Tru Physics

Momentum

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Tru Physics
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Introduction

Momentum is a fundamental concept in physics. It is a vector quantity and is expressed classically as the product of an objects mass and velocity.

Definition of Momentum

The momentum \vec{p} of an object can be calculated using the formula:

\vec{p} = m\vec{v}

where m is the mass of the object and \vec{v} is its velocity.

Conservation of Momentum

One of the key principles involving momentum is the law of conservation of momentum. In a closed system (i.e., a system with no external forces), the total momentum before and after an event (like a collision or explosion) remains the same.

\vec{P}_{\text{initial}}^{\text{total}} = \vec{P}_{\text{final}}^{\text{total}}

Impulse

Impulse is the change in momentum of an object when a force is applied over a time interval. It can be calculated as the product of the average force and the time interval over which it is applied.

\vec{J} = \Delta \vec{p} = \vec{F}_{\text{avg}} \Delta t

Momentum in Special Relativity

In the theory of special relativity, momentum is redefined to be:

\vec{p} = \gamma m \vec{v}

where \gamma \equiv \dfrac{1}{\sqrt{1 - \frac{v^2}{c^2}}} is the Lorentz factor, v is the velocity of the object, and c is the speed of light. This modification to the definition of momentum ensures that momentum is still conserved in high-speed (near the speed of light) interactions. As speeds much less than c, \gamma is negligible.

Angular Momentum

Angular momentum is a measure of the quantity of rotation of a system. For a particle, it can be defined as the cross product of the particle’s position vector \vec{r} and its momentum \vec{p}:

\vec{L} = \vec{r} \times \vec{p}

Like linear momentum, angular momentum is also conserved in a closed system.

Understanding the concept of momentum and its conservation laws is crucial for solving problems in mechanics and is a foundation for more advanced topics in physics.

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