If you have played cricket, you would have experienced that catching a ball which is moving with a higher velocity is more difficult than catching a ball which is moving with lesser velocity. Similarly, it is easier to catch a table tennis ball as compared to a cricket ball when both are dropped from the same height. Thus, we see that moving bodies postess a physical quantity associated with their motion which determine how much force is required to bring them to rest . This quantity which depends on the mass and velocity of the moving body is called momentum and is defined as momentum (p) = mass (m) × velocity (v)
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Definition of momentum – The momentum of a system is conserved ( does not change) when there is no net external force.
Units of Momentum
SI unit of momentum is kg m s-1 or Newton. Cgs unit of momentum is g cm s-1
Law of conservation of momentum
The total momentum of a system of bodies remains constant unless there is a net external force acting on the system.
Angular momentum
A object moving in a straight line has linear momentum(\(\vec{P}\)) . When a object rotates about some axis, there is momentum associated with rotational motion called the angular momentum \(\vec{L}\) Just as net external force is required to change the linear momentum of an object a net external torque is required to change the angular momentum of an object.
S.I. unit of angular momentum \(\vec{L}\) is kg m2 s-1
Note: When net external torque is zero angular Momentum can be consverd
Type of angular momentum
The angular momentum is classified in following three types.
Angular momentum of a particle about some point
Let Assume that a particle Q of mass m is moving with linear momentum \(\vec{P}\) = m\(\vec{v}\). Its angular momentum L about point O is defined as:
\(\vec{L}\)= \(\vec{r}\) × \(\vec{P}\) = \(\vec{r}\) × (m\(\vec{v}\)) = m( \(\vec{r}\)×\(\vec{v}\) )
Here, \(\vec{r}\)is the radius vector of particle Q about O at that instant of time.
The magnitude of Angular momentum \(\vec{L}\) is L = mvr sinθ = mvr⊥
Angular Momentum of a rigid body rotating about a fixed axis
Suppose a particle P of mass m is going in a circle of radius r and at some instant the speed of the particle is v. For finding the angular momentum of the particle about the axis of rotation, the origin may be chosen anywhere on the axis.
We choose it at the centre of the circle. In this case \(\vec{r}\) and \(\vec{P}\)are perpendicular to each other and \(\vec{r}\)×\(\vec{P}\) is along the axis. Thus, component of \(\vec{r}\)×\(\vec{P}\) along the axis is mvr itself. The angular momentum of the whole rigid body about AB is the sum of components of all particles, i.e.,
L = Iω
Here, I is the moment of inertia of the rigid body about AB.
Conservation of Angular Momentum
The conservation of angular momentum is a fundamental principle in physics that states that the total angular momentum of a closed system remains constant if no external torques are acting on it. Angular momentum is a vector quantity that describes the rotational motion of an object or a system of objects. Angular momentum (L) is defined as the product of the moment of inertia (I) and the angular velocity (ω) of an object: L = Iω where: – L is the angular momentum, – I is the moment of inertia (a measure of an object’s resistance to changes in its rotational motion), and – ω is the angular velocity (the rate of change of the angle through which an object rotates in a given time). According to the conservation of angular momentum, if the net external torque acting on a system is zero, the total angular momentum of the system remains constant over time. Mathematically, this can be expressed as: L₁ + L₂ + L₃ + … = constant where L₁, L₂, L₃, etc., are the individual angular momenta of the objects within the system. This principle has various practical applications. For example, it explains why a spinning ice skater can increase or decrease their rotational speed by changing their body position. By extending their arms outward, they increase their moment of inertia, resulting in a decrease in their angular velocity to maintain the constant angular momentum. Conversely, when they bring their arms closer to their body, their moment of inertia decreases, causing an increase in their angular velocity. The conservation of angular momentum is also observed in many other phenomena, such as planetary motion, spinning tops, and the motion of galaxies. It is a fundamental concept in understanding rotational dynamics and plays a crucial role in explaining and predicting the behavior of rotating systems in physics.
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