16.6:
Principle of Impulse and Moment
Consider a rigid body undergoing a general planar motion, a combination of translational and rotational motion.
Newton's second law gives the equation of translational motion for a rigid body. Multiplying it with time interval dt and integrating it over the limits of the integration yields an equation for the principle of linear impulse.
The principle of linear impulse articulates that the change in momentum of an object is directly proportional to the impulse exerted on it. This equation can be represented using three rectangular components.
On the other hand, the equation for rotational motion can be written as the time derivative of the product of the moment of inertia about the center of mass of the object and its angular velocity.
Here, the moment of inertia is constant; transforming this equation into an integral form results in the principle of rotational impulse.
Here, the product of the moment of inertia and the angular velocity of the rigid body is equal to the angular momentum, which can be depicted using three rectangular components.
16.6:
Principle of Impulse and Moment
When one considers a rigid body undergoing a plane motion, which is essentially a blend of translational and rotational movement, the application of Newton's second law gives the formula for the translational movement of such a body. If this equation is multiplied by a time interval, dt, and then integrated over the limits of integration, it results in an equation that embodies the principle of linear impulse.
Here subscript G represents the center of mass of the object.
The principle of linear impulse is a concept that communicates that the alteration in an object's momentum is proportional to the impulse applied to it. This particular equation can be represented through the use of three rectangular components.
In contrast, the equation for rotational motion is expressed as the rate of change over time (time derivative) of the product of the moment of inertia about the center of mass of the object and its angular velocity. In this context, the moment of inertia remains constant. If this equation is transformed into an integral form, it results in the principle of rotational impulse.
In this principle, the product of the moment of inertia and the angular velocity of the rigid body equates to the angular momentum. This relationship can be illustrated using three rectangular components. Thus, this narrative explores the intricate dynamics of a rigid body in plane motion, highlighting the principles of linear and rotational impulse.