Dimension of Angular Momentum: Angular momentum can be defined as a vector, which is obviously the product of an object’s moment of inertia times its rotational speed (in radians/second) about a particular axis. When calculating angular momentum as the product of the moment of inertia and angular velocity, the angular velocity must be expressed in radians per second, where radians take a dimensionless unit value.
To find the angular velocity o, we use the law of conservation of angular momentum and note that the moment of inertia and initial angular velocity is given. Because angular momentum is the product of the moment of inertia and angular velocity, if angular momentum remains constant (conserved), then the angular velocity (spin velocity) of the spinning skater must increase.
The greater the mass and the farther away from the centre of rotation (the longer the moment arm), the greater the moment of inertia and hence the greater the angular momentum for a given angular velocity. Since the moment of inertia is a decisive part of the angular momentum of rotation, the latter necessarily includes all the complications of the former, calculated by multiplying the elementary units of mass by the squares of their distances from the centre of rotation.
The magnitude of the angular momentum of an object in orbit is equal to its linear momentum (the product of its mass m and its linear velocity v) multiplied by the distance r along the perpendicular from the centre of rotation to a line drawn in the direction of its instantaneous and through moving through the centre of gravity of objects, or just mvr. The orbital angular velocity of the particle point is always parallel and directly proportional to the orbital angular velocity vector o, on which the constant measurement depends on both particle weight and its distance from its source.
Consider a particle with mass m and momentum P, which is located at a distance r from the origin O. This expression is exactly the same as the relationship between force and momentum F = Dp/Dt. In the stucco and door example, the door hinge is a natural point that can be seen as a plank, and the piece of stucco changes its angle if it is seen by someone standing on the hinge, fig. c. For this reason, the conserved quantity we are studying is called angular momentum.
Using the principle of uniformity of size, it is possible to derive a new relationship between physical quantities if the dependent quantities are known. This concept is better known as the principle of dimensional uniformity.
The dimensions of a given quantity tell us how and in what way various physical quantities are related to each other. Focus on how different physical quantities are used to derive a formula and are used to calculate it.
Reducing the size of an object by a factor of n causes its angular velocity to increase by a factor of n. The size of the earth M is 5.979 x 1024 kg, and the Earth Radius is R 6.376 x 106 m. The larger angular velocity or is, of course, exactly one revolution per day, but we have to convert them or to radians per second in order to calculate in SI units. The mass of the turntable is 120 kg, the radius is 1.80 m, and the angular velocity is 0.500 rpm.
It turns out there is a way to define angular momentum as a vector, but in this section, the examples will be limited to a single plane of rotation, i.e. effectively two-dimensional situations. They are isolated from influences that change rotation, that is, they are closed systems. Although angular and linear momentum is exact analogs, they have different units and cannot be directly converted to each other like forms of energy. Later, the concept of angular will be extended to systems of particles, including rigid bodies.
A rigid rotating object, for example, continues to rotate at a constant speed and in a fixed orientation unless an external torque is applied to it. Stopping won’t violate the law of conservation of momentum, because the rotations don’t add anything to the Earth’s momentum. It would be a violation of the conservation of angular momentum if the skater kept spinning at the same speed, i.e. giving the same amount of time to each spin because her contribution to her angular momentum would decrease and no other part of her body would increase her momentum. Issues for Discussion * The preservation of the simple old moment p can be seen as a greatly expanded and modified offspring of Galileo’s original principle of inertia, therefore no force will be required to keep an object in motion.
The kinetic energy of the rotation is much larger than the initial value. Knowing the angular acceleration, the final angular velocity and kinetic energy of the rotation can be calculated. Angular acceleration can use Newton’s second law or the rotational counterpart of Newton’s second law.
Since, Angular Momentum = Angular Velocity × Moment of Inertia . . . . (1)
Because, Angular Velocity = Angular displacement × [Time]-1 = [M0L0T0] [T]-1
Therefore the dimensional formula of Angular Velocity = M0L0T-1 . . . . . . (2)
And, Moment of Inertia, M.O.I = Mass × (Radius of Gyration)2
Therefore the dimensional formula of M.O.I = M1L2T0 . . . . . (3)
Here, substituting equation (2) and (3) in equation (1) we get,
From (1)
M = [M0L0T-1] × [M1L2T0]-1 = [M1L2T-1].
So, the angular momentum is dimensionally represented as [M1L2T-1].
Keep in note that where M is mass, L is length, T is time.
Angular momentum is a physical quantity that represents the rotational motion of an object. It is the product of an object's moment of inertia and its angular velocity. Mathematically, it is expressed as: where:
The dimensions of angular momentum help in understanding how it behaves under different physical conditions, such as in rotational motion. They also help in verifying the consistency of equations in rotational mechanics and ensuring the correctness of physical laws, such as conservation of angular momentum.
Angular momentum is the rotational equivalent of linear momentum. While linear momentum is the product of mass and velocity (), angular momentum involves rotational quantities such as moment of inertia and angular velocity. In a similar way, angular momentum describes the rotational motion of an object, while linear momentum describes its straight-line motion.
No, the dimension of angular momentum is independent of the reference frame. However, its magnitude and direction may change depending on the observer’s position and velocity. The dimension always remains the same across all frames of reference.
Angular momentum plays a crucial role in various real-world phenomena, such as: