Angular Impulse

Momentum

Before diving into angular impulse, we must discuss a related topic, momentum. Momentum is a vector quantity associated with objects in motion. It can be linear or angular depending on the motion of a system. However, for this article, we will focus on angular momentum.

The mathematical formula corresponding to this definition is $$L=I\omega$$ where \( \omega \) is angular velocity measures in \( \mathrm{\frac{rad}{s}} \) and \( I \) is inertia measured in \( \mathrm{kg\,m^2}. \) Angular momentum has SI units of \( \mathrm{kg\,\frac{m^2}{s}} \).

Note that this formula can only be used when the moment of inertia is constant. If the moment of inertia is not constant, we have to look at what is causing the rotational motion, which we know to be torque, the angular equivalent of force.

The equation for torque can be written in the same form as Newton's second law, \( F= ma, \) and is expressed as \( \tau=I\alpha, \) where \( I \) is the moment of inertia and \( \alpha \) is angular acceleration.

As angular momentum corresponds to rotational motion, motion associated with objects traveling in a circular path about a fixed axis, angular impulse also corresponds to this type of motion.

Angular Impulse Definition

Angular impulse, a vector quantity, describes how torque, the rotational analog of force, affects a system with respect to time.

Angular Impulse Formula

The mathematical formula corresponding to the definition of angular impulse is

$$\Delta \vec{J_{rot}}= \int_{t_o}^{t}\vec{\tau}(t)dt,$$

which can be simplified to

$$J=\tau\Delta{t},$$ when \( \tau \) doesn't vary with time.

Note \( \tau \) is torque measured in \( \mathrm{Nm} \) and \( t \) is time measured in \( \mathrm{s}. \)

Angular Impulse SI Unit

The SI unit for angular impulse is Newton-seconds which is abbreviated as \( \mathrm{Nm\,s}. \)

Impulse-Momentum Theorem

Impulse and momentum are related by the impulse-momentum theorem. This theorem states that the impulse applied to an object is equal to the object's change in momentum. For rotational motion, this relationship is described by the equation \( J=\Delta{L}. \) However, we must remember that although related by this theorem, each represents a quantity described by distinct definitions and formulas. Angular impulse refers to the change in an object's momentum while angular momentum refers to the product of an object's moment of inertia and angular velocity.

Momentum, Impulse, and Newton's Second Law of Motion

Newton's second law of motion can be derived from the impulse-momentum relationship. To complete this derivation, we must use the equations corresponding to the impulse-momentum theorem in conjunction with the individual formulas of momentum and impulse. Let us derive Newton's second law for rotational motion starting with the equation \( J=\Delta{L} \) and rewriting it as \( \tau\Delta{t}=I\Delta{\omega} \).

$$\begin{align}J&=\Delta{L}\\\tau\Delta{t}&=\Delta{L}\\\tau\Delta{t}&=I\Delta{\omega}\\\tau&=\frac{I\Delta{\omega}}{\Delta{t}}\\\end{align}$$

Be sure to recognize that \( \frac{\Delta{\omega}}{\Delta{t}} \) is the definition of angular acceleration so the equation can be written as

$$\begin{align}\tau&= I{\alpha}\\\end{align},$$ which we know to be Newton's second law in angular form. As a result of this relationship, we can define torque in terms of momentum. Torque is the rate at which the angular momentum of an object changes with respect to time.

Importance of Impulse-Momentum Theorem

The impulse-momentum theorem is important because we can make connections between how forces act on an object and how this affects the motion of an object. Some real-life applications of this theorem include the determination of safe stopping and following distances for vehicles and the designing of fire equipment such as large nets and giant inflatable mattresses used by firefighters. However, in the real world, as forces are not always constant, things are more complex without this theorem. For example, non-constant forces caused by humans and engines. As all people and engines are built differently, several factors need to be considered as they directly impact the overall force. Consequently, determining how multiple factors would impact overall force would be fairly difficult if not for this theorem.

Failures of the Impulse-Momentum Theorem

The impulse-momentum theorem does not fail due to the fundamental concept of conservation of momentum. Conservation of momentum states that momentum is conserved as it cannot be created nor destroyed but only changed as a result of forces. For example, let us consider a collision between two objects. As stated by conservation of momentum, we know that the forces acting on the objects will be equal and opposite in direction during a specific time interval. It then follows that the impulse of the objects will also be equal in magnitude and opposite in direction. Therefore, by applying the theorem, it can be concluded that the objects also must undergo equal and opposite momentum changes,

Angular Impulse and Momentum Examples

To solve impulse and momentum problems, the equations for angular impulse and angular momentum can be applied to different problems. As we have defined angular impulse and angular momentum, let us work through some examples to gain a better understanding of impulse and momentum. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to provide a visual aid.

Examples

Let's apply our new knowledge to some examples.

Let's try a slightly more difficult example.