[Music] consider a solid spear a solid cylinder and a hollow cylinder having same masses are rolled on a ramp which one will reach the bottom first let us check via reaches the bottom first do you know why - no this reason you have to understand for this moment of inertia let us learn about woman DAF in a shelf here there are two wheels with the same mass and different radians if they are made to rotate on the road each one is harder to rotate the radius of alia is greater than the radius of will be will be accelerate faster than we lived since Velia has more radius the mass distributed about this interruptible a is greater than that of the VB so Amanda of inertia of a is greater than moment of inertia be moment of inertia is a scalar value which tells us how difficult it is to change the rotational velocity of the object here only given rotational axis if moment of inertia is relatively large it is very difficult to accelerate in angular direction if the moment of inertia is small it is relatively easy to rotate angularly therefore with less moment of inertia accelerate faster now let us check how woman of inertia changes with respect to some physical quantities first we can check what happens to the moment of inertia if mass of the body changes with respect to an axis consider a wheel of mass 20 kilogram with radius point 2 meter formula to find out the moment of inertia is different for different shapes and different axes here the moment of inertia of the wheel about the axis of rotation is M R square which is equal to 20 into 0.2 square which is equal to point 8 program meter square hence the moment of inertia about an axis through the center in these cases point 8 program meters per let us take another wheel of mass 20 fake la Graham with same radius point 2 meter the moment of inertia of the wheel about the axis of rotation is again M R square which is equal to 25 into 0.2 square equal to 1 kilogram meter square therefore the moment of inertia about the axis through the center in these cases 1 kilogram meter square if the mass increases the moment of inertia also increases so it is much harder to rotate an object of greater mass with same radius now consider two wheels with the same moss and different radius one wheel with mass 20 kilogram and radius point 2 meter we already calculate the moment of inertia about its axis which is equal to 0.82 Agra meter square now let us take the second wheel which he has the same mass 20 kilogram and radius point 5 meter the moment of inertia of the wheel about the axis of rotation is same R square which is equal to 20 into 0.5 square which is equal to 5 kilo gram meter square therefore moment of inertia about the axis through the center in this case is 5 kilometer square if the radius increases the moment of inertia also increases so it is much harder to rotate an object with same Maus and greater radius now let us try what happens to the moment of inertia if the axis of the rotation changes this is a rod of length L if he grab the rod in the middle and tried to spin it it is easy to spin mass of the rod is distributed equally from the centre of rotation if we grab the rod at the end and try to spin it it is extremely difficult to rotate if the rod is rotated about an axis through the center of the rod the formula to calculate moment of inertia about the axis through the center of the rod is ml square by 12 if the rod is rotated about an axis through an end all the mass is distributed very far from the axis of rotation the formula to calculate the moment of inertia about an axis through an end is M L square by 3 ml square by 3 is greater than ml square by 12 for a given rod therefore moment of inertia of a spinning rod about an axis through an end of the rod is greater than moment of inertia of a rod spinning about an axis through the center of the rod so it is extremely difficult to rotate through an axis in which mass is distributed very far from the axis of rotation comparing these three examples we can come to a conclusion that moment of inertia depends on mass of the object radius of the object and the axis of rotation and each shape has different formula to calculate moment of inertia moment of inertia is a property of any object which can be rotated it is a scalar value which tells us how difficult it is to change the rotational velocity of the object they are only given rotational axis the moment of inertia of an object depends on Mars it also depends on the distribution of mass related to the axis of rotation now if you look into our initial problem the moment of inertia of a solid sphere about its central axis is 2 by 5 M r-square moment of inertia of a solid cylinder about its central axis ease of mr square the moment of inertia of a hollow cylinder about its central axis mr square if mass is same in three cases mr square is greater than half mr square is greater than 2 by Phi M r-square spear has lesser moment of inertia among these three so it reaches the bottom first [Music]