Name: impulse and angular momentum in rigid bodies
Uploaded: 2025-05-22T09:06:17.000Z
Duration: 1 min 15 s
Channel: JoVE

Consider a rigid body undergoing a general planar motion, a combination of translational and rotational motion.
Newton&#39;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.

principe van impuls en moment

Principle of Impulse and Moment

Wanneer men een star lichaam beschouwt dat een vlakke beweging ondergaat, wat in wezen een mengsel is van translatie- en rotatiebeweging, geeft de toepassing van de tweede wet van Newton de formule voor de translatiebeweging van zo'n lichaam. Als deze vergelijking wordt vermenigvuldigd met een tijdsinterval, dt, en vervolgens wordt geïntegreerd over de integratiegrenzen, resulteert dit in een vergelijking die het principe van lineaire impuls belichaamt.

<img alt="Equation 1" src="/files/ftp_upload/15550/15550_PT_equation_001.svg" style="text-align: center; height: 50px;">
Hier vertegenwoordigt subscript G het massamiddelpunt van het object.

Het principe van lineaire impuls is een concept dat communiceert dat de verandering in het momentum van een object evenredig is aan de impuls die erop wordt uitgeoefend. Deze specifieke vergelijking kan worden weergegeven door het gebruik van drie rechthoekige componenten.

Daarentegen wordt de vergelijking voor rotatiebeweging uitgedrukt, als de mate van verandering in de tijd (tijdsafgeleide) van het product van het traagheidsmoment rond het massamiddelpunt van het object en zijn hoeksnelheid. In deze context blijft het traagheidsmoment constant. Als deze vergelijking wordt omgezet in een integrale vorm, resulteert dit in het principe van rotatie-impuls.

<img alt="Equation 2" src="/files/ftp_upload/15550/15550_PT_equation_002.svg" style="text-align: center; height: 50px;">
In dit principe is het product van het traagheidsmoment en de hoeksnelheid van het starre lichaam gelijk aan het impulsmoment. Deze relatie kan worden geïllustreerd aan de hand van drie rechthoekige componenten. Dit verhaal onderzoekt dus de ingewikkelde dynamiek van een star lichaam in vlakke beweging, waarbij de principes van lineaire en roterende impulsen worden benadrukt.

Principe van impuls en moment

When one considers a rigid body undergoing a plane motion, which is essentially a blend of translational and rotational movement, the application of ...

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3-Dimensional Kinetics of a Rigid Body

Driedimensionale kinetiek van een star lichaam

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impulse and angular momentum in rigid bodies

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion.

For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the body. The result can also be written in three rectangular components.

Rotational motion is treated in a similar way. The time rate of change of the product of the moment of inertia about the center of mass and the angular velocity gives the rotational equation. Here, the moment of inertia is constant. When this form is written as an integral, it becomes the principle of rotational impulse.

In the rotational case, the product of the moment of inertia and angular velocity is the angular momentum. This relationship can also be shown with three rectangular components. Together, these ideas describe the linear and rotational impulse behavior of a rigid body in plane motion.

Impulse and Angular Momentum in Rigid Bodies

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moments and product of inertia

Moment and Product of Inertia Basics

Moment of inertia and product of inertia describe how mass is distributed in a rigid body. They help show how the body responds about coordinate axes and planes. For a small mass element, the moment of inertia is found by multiplying the element's mass by the square of the shortest distance to one of the three coordinate axes. That same idea is extended to the whole body by integration. Adding the contributions of all the small mass elements gives the body's moment of inertia about that axis.

inertia tensor

Principal Axes and Moments of Inertia

The inertia tensor describes how mass is spread out in a rigid body and how that body resists rotation. It is written as a 3×3 matrix, and each entry represents a moment of inertia about a particular axis. This makes the tensor a compact way to study rotational behavior in three dimensions. The diagonal entries show the moments of inertia about the principal axes. These principal axes are the directions where the object has the least resistance to rotation. A smaller moment of inertia along...

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Angular momentum relative to point P is found by adding the motion of every mass element in a rigid body. The body has mass m and a center of mass at point G. It rotates in an inertial reference frame, and the angular momentum at P comes from the cross product of each position vector and its linear momentum. To build the full expression, the velocity of each mass element is written as two parts. One part is the translational velocity of the body. The other part is the relative velocity caused...

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Angular Momentum in Principal Axes

Angular momentum in a rigid body can be written using rectangular coordinates. The body’s angular momentum comes from the cross-product of each mass element’s position vector with the cross-product of the body’s angular velocity and that position vector. This gives a compact way to describe how a solid structure moves in rotation. To work with the equation more easily, the XYZ axes can be chosen as a new set of rectangular axes. These axes may be inclined at any angle to the reference frame.

principle of impulse and moment

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion. For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the...

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Rigid body kinetic energy in planar motion can be written using the body’s center of mass and its rotation. The solid object moves in general planar motion, and its center of mass is marked at point G. To find the kinetic energy of the i-th particle, the relative velocity definition is used with the position vector rA from point A to that particle. The kinetic energy of the whole body is built from the particles. First, a scalar product is taken. Then the expression is written in integral form.

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Rigid body motion is described with both translational motion and rotational motion about the center of mass, point G. The center of mass is the point where Newton's Second Law applies to translation. It gives the linear motion of the body. The same body also has rotational motion about point G. The combined moments about the center of mass equal the rate of change of angular momentum. A moment is the turning effect caused by a force, and an external force applied away from the center of mass...

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Euler’s equations of motion describe how a rigid body rotates when measured in a moving frame attached to the body. The body rotates with angular velocity ω in an inertial frame, and the attached frame rotates with angular velocity Ω. Together, these two frames help describe the motion of the rigid body around its center of mass. The total moment about the center of mass is found by adding two parts. One part is the rate of change of angular momentum about the center of mass in the rotating...

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Torque-free motion describes the movement of a rigid body when no external torques act on it. It can be seen in places with no outside forces or friction, such as outer space. Mars is one example of this kind of motion because it rotates in space without external torque. Mars is an axisymmetric object, which means it has an axis of symmetry. That symmetry defines the rotating z-axis. The rotating frame is set so that Mars&amp;#39;s center of mass is at the origin. This setup makes the moments...

key terms and concepts

Learn how linear impulse and rotational impulse describe rigid body motion in a plane, including center of mass, angular velocity, and angular momentum.

Impulse and Angular Momentum in Rigid Bodies | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Impulse and Angular Momentum in Rigid Bodies

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