Answer
Verified
411k+ views
Hint : We know the definitions of angular velocity and moment of inertia of a body, and also know how they are connected to angular momentum. Again, as we know the angular momentum of a body is always conserved, so, from their relation, we can easily find out how moment of inertia affects angular velocity.
Complete Step-by-step solution:
The rotational equivalent of linear momentum is called angular momentum. Angular momentum can be defined mathematically as, $L=I\omega$ , where,
$L$ = angular momentum,
$I$ = moment of inertia, and,
$\omega$= angular velocity of the body or system.
Moment of inertia – The moment of inertia of a particle of mass $m$ about a line or axis is defined as, $I=m{{r}^{2}}$ , where $r$ is the distance from mass to the center or axis of motion. This quantity is used to express the tendency of a body to defend angular acceleration.
Angular velocity – The quantity angular velocity refers to the speed at which an object rotates or revolves with respect to another point. It means, the angular velocity of an object changes over time. It can be mathematically defined as, $\omega =\dfrac{d\theta }{dt}$ , where $\theta$ is the angle of rotation.
Now, we know the principle of conservation of angular momentum, which can be stated as – The total angular momentum of a particle remains constant in time if the total (external) torque acting on the particle is zero.
So, from the relation, $L=I\omega$ , if $L$ is conserved, then $I$ and $\omega$ must be inversely proportional to each other. It means, if the moment of inertia of a body increases, angular velocity of that body must decrease, and if moment of inertia of a body is decreased, angular velocity of that body must increase.
Note:
Angular momentum of a body has an unit kilogram meters squared per second ( $kg{{m}^{2}}/\sec$ ), moment of inertia of a body has an SI unit kilogram meters squared ( $kg{{m}^{2}}$ ), and angular velocity of a body has the SI unit as ${{s}^{-1}}$ .
Complete Step-by-step solution:
The rotational equivalent of linear momentum is called angular momentum. Angular momentum can be defined mathematically as, $L=I\omega$ , where,
$L$ = angular momentum,
$I$ = moment of inertia, and,
$\omega$= angular velocity of the body or system.
Moment of inertia – The moment of inertia of a particle of mass $m$ about a line or axis is defined as, $I=m{{r}^{2}}$ , where $r$ is the distance from mass to the center or axis of motion. This quantity is used to express the tendency of a body to defend angular acceleration.
Angular velocity – The quantity angular velocity refers to the speed at which an object rotates or revolves with respect to another point. It means, the angular velocity of an object changes over time. It can be mathematically defined as, $\omega =\dfrac{d\theta }{dt}$ , where $\theta$ is the angle of rotation.
Now, we know the principle of conservation of angular momentum, which can be stated as – The total angular momentum of a particle remains constant in time if the total (external) torque acting on the particle is zero.
So, from the relation, $L=I\omega$ , if $L$ is conserved, then $I$ and $\omega$ must be inversely proportional to each other. It means, if the moment of inertia of a body increases, angular velocity of that body must decrease, and if moment of inertia of a body is decreased, angular velocity of that body must increase.
Note:
Angular momentum of a body has an unit kilogram meters squared per second ( $kg{{m}^{2}}/\sec$ ), moment of inertia of a body has an SI unit kilogram meters squared ( $kg{{m}^{2}}$ ), and angular velocity of a body has the SI unit as ${{s}^{-1}}$ .
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE