Conservation of Angular Momentum: Law, Calculation, and Applications

Conservation of angular momentum states that the total angular momentum of an isolated system remains constant.

• The quantity of motion contained in a body is known as momentum.
• It is given by the product of the mass and velocity of the body.
• When the momentum of the body is in a straight line then it is referred to as linear momentum.
• Angular momentum is a rotating analog of linear momentum.
• Angular momentum is a vector quantity and its SI unit is kg m²/s.

According to the law of conservation angular momentum,

“Angular momentum of a rotating body is constant if there is no external torque acting on the object”

i.e. if external torque, τ = 0, then L = constant

Key Terms: Angular momentum, Law of conservation of angular momentum, Linear momentum, Torque, Velocity, Rotational motion, Angular velocity

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Angular Momentum

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The rotating analog of linear momentum is called angular momentum. 

• It is also known as the moment of momentum or rotational momentum. 
• It is a property of a rotating body determined by the product of the rotating moment of inertia of the object and its angular velocity. 
• It is a vector quantity, which means that the direction is taken into account in addition to the magnitude.
• It is an important physical quantity since it is a conserved quantity. 
• In a closed system, the total angular momentum remains constant.
• It is denoted by \(\vec{L}\)
• The SI unit of angular momentum is kg m²/s.
• The dimensional formula of angular momentum is [M L² T^-1].

The formula of angular momentum for a point object is given by

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

Where

• L is the angular momentum
• r is the radius of the circular path
• p is the linear momentum of the body

The formula of angular momentum for an extended object is given by

\(\vec{L} = \vec{I} \times \vec{\omega}\)

Where

• I is the moment of inertia of the rotating object
• ω is the angular velocity

Angular Momentum

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Relationship Between Torque and Angular Momentum

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The angular momentum of an object having linear p rotating about a point in a circular path of radius r is given by

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

Differentiating both sides with respect to time t, we get

\(\frac{d}{dt} \vec{L} = \frac{d}{dt} (\vec{r} \times \vec{p})\)

⇒ \(\frac{d}{dt} \vec{L} = \vec{r} \times \frac{d}{dt} \vec{P} + \vec{P}\frac{d}{dt} \vec{r}\)

But \(\frac{d}{dt} \vec{r} = \vec{v} \)

And \(\vec{P} = m \vec{v}\)

Where

• m is the mass of the rotating object
• v is the linear velocity of the object

On substituting the values, we get

\(\frac{d}{dt} \vec{L} = \vec{r} \times \frac{d}{dt} \vec{P} + (m\vec{v} \times \vec{v})\)

Since (\(m \vec{v} \times \vec{v}\))=0, therefore

\(\frac{d}{dt} \vec{L} =\vec{r} \times \frac{d}{dt} \vec{p}\)

Also, \(\frac{d}{dt} \vec{P} = \vec{F}\)

⇒ \(\frac{d}{dt} \vec{L} = \vec{r} \times \vec{F}\)

But \(\vec{r} \times \vec{F} = \tau\), the torque acting on the rotating object

Therefore

\(\frac{d}{dt} \vec{L} = \vec{\tau}\)

Hence the time rate of change of angular momentum of a particle is equal to the torque acting on it.

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Law of Conservation of Angular Momentum

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Angular momentum, like other aspects of physics, is subject to the laws of conservation, which define that certain properties of a physical system remain constant even while that system evolves over time. 

• In other words, unless acted upon by an external force, that quality remains constant. 
• Energy, electric charge, particle physics, linear momentum, and angular momentum are all governed by conservation laws.
• The law of conservation, as applied to angular momentum, states that the momentum of a rotating object does not change until some form of external torque is applied. 
• Torque, in this case, can refer to any external force acting on an object to cause it to rotate or spin.

According to the law of conservation of angular momentum,

“If the external torque on a system is zero, the total angular momentum of the system is conserved i.e. remains constant.”

The relation between angular momentum and torque is given by

\(\vec{\tau} = \frac{d}{dt} \vec{L}\)

If the external torque on the system is zero i.e. =0, then

\(\frac{d}{dt} \vec{L}=0\)

 i.e. change in linear momentum is zero

\(\vec{L}\) = constant

Where L is the angular momentum of the body.

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Conservation of Angular Momentum Calculation

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The angular momentum of a system is conserved as long as there is no net external torque working on the system. Because of the law of conservation of angular momentum, the earth has been revolving on its axis since the formation of the solar system.

There are two ways to calculate the angular momentum of an object

• If the object is a point object, then angular momentum is given by the cross product of the position vector and linear momentum i.e.

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

• If the object is an extended object, then angular momentum is given by the cross product of the moment of inertia of the rotating object and its angular velocity i.e.

\(\vec{L} = \vec{I} \times \vec{\omega}\)

In both cases, if no external torque is acting on the body then the initial angular momentum of the body will be equal to the final angular momentum i.e. angular momentum will remain conserved if no external torque acting on the object.

Angular Momentum Formula

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Applications of Conservation of Angular Momentum

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The following are examples of the conservation of angular momentum

• Based on the conservation of momentum, an ice skater performs the feasts.
• Suppose an ice skater is rotating with her arms and legs stretched outwards. When she folds her arms and brings the stretched legs close to each other, her moment of inertia decreases, and hence her angular velocity increases.
• The angular velocity of the revolution of a planet around the sun in an elliptical orbit increases, when the planet comes closer to the sun and vice versa. This is because when the planet comes closer to the sun, its moment of inertia about the sun decreases, therefore its angular velocity increases.
• The conservation of angular momentum is also used in the design of roller coasters and other amusement park rides to ensure that riders are not thrown from the ride.
• When a diver tucks their body into a ball before entering the water, they decrease their moment of inertia, which allows them to spin faster.

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Things to Remember

• The momentum of a body is given by the product of its mass and velocity.
• Angular momentum is a property of a rotating body determined by the product of the rotating moment of inertia of the object and its angular velocity i.e.

L = Iω

• Angular momentum is a scalar quantity and its SI unit is kg m²/s.
• The dimensional formula of angular momentum is [M L² T^-1].
• According to the law of conservation of angular momentum, the angular momentum of a body will remain constant if no external torque acts on the body i.e. L₁ = L₂ or L = constant.
• If I₁ and I₂ are the initial and final moment of inertia, and ω₁ and ω₂ are their initial and final angular velocities, if no external torque is applied, then 

I₁ω₁ = I₂ω₂

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