Angular Impulse

Angular impulse is a concept in rotational dynamics that is analogous to linear impulse in linear dynamics. It is a measure of the change in angular momentum of a system over a certain time period due to the application of a torque.

Understanding Angular Impulse

Angular impulse, denoted by ( \vec{J} ), is defined as the integral of torque ( \vec{\tau} ) with respect to time ( t ):

[ \vec{J} = \int \vec{\tau} \, dt ]

This equation tells us that the angular impulse is equal to the area under the torque-time graph.

Relation to Angular Momentum

Angular impulse is directly related to the change in angular momentum ( \Delta \vec{L} ) of a system:

[ \vec{J} = \Delta \vec{L} = \vec{L}{final} - \vec{L}{initial} ]

Where:

• ( \vec{L}_{final} ) is the final angular momentum
• ( \vec{L}_{initial} ) is the initial angular momentum

Formula for Angular Impulse

If the torque is constant over the time interval ( \Delta t ), the angular impulse can be simplified to:

[ \vec{J} = \vec{\tau} \Delta t ]

Important Points and Differences

Here is a table summarizing the key points and differences between angular impulse and other related concepts:

Concept	Definition	Formula	Units	Relation to Other Concepts
Angular Impulse	Change in angular momentum due to an applied torque	( \vec{J} = \int \vec{\tau} \, dt )	N·m·s or kg·m²/s	( \vec{J} = \Delta \vec{L} )
Angular Momentum	Product of moment of inertia and angular velocity	( \vec{L} = I \vec{\omega} )	kg·m²/s	( \vec{L} = \vec{r} \times \vec{p} )
Torque	Rotational equivalent of force	( \vec{\tau} = \vec{r} \times \vec{F} )	N·m	( \vec{\tau} = I \vec{\alpha} )
Moment of Inertia	Resistance to change in rotational motion	( I = \sum m_i r_i^2 )	kg·m²	Used in ( \vec{L} ) and ( \vec{\tau} ) formulas

Examples

Example 1: Constant Torque

A disk with a moment of inertia ( I = 2 \, \text{kg·m}^2 ) is subjected to a constant torque of ( 10 \, \text{N·m} ) for ( 5 ) seconds. Calculate the angular impulse and the change in angular momentum.

Solution:

Using the formula for constant torque:

[ \vec{J} = \vec{\tau} \Delta t = 10 \, \text{N·m} \times 5 \, \text{s} = 50 \, \text{N·m·s} ]

Since ( \vec{J} = \Delta \vec{L} ), the change in angular momentum is also ( 50 \, \text{kg·m}^2/\text{s} ).

Example 2: Variable Torque

A torque that varies with time is applied to a wheel. The torque function is given by ( \tau(t) = 5t \, \text{N·m} ), where ( t ) is in seconds. Calculate the angular impulse over ( 3 ) seconds.

Solution:

Using the integral definition of angular impulse:

[ \vec{J} = \int_0^3 5t \, dt = \left[ \frac{5}{2}t^2 \right]_0^3 = \frac{5}{2}(3)^2 = \frac{45}{2} \, \text{N·m·s} = 22.5 \, \text{N·m·s} ]

The angular impulse over the 3 seconds is ( 22.5 \, \text{N·m·s} ).

Example 3: Impulse-Momentum Theorem

A flywheel with a moment of inertia of ( 0.5 \, \text{kg·m}^2 ) is initially at rest. A torque of ( 20 \, \text{N·m} ) is applied for ( 2 ) seconds. What is the final angular velocity of the flywheel?

Solution:

First, calculate the angular impulse:

[ \vec{J} = \vec{\tau} \Delta t = 20 \, \text{N·m} \times 2 \, \text{s} = 40 \, \text{N·m·s} ]

Since the initial angular momentum is zero (initially at rest), the final angular momentum is equal to the angular impulse:

[ \vec{L}_{final} = \vec{J} = 40 \, \text{kg·m}^2/\text{s} ]

Using the relation ( \vec{L} = I \vec{\omega} ):

[ \vec{\omega}{final} = \frac{\vec{L}{final}}{I} = \frac{40 \, \text{kg·m}^2/\text{s}}{0.5 \, \text{kg·m}^2} = 80 \, \text{rad/s} ]

The final angular velocity of the flywheel is ( 80 \, \text{rad/s} ).

These examples illustrate how angular impulse is used to determine the change in angular momentum of a system and how it can be applied to solve problems in rotational dynamics.