How To Find Angular Momentum With Mass: Detailed Explanations

This article discusses about how to find angular momentum with mass. Angular momentum with mass is simply the angular momentum of the entire body which has a definite mass.

This article discusses about angular momentum as well as linear momentum of bodies having definite amount of mass. The mass of the object affects the value of angular momentum drastically as angular momentum is directly proportional to the mass.

What is angular momentum?

The angular momentum is defined as the product of three quantities- mass of the object, velocity of the object and the radius of rotation of the object.

When we talk about angular momentum we need to understand that the motion of the object is rotational and not linear. The rotation may take about a point or an axis. Although axis will provide us with a better information on frame of reference. We shall study more about angular momentum in this article.

What is linear momentum?

Like angular momentum, linear momentum is a product of two quantities- Mass of the object and the velocity of the object. For an object to have linear momentum the object has to have a non zero value of velocity.

The linear momentum of an object implies that the object has a non zero value of linear velocity. Without linear velocity, the value of linear momentum will be zero. We shall study about different examples of both angular momentum and linear momentum in further sections of this article.

how to find angular momentum with mass
Image: Transfer of momentum taking place in the game of Billiards

Image credits: No-w-ay in collaboration with H. Caps, BillardCC BY-SA 4.0

How to calculate angular momentum per unit mass

The formula of angular momentum has three quantities- mass of the object, velocity of the object and the radius of the rotation.

For a unit mass, the formula of the angular velocity becomes, L= v.r where L represents the angular momentum , v represents the value of velocity of the object and r is the radius of the rotation.

Angular momentum and mass relation

As discussed in the above section, angular momentum and mass are directly related to each other. Mass directly affects the value of angular momentum.

The relationship between angular momentum and mass is that the angular momentum of an object is directly proportional to the mass of object. The formula of the angular momentum is, L= mvr where L is the angular momentum of the object, m is the mass of the object, v is the velocity of the object and r is the radius of the object.

What happens to angular momentum when mass doubles

The angular momentum has a direct proportionality relationship with mass. So as mass increases the angular momentum also increases and likewise the angular momentum decreases as the mass decreases.

So when the mass doubles, the value of angular momentum becomes twice as the formula becomes 2mvr from mvr that means it becomes 2L from L. Hence we conclude that the value of angular momentum is doubled if the value of mass is also doubled.

Angular momentum examples

We can see angular momentum examples in our daily lives. The examples of of angular momentum are given in the list below-

  • Ballerina – A ballerina rotates on her toes. The angular speed of the ballerina depends on the movement of her hands. The hands when spread apart reduces the angular speed. This is because the radius of rotation increases. When the ballerina keeps her hands closer to the body, her angular speed increases. Hence this is a perfect example to describe the law of conservation of momentum.
  • Ice skater – An ice skater’s movements are similar to a ballerina when he performs stunts. The ice skater rotates on his skates such that the angular speed varies with his hand movements. The ice skater moves rotates with a higher angular velocity when his hands are closer to the body, this is because the radius of the rotation decreases. Likewise the angular speed decreases when his hands are spread wide apart. This happens due to the fact that radius of rotation increases. This is also a perfect example of conservation of angular momentum.
  • Basketball player spinning ball on his finger – We all must have seen basket ball players spinning basketballs on their fingers. When the ball rotates the ball attains some amount of angular momentum hence this is also an example of angular momentum which we see in our daily lives.
  • Rotation of Earth – The rotation of Earth’s surface allows the surface to have a certain amount of angular momentum. The value of angular momentum can be calculated using the formulae discussed above.
  • Fan blade – A fan’s blade has angular momentum once the fan is switched ON. It is a notable fact that the fans with smaller blades rotate faster and fans with larger blades rotate slower. This is because the smaller blades have lower radius of rotation and hence greater angular velocity, the blades having more length will have lesser angular velocity . This is also an example of conservation of angular momentum.
  • Satellite revolving around the Earth – Satellite revolving around the Earth is an example of angular momentum as the satellite will gain some angular momentum only then it can orbit around the Earth. The satellite stays in orbit by using equilibrium of two forces- gravitational pull from Earth and centrifugal force acting on the satellite.
  • Spinning disc– When we spin a disc, it starts rotating about a common axis of rotation. The disc attains angular momentum as it starts rotating.
  • Toys– Some toys have rotating elements inside it. These rotating elements have angular momentum due to which they rotate. In toys also the angular momentum is conserved.
  • Rotary pumps– In rotary pumps the rotating parts inside the pump have an angular momentum associated with it.
  • Cart wheel – The cart wheel when rotates attains angular velocity. As a result of which it attains angular momentum also.
  • Giant wheel– A giant wheel is nothing but an amusement park ride in which people sit inside cabins. These cabins move up in a circular trajectory. This way the giant wheel attains angular velocity. The speed of rotation is very slow but the radius of rotation is very large. So the angular momentum is also very large this way.

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