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Momentum: The product of an object’s mass and velocity is its momentum. Momentum is a property of anything that has mass and moves. It is a vector quantity in that it has both a magnitude and a direction. A bicycle moving down a hill is an example of an object in momentum.
Velocity: The velocity of a point is a quantity that describes how fast and in which direction it is moving. For a circular path, a point always moves in a direction that is tangential to its path. The time rate at which the point moves along its route is measured by the magnitude of the velocity.
Just like linear momentum and velocity in rectilinear motion, we can define angular momentum and angular velocity in rotational motion.
Angular velocity:
Angular velocity is the rate at which an object rotates or circles around an axis, or the pace at which the angle between two bodies changes.
The Greek letter omega (ω) is commonly used to represent angular velocity and is also used to represent frequency.
The angular acceleration is the rate at which the angular velocity changes over time, and it is commonly given in radians per second.
The unit of angular velocity is revolutions per minute (rpm).
Formula of angular velocity:
There are three formulae used to calculate angular velocity-
The pace at which an object’s location angle changes with respect to time is known as angular velocity.
So,
- ω = θ/t
Where,
ω = angular velocity
θ = position angle
t = time
- ω = s/rt
Where,
s = length of arc
r = radius of circle
t = time
- ω = v/r
Where, v = linear velocity
r = radius of circle
Average angular velocity
The average angular velocity is calculated by dividing the change in position of the angle by the change in time. The angular velocity is a vector pointing in the direction of the rotational axis.
The angular velocity unit is radians/s.
ωavg = ∆θ/∆t = θ2-θ1/t2-t1
Where,
ωavg = average angular velocity
∆θ = change in position of angle
∆t = change in time.
θ2 & θ1 = final and initial angular coordinate respectively
t1 & t2 = initial and final time respectively
Instantaneous velocity
The instantaneous velocity is also known as just velocity. It is a statistic that indicates how quickly an item is moving along its path. When the duration (and consequently the distance) between two points on a path approaches 0, it’s the average velocity between them.
The direction of angular velocity
The angular velocity is measured along the axis of rotation and points away from you for a clockwise rotating item, and towards you for a counterclockwise rotating object. The right-hand rule describes its direction.
Right-hand rule
An object’s angular displacement occurs when it rotates. Since a point on a revolving wheel changes its direction frequently as it rotates, determining the direction of angular velocity is challenging. On a rotating wheel, only the axis of rotation has a fixed direction. The right-hand rule can be used to determine the direction of angular velocity along this axis. It says that if you grasp the axis in your right hand and curl your fingers in the object’s rotational direction, your thumb will point in the direction of angular velocity.
Angular Momentum
Angular momentum is a significant quantity since it is conserved. This means that the total angular momentum of a closed system remains constant. Both the direction and the amplitude of angular momentum are conserved.
It is the attribute of any rotating object calculated by multiplying the moment of inertia by the angular velocity.
Formula for Angular Momentum:
The distance of an item from a rotation axis multiplied by the linear momentum yields angular momentum (L)
L = mvr
Where,
L is angular momentum
m is mass
v is velocity
r is radius
So long as there is no resulting torque on a system of particles, its angular momentum is conserved.
The resulting torque operating on a system of particles is equal to the rate of change of angular momentum of the system.
The SI unit of angular momentum is kg m² s-1.
Dimensional formula of angular momentum:
Any physical quantity’s dimensional formula describes how and which of the base quantities are included in that quantity.
The dimensional formula of angular momentum is written as M¹L²T-1
Where,
M is mass
L is length
T is time
Derivation of the dimensional formula of angular momentum:
Angular Momentum = Angular Velocity × Moment of Inertia …..(1)
Angular Velocity = Angular displacement × (time)-1
The Dimensional formula for angular Velocity is M⁰L⁰T-1 ….(2)
Moment of Inertia = Mass × (Radius of gyration)²
The dimensional formula for the moment of inertia = M¹L²T⁰
By substituting (2) & (3) into (1) we get,
M = [M⁰L⁰ T-1] × [M¹ L²T⁰]-1 = M¹L²T-1
Hence, the dimensional formula of angular momentum is M¹L²T-1.
Importance of angular momentum:
Since the angular momentum is a conserved quantity, it is important in physics. Unless an external torque acts on a system, its angular momentum remains constant. The rate at which the angular momentum is transmitted into or out of a system is known as torque.
The resistance of a rigid body to a change is measured by its change of inertia.
The conservation of angular momentum explains a wide range of human and natural events.
Motorcycles, frisbees, and rifled bullets all have useful properties due to angular momentum conservation.
Conservation, in general, limits but does not dictate a system’s possible motion.
Conclusion:
A person holding a spinning bicycle wheel on a revolving chair is an example of angular momentum conservation. The individual then flips the bicycle wheel over, causing it to spin in the opposite direction.
The wheel has angular momentum in the upward direction at first. The angular momentum of the wheel reverses direction as the operator spins it over. Since the human-wheelchair system is an isolated system, total angular momentum must be conserved, and the person rotates in the opposite direction as the wheel. The angular momentum vector total is the same, and momentum is conserved.
