Difference Between Linear Momentum and Angular Momentum

Main Difference – Linear Momentum vs. Angular Momentum

Momentum is a property of moving objects that have mass. Often, we talk about two types of momenta: linear and angular. The main difference between linear momentum and angular momentum is that linear momentum is a property of an object which is in motion with respect to a reference point (i.e. any object changing its position with respect to the reference point) while angular momentum is a property of objects which are not only changing their position but also the direction of their position with respect to a reference point (i.e. they are not moving in a straight line).

What is Linear Momentum

Linear momentum of an object is the product of the object’s mass and velocity. Linear momentum is a vector quantity, and the direction of momentum is taken to be the direction of the object’s velocity. If the mass of the object is m and the velocity of the object is \vec{v}, then the linear momentum \vec{p} is given by:

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

Linear momentum is a conserved quantity: the total linear momentum of particles in a system is conserved if no external forces act on the system. If there is a resultant external force on the system, then the momentum changes, so that the rate of change of momentum is equal to the resultant external force:

\sum \vec{F}=\frac{\mathrm{d}\vec{p}}{\mathrm{d}t}

The SI units for measuring linear momentum is kg m s-1. We have discussed linear momentum at length in this article.

What is Angular Momentum

For an object with mass m moving at a velocity \vec{v}, the angular momentum \vec{L} with respect to a reference point is defined using the cross product as:

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

where \vec{r} is the position vector of the object that describes the object’s position with respect to the reference point. The units for measuring angular momentum is kg m2 s-1. Since angular momentum is defined in terms of a cross product, the direction of the angular momentum vector is taken to be in a direction perpendicular to both the particle’s position vector \vec{r} and its velocity vector \vec{v}.

Difference Between Linear Momentum and Angular Momentum - Angular_momentum

Defining angular momentum

Using the definition above, we can come up with an expression for calculating angular velocity of a rigid body which is rotating about an axis that is at right angles to the plane in which the particles are rotating. The rigid body is made of many particles, and the sum of angular momenta of all the particles gives the total angular momentum of the rigid body. Then, in terms of the masses and velocities of individual particles, we can write the total angular momentum as:

L=\sum_{i} r_im_iv_i

Difference Between Linear Momentum and Angular Momentum - Angular_momentum_of_a_rigid_body

Finding the angular momentum of a rigid body

Note that since the axis of rotation is perpendicular to the plane in which the particles are rotating, the cross product boils down to a simple multiplication. We can write the linear velocity v of the particles in terms of their angular velocities \left( \omega\right):

L=\sum_{i} {r_i}^2m_i{\omega}_i

Since the object is rigid, all particles rotate in unison. This means that the angular velocity for all particles is common. Then,

 L=\sum_{i}\left( {r_i}^2m_i\right){\omega}

The quantity L=\sum_{i}\left( {r_i}^2m_i\right) is the object’s  ,I. Then, we can write the angular momentum in the object as:

 L=I\omega

Like linear momentum, angular momentum is also a conserved quantity. The angular momentum of a system of particles is conserved if no external torques act on the system. If there is a resultant external torque, the angular momentum changes so that the resultant torque is equal to the rate of change of angular momentum of the object:

\sum \vec{\tau}=\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}

Difference Between Linear Momentum and Angular Momentum

Type of Motion

Linear momentum is a property of objects which are changing their position with respect to a reference point.

Angular momentum is a property of objects which are changing the angle of their position vector with respect to a reference point.

Conservation

Linear momentum of a system of particles is conserved so long as there is no resultant force on the system.

Angular momentum of a system of particles is conserved so long as there is no resultant torque on the system.

Rate of Change

The rate of change of linear momentum of a system of particles is equal to the resultant force acting on the system.

The rate of change of angular momentum of a system of particles is equal to the resultant torque acting on the system.

SI Units

Linear Momentum is measured in units of kg m2 s-1.

Angular momentum is measured in units of kg m2 s-1.

About the Author: Nipun


Related pages


adverb of reason definitionexamples of caesura in literaturearchaeology anthropology definitionaccommodation and assimilation examplesmetaphors and analogiesconfectionery definedifference between conductors and insulatorssomatic cells vs gametesdifferent types of informal lettersunicameral governmentwhat is the difference between passive transport and active transportcarpe diem poemshow to calculate cpi indexchemical formula for formaldehydewhat groan meanstwo examples of pure substancesrutherford gold foil experiment explanationjean piaget schema theoryhow to write a monologuedifferences between anaerobic and aerobic respirationflat white coffee vs lattesepsis versus septicemiastanza literary definitiondefine fate and destinyanother word for madamc3 and c4 plants definitionpteridopytesvitamin c in lime vs lemondifference between starch and glucosewhat is the difference between norepinephrine and epinephrinedefine upthrustcompound pendulum theorywhat is the difference between optical and laser mouseexamples of round characterswhat is the difference between a autobiography and a biographydifference between a dove and a pigeonwhat does smriti meancodons and anti codonstypes of bullet trainschief bridesmaid vs maid of honorray diagram for telescopedifference between archaea and eubacteriadefinition enunciationdefinition of command economiesdefine metallic bondinghomophone for stairsdifference between rolling friction and sliding frictionscanning electron microscope wikidifference between qualcomm and snapdragon processordefinition coenzymethe difference between tension and compressionconscious vs subconsciouscharacteristics of comedycommon collective nounsepithelium simplehow does a cation differ from an anionmeaning of predicate adjectivecollenchyma functionpredicative adjectivesliteral meaning and figurative meaningwhat is the difference between thrombosis and embolismdifference between byte and bitsense and antisense dna strandscalculate cpi examplemicrotubules microfilaments and intermediate filamentshallucinations vs delusionsmadam meaning in usadifference between calorimeter and spectrophotometerwhat is the difference between bigamy and polygamywhat is the difference between pre mrna and mrnaexamples of heteronymunpolarised lighthomophone homonym differencedifference between separation and divorce in indiathe difference between pollination and fertilizationldpe and hdpe differencefunction of radicle in seeddefine comedy in literaturederive an expression for centripetal accelerationdispersion forces definition