Angular momentum (L) is the rotational equivalent of linear momentum. It is a vector quantity that describes the rotational momentum of a rotating object. Mathematically, the angular momentum can be expressed as the product of an object’s moment of inertia with respect to the axis of rotation and angular velocity.
Angular momentum can be applied for circular motion, but it can also be applied for non-circular motion when the direction of motion perpendicular to the radius vector is studied.
The direction of angular momentum
The direction of angular momentum can be determined using the right-hand rule, where four fingers represent the direction of motion while the thumb represents the direction of angular velocity momentum.
Figure 1. The right-hand rule used for finding the direction of angular momentum. Source: Oğulcan Tezcan, StudySmarter.
If the direction of motion is anticlockwise, the angular momentum is positive.
If the direction of motion is clockwise, the angular momentum is negative.
Angular momentum formula
As shown in figure 2, a point particle is moving in a circular motion. A plane is created between the linear velocity v and position vector r. The magnitude of angular momentum for a rotating point particle is the product of the mass of the particle m, the position and velocity vectors, and the angle θ between them.
The angle θ created between the two vectors r and v can be in a range of 0 ≤ θ ≤ 180 degrees. Hence, the formula of angular momentum can also be written in terms of momentum, as shown below, where m is the mass and p is the linear momentum.
\[L = [kg \space m^2/s] = m \cdot u \cdot r \cdot \sin \theta = p \space r \space \sin \theta\]
Figure 2. Angular momentum.
This can be further modified to better fit the cases of circular motion by utilising angular velocity instead of linear velocity. As shown in figure 3, the angular momentum (L) of a point particle moving in a circular trajectory is perpendicular to the plane formed by the radius vector r and the linear velocity vector v.
Hence, by substituting angular velocity instead of linear velocity in the formula below, we get a more suitable equation, which can also be written in terms of the moment of inertia. Angular momentum is measured in kgm2/s.
The equation below shows that angular momentum has the same form as linear momentum. However, the moment of inertia I is the reciprocal of mass m, while angular velocity ω is the reciprocal of linear acceleration a in linear motion.
\[v = r \cdot \omega L [kg \space m^2/2] = m \cdot r \cdot v = m \cdot r \cdot (r \cdot \omega) = m \cdot r^2 \cdot \omega\]
Figure 3. Component vectors forming angular momentum.
Angular motion and Newton’s laws
Newton’s second law of linear motion states that the acceleration αt of an object moving linearly is proportional to the net force F acting on the body and has a magnitude that is inversely proportional to its mass, as described by the equation below.
\[ \sum F [Newtons] = m [kg] \cdot a_t [m/s^2] \qquad a_t [m/s^2] = \frac{\sum F}{m}\]
This can also be applied in terms of angular momentum, where an object is rotating.
Newton’s second law for angular motion states that the angular acceleration (α) of a rotating object is directly proportional to the sum of external torque (T) acting on the object’s axis of rotation.
On the other hand, angular acceleration is inversely proportional to the moment of inertia (I) with respect to the axis of rotation. This is shown in the equation below, where a is the rotational acceleration, while I is the moment of inertia, which is the reciprocal of mass in linear motion.
\[\sum T = I \cdot \alpha \qquad \alpha = \frac{\sum T}{I}\]
Newton’s law generalised for angular motion
Newton’s second law can be expressed in terms of linear momentum in cases of linear motion when mass is constant. Similarly, for angular motion, when the moment of inertia is constant, Newton’s law can also be expressed in terms of angular momentum.
The rate of change of angular momentum of a body with respect to some point in space is equal to the sum of external torque impacting the body with respect to that point, as shown below:
\[\Delta T = \frac{\Delta L}{\Delta t}\]
Here, torque is the product of the applied force multiplied by the perpendicular distance to the axis of rotation and the sine of the angle between them. Torque is measured in Newton metres.
A metal disc is rotating with an angular velocity of 18 rad/s. The disc has a moment of inertia of 0.05 kgm2. Calculate the angular momentum.
Using the formula for angular momentum and substituting the given variables for angular velocity and the moment of inertia, we get:
\(L = I \cdot \omega = 0.05 kg \space m^2 \cdot 18 rad/s = 0.9 kg \space m^2/s\)
A small ball weighing 0.3 kg is rotating around an axis located 0.2 m away at a rate of 5 rad/s. Determine the angular momentum of the ball.
The angular momentum is the product of the moment of inertia and angular velocity. This gives us:
\(I = m \cdot r^2 = 0.3 \cdot 0.2^2 = 0.012 kg \space m^2\)
\(I = m \cdot r^2 = 0.3 kg \cdot (0.2 m)^2 = 0.012 kg \space m^2\)
We substitute the given angular velocity to determine the angular momentum.
\(L = I \cdot \omega = 0.012 kg \space m^2 \cdot 5 s^{-1} = 0.06 kg \space m^2/s\)
Conservation of angular momentum
As Newton’s generalised law for angular motion states, the sum of external torque is equal to the rate of change of angular momentum.
However, if the sum of external forces acting on a body or system with respect to a point in space is zero, the conservation of angular momentum states that the total angular momentum of a body with respect to that point is conserved and remains constant. This can be expressed mathematically as follows:
\[\Delta M = \frac{\Delta L}{\Delta t} = 0 \Rightarrow \Delta L = 0 \qquad L_{initial} = L_{Final}\]
A disk with a moment of inertia of 0.02 kg⋅m2 is rotating without any external forces at the rate of 5 rad/s. Suddenly, a coin is dropped on the disk, causing its moment of inertia to increase to 0.025 kg⋅m2. Determine the angular velocity after the impact.
We begin by using the conservation of angular momentum.
\[\Delta M = \frac{\Delta L}{\Delta t} = 0 \Rightarrow \Delta L = 0 \qquad L_{initial} = L_{Final}\]
Then, we need to find the angular momentum before and after the event.
\(L_{before} = I_1 \cdot \omega_1 = 0.02 kg \space m^2 \cdot 5 s^{-1} = 0.1 kg \space m^2/s\)
By equating the initial and final angular momentum, we can determine the final angular velocity:
\(L_{before} = L_{after} 0.1 kg \space m^2/s = I_2 \cdot \omega_2\)
We rearrange to make ω2 the subject.
\(0.025 kg \space m^2 \cdot \omega_2 = 0.1 kg \space m^2/s \Rightarrow \omega_2 = \frac{0.1 kg \space m^2/s}{0.025 kg \space m^2} \Rightarrow \omega_2 = 4 rad/s\)
Angular Momentum - Key takeaways
Angular momentum is the reciprocal of linear momentum for angular motion.
Angular momentum has the same form as linear momentum, where the moment of inertia is the reciprocal of mass, and angular velocity is the reciprocal of linear acceleration.
Newton’s law of motion can also be expressed in terms of the moment of inertia and generalised for angular motion.
When no external forces act on a body, the angular momentum remains constant.
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