation sign.
But we know that V i = r i times omega, where r i is distance of particle m i from axis of rotation and omega is angular velocity which is constant for this case. Substituting this formulae and taking constant outside we get. Kinetic Energy of rotating body = ½ omega square into summation of m i times r i square for all the particles. This summation of m i times r i square for all the particle is called as moment of inertia. And basically originated and defined to find Kinetic energy of rotating particle. When m i tends to zero, this summation can be written as integral of r square dm. This is the correct defination of moment of inertia.
Now we now know the origin of moment of inertia and why it is defined in such a way and why there was need to define some quantity like this.
Now we come to the next topic in understanding moment of inertia. How to feel it. Let’s say I have an object whose Moment of Inertia is say 1 kg meter square. Then what is the significance of this number. Let us try to decode this. We know Moment of inertia is integral r square dm or summation of m i times r i square. Now if we take a point mass at 1 meter distance from the axis of rotation then numerical value of moment of inertia is equal to mass of point object located at distance of 1 meter from axis of rotation. Now if moment of inertia is equal to say 5 kg meter square then it is equivalent to having mass of 5 kg at distance 1 meter from axis of rotation.
Now that is fine, but how do i feel it. Still not having a feel of higher or lower moment of inertia. Let’s do some practical To feel how moment of inertia and how it increase with increase in distance from axis of rotation, Just attach a string to mass and try to rotate it with hand at different radius. Feel the difference. When mass moves away from the axis of rotation we feel more weight in out hand. Not having stone and string with you now. Then take some mass in you palm and rotate it about you arm slowly and then increase the radius (keeping rpm constant) and feel how energy required to rotate changes with different radius and heavy ness. As radius increase we need to put significant energy to maintain speed constant and we feel more exhausted. Right
Time to start with application. Moment of inertia finds its uses at many places. Every motor vehicle uses a component called flywheel. It uses a specially designed metallic wheel. These wheel is designed to have higher moment of inertia. Higher the moment of inertia higher is the Kinetic energy it can store for same angular velocity.
Why it is used. Most of the motor vehicle engine running of gas/petrol/diesel now are 4 stroke. In first stroke engine sucks air and fuel, in second stroke, compresses this fuel and air mixture. combustion takes place (in diesel engine) or initiated (in petrol engine) at the end of second stroke or at the initiation of third stroke. After combustion temperature and pressure of gas increases Third stroke begins with the expansion of this heated gases and part of the energy released from combustion is converted into mechanical energy used for driving vehicle. Fourth stroke is expulsion of combustion products out of the engine. Now out of this 4 strokes, only 3rd stroke which is expansion of hot pressurized gases gives power from the engine while other 3 strokes are doesn’t give any energy , in fact some of them actually consumes energy. Overall energy given by engine is more that energy consumed and hence we get net power out of engine. It is worth to note that if this IC engine is directly connected to Wheels of automobile then the fluctuation in energy output with different storks will cause tremendous jerk and load and motion of vehicle will not be smooth. This problem is solved by using a flywheel. It stores energy in form of Kinetic energy, receivers energy from engine during the power stroke and supply energy to engine if required during the remaining 3 strokes And at the same time it also provides continuous energy for movement of vehicle. By this way it dampens the fluctuation of energy that is provided from engine to driving wheels and makes our journey comfortable
It is also used in lot of sports. One such beautiful application is seen when swimmer dives into swimming pool while performing acrobatic stunts. Just after starting dive, swimmer performs variety of acrobatic stunts by rolling their body with high angular speed. However in order to land perfectly into water, they need to slow down their angular speed. To achieve this when they reach near water surface they spread their body so as to increase moment of inertia. Since kinetic energy has to be conserved increasing moment of inertia reduces angular velocity. This helps diver to land nicely into water without any injury.
Lets do some practical
Try to balance a steady top on its tip. Finding it difficult? It is almost impossible to make top stand on its tip when it is stationary. However that is not the case with spinning top. Right. Without any extra effort spinning top balances itself very nicely on its tip. The reason is moment of inertia. Let us see why?. When top is spinning its angular momentum (about axis of spin) is quite high compared to couple acting on the top due to gravity. By the way, Angular moment of spinning object is Moment of inertia about spinning axis times angular velocity. Higher the moment of inertia higher the angular momentum. Like conservation of momentum, there is law which says that angular momentum should also be conserved. So when top is spinning about its axis at higher angular velocity this angular momentum is higher and by conservation of angular momentum the spinning top is then more stable against small torques like the action of gravity on the top. However when angular velocity of top reduces, there comes a time when torque acting on it due to gravity on top becomes significant and this is the time when it starts falling and ultimately end up on ground. This shows that angular momentum (result of higher moment of inertia and angular velocity) helps in stabilizing the spinning top on its tip and prevent its fall onto ground.
Third application of moment of inertia is seen in satellite.A reaction wheel is a type of flywheel used primarily by spacecraft for three axis attitude control, which doesn't require rockets or external applicators of torque. They provide a high pointing accuracy and are particularly useful when the spacecraft must be rotated by very small amounts, such as keeping a telescope pointed at a star. A reaction wheel is sometimes operated as (and referred to as) a momentum wheel, by operating it at a constant (or near-constant) rotation speed, in order to imbue a satellite with a large amount of stored angular momentum. Doing so alters the spacecraft's rotational dynamics so that disturbance torques perpendicular to one axis of the satellite (the axis parallel to the wheel's spin axis) do not result directly in spacecraft angular motion about the same axis as the disturbance torque; instead, they result in (generally smaller) angular motion (precession) of that spacecraft axis about a perpendicular axis. This has the effect of tending to stabilize that spacecraft axis to point in a nearly-fixed direction, allowing for a less-complicated attitude control system. Satellites using this "momentum-bias" stabilization approach include SCISAT-1; by orienting the momentum wheel's axis to be parallel to the orbit-normal vector, this satellite is in a "pitch momentum bias" configuration.
Finally moment of inertia is used in design of all structures including Famous Eiffel Tower. To understand this, let us see another way to feel moment of inertia. Moment of inertia also can be defined in terms of distance. If we fix mass as 1 kg then moment of inertia is the distance (which is square root of Moment of inertia) at which 1 kg of point mass is located. From this we know that higher the moment of inertia, farther apart is the distribution of mass from the axis of rotation. With this feel let us do some simple experiment.
Try to bend a ruler in the first way as shown in the figure, It is easy to bend. Now orient the ruler as shown in second figure. And try to bend it. Finding Difficult to bend it to the same amount. Bending ruler in the second case is difficult. Why?. Answer to this question is related to how far mass of bending object is distributed from the axis of bending as shown. This property turns out to be proportional to moment of inertia. Bending axis for each of the case is shown.Fibers or mass below the bending axis will feel tension due to external force while fibers of mass above bending axis will be in compression due to external bending force. For the first case, mass if distributed near the bending axis and moment of inertia in first case say MI 1 is lower then the moment of inertia for the case 2 say MI 2. Moment of inertia for the second case MI 2 is higher compared to MI 1 as mass in case 2 is distributed relatively away from bending axis. And according to science, resistance to bending by external forces is directly proportional to moment of inertia about the bending axis. Higher the moment of inertia, higher is the resistance to bending. Hence by designing structure in particular way i.e. increasing moment of inertia about bending axis, engineers can design stronger beams.
This relation of moment of inertia with resistance to bending or strength of beam is used to design each and every structure in the world. See the beam cross section is designed in such a way that it gives higher moment of inertia. From the above principle we can also save huge money and natural resources. We know that mass located near the bending axis contributes less to moment of inertia compared to same mass located further away from axis. This means instead of putting mass near axis which contributes less in strength we can put the same masses away for better strength. The same is depicted here, by removing mass which doesn’t provide bending strength thus results in saving of natural resource and money.
Even design of Eiffel tower is based on this principle. Infact one of the main structural component of eiffel tower is beams and to provide such an enormous strength to beam so that it can sustain loads of such a massive structure, use of beams with higher moment of inertia becomes is inevitable.
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