To find a change in angular momentum, you determine the initial and final angular momentum values and calculate their difference.
Understanding the change in angular momentum is crucial in physics, particularly when dealing with rotating or orbiting systems. It follows a straightforward process involving calculating angular momentum at two different points in time or under different conditions and then finding the difference.
How to Find the Change in Angular Momentum (ΔL)
The change in angular momentum (ΔL) is simply the difference between the final angular momentum (Lf) and the initial angular momentum (Li). The fundamental method involves calculating these two values and subtracting the initial from the final.
Here are the steps to follow, incorporating the information from the provided reference:
Step 1: Determine the Type of Rotation
The first step is to understand the system you are analyzing.
- Reference Step 1: Determine if the rotating object is rotating about another object or rotating about an axis.
This distinction helps you determine the appropriate formula for calculating angular momentum.
- Rotation about an axis (like a spinning top or a figure skater) typically involves the object's moment of inertia ($I$) and its angular velocity ($\omega$), where angular momentum $L = I\omega$.
- Rotation about another object (like a planet orbiting a star or a ball on a string being swung around) involves the object's position relative to the axis of rotation ($\mathbf{r}$) and its linear momentum ($\mathbf{p} = m\mathbf{v}$), where angular momentum is the vector cross product $\mathbf{L} = \mathbf{r} \times \mathbf{p}$.
Step 2: Calculate the Initial Angular Momentum (Lᵢ)
Next, calculate the angular momentum of the system at the beginning of the time interval you are interested in.
- Reference Step 2: Calculate the initial angular momentum.
Use the appropriate formula based on the type of rotation determined in Step 1 and the initial conditions (initial angular velocity, initial position, initial velocity, etc.).
Step 3: Calculate the Final Angular Momentum (Lբ)
Calculate the angular momentum of the system at the end of the time interval or under the final conditions.
- Reference Step 3: Calculate the final angular momentum.
Again, use the appropriate formula based on the type of rotation, but this time using the final conditions (final angular velocity, final position, final velocity, etc.).
Step 4: Calculate the Change in Angular Momentum (ΔL)
Finally, calculate the difference between the final and initial angular momentum values.
- Reference Step 4: Calculate the change in angular momentum, $\Delta L = L_f - L_i$.
The result, ΔL, represents the change in angular momentum over the specified interval.
Summary of Steps
| Step | Action | Formula/Concept |
|---|---|---|
| 1 | Identify Rotation Type | About axis ($L=I\omega$) or about another object ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) |
| 2 | Calculate Initial Angular Momentum (Lᵢ) | Use relevant formula with initial conditions |
| 3 | Calculate Final Angular Momentum (Lբ) | Use relevant formula with final conditions |
| 4 | Calculate Change (ΔL) | $\Delta L = L_f - L_i$ |
Practical Insights
- A change in angular momentum is caused by an external torque acting on the system. Newton's second law for rotation states that the net external torque ($\tau_{net}$) equals the rate of change of angular momentum ($\frac{d\mathbf{L}}{dt}$).
- If the net external torque is zero, the angular momentum of the system is conserved (ΔL = 0). This is the principle of conservation of angular momentum, seen in examples like a figure skater spinning faster as they pull their arms in.
By following these steps, you can systematically determine the change in angular momentum for a given system.
