It is the characteristic of the rigid body and the axis about which it rotates. The parameter I is independent of the magnitude of the angular velocity. Let Σ m i(r i ) 2 be I which is a new parameter characterising the rigid body known as the Moment of Inertia. Therefore, taking ω out of the sum, we get, We know that angular acceleration ω is the same for all particles. ∴ K = Σ k i = 1/2 ( Σ m i(r i ) 2ω 2 ) where n is the number of particles in the body. Where m i is the mass of the particle. The total kinetic energy K of the body is thus the sum of the kinetic energies of individual particles. What is the analogue of mass in rotational motion? To answer this question, we have to derive the equation of kinetic energy in rotational motion.Ĭonsider a particle of mass m at a distance from the axis with linear velocity = v i = r iω. Therefore, the kinetic energy of this particle is, I c = (1/12) ML 2 Kinetic Energy in Rotational Motion Therefore, the parallel axis theorem of the rod is: The distance between the end of the rod and its centre is: Let us see how the Parallel Axis Theorem helps us to determine the moment of inertia of a rod whose axis is parallel to the axis of the rod and it passes through the center of the rod. H 2 = square of the distance between 2 axes The moment of inertia of a body about an axis parallel to the body passing through its centre is equal to the sum of moment of inertia of the body about the axis passing through the centre and product of the mass of the body times the square of the distance between the two axes. As a result, the following theorems can be used to calculate the moment of inertia along any given axis: After selecting two distinct axes, you will notice that the object resists the rotational change differently. The moment of inertia, as we all know, is affected by the axis of rotation. I = 0.168 kg m 2 Moments of Inertia for Different Objects I = Σ m ir i 2 = m Σ r i 2 = 0.3 ….(Converting the distance of the particles to metre) What is the moment of inertia of the system about the given axis? Each particle has a mass of 0.3 kg and they all lie in the same plane. Where r i is the perpendicular distance from the axis to the i th particle which has mass m i.Ī system of point particles is shown in the following figure. R = (perpendicular) distance between the point mass and the axis of rotation Moment of Inertia of a System of Particlesįor a system of point particles revolving about a fixed axis, the moment of inertia is: R = Distance from the axis of the rotation.Īnd the Integral form of MOI is as follows:ĭm = The mass of an infinitesimally small component of the body
Moment of inertia is the property of the body due to which it resists angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation. Each particle in the body moves in a circle with linear velocity, that is, each particle moves with an angular acceleration. In rotational motion, a body rotates about a fixed axis. So we have studied that inertia is basically mass. Because the heavier one has more mass, it resists change more, that is, it has more inertia. For instance, it is easier to throw a small stone farther than a heavier one. More the mass of a body more is the inertia. But what causes inertia in a body? Let’s find out.
What is Inertia? It is the property of a body by virtue of which it resists change in its state of rest or motion.
Kinematics of Rotation Motion about a Fixed Axis.Dynamics of Rotational Motion About a Fixed Axis.Angular Momentum in Case of Rotation About a Fixed Axis.Angular Velocity and Angular Acceleration.Theorems of Parallel and Perpendicular Axis.Browse more Topics Under System Of Particles And Rotational Dynamics Understand the Theorem of Parallel and Perpendicular Axis here in detail. Therefore, it gets pushed backwards, that is, it resists change in its state. As soon as you board the moving train, your lower body comes in contact with the train but your upper body is still at rest. That is because before boarding the train you were at rest. Similarly, when you board a moving train, you experience a force that pushes you backwards. Therefore, when the bus stopped, your lower body stopped with the bus but your upper body kept moving forward, that is, it resisted change in its state. Your lower body is in contact with the bus but your upper body is not in contact with the bus directly. When the bus stopped, your upper body moved forward whereas your lower body did not move. What did you experience at this point? Yes. After a few minutes, you arrive at a bus stop and the bus stops.
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