16.8
Equation of Motion for a Rigid Body
The movement of a rigid object can be understood through the equations that explain both translational and rotational motion about the center of mass…
The motion of a rigid body can be described using equations for translational motion and rotational motion about the center of mass.
Newton's Second Law gives the equation of motion for translational motion for a center of mass, G, of the body.
The summation of all the moments created about point G is equal to the rate of change of angular momentum of the body.
If an external force is applied at point A, other than point G, it creates a moment that causes the body to rotate.
The angular momentum of point A can be expressed as a vector product of its relative position and relative velocity with respect to point G. The time derivatives of angular momentum give the moment of point A.
Summing over all points within the rigid body, the total moment of the system about point G is calculated.
Using the relative acceleration definition and the distributive law for vectors gives the equation for the total moment about point G due to an external force.
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Q1: How does Newton's Second Law apply to rigid body motion?
Newton's Second Law governs the translational motion of a rigid body's center of mass, point G. The sum of all external forces equals the mass times the acceleration of the center of mass. This equation describes how the body moves through space without accounting for rotation about its center.
Q2: What happens when an external force is applied away from the center of mass?
When an external force is applied at a point other than the center of mass, it creates a moment that causes the body to rotate. This moment is calculated as the vector product of the relative position and the force applied at that point. The resulting rotation occurs about the center of mass.
Q3: How is angular momentum related to the moment applied to a rigid body?
The time derivative of angular momentum equals the total moment applied to the rigid body about its center of mass. By summing the moments from all external forces acting on the body, you obtain the rate of change of angular momentum. This relationship connects rotational dynamics to angular momentum about an arbitrary axis.
Q4: What is the vector product used for in rigid body motion equations?
The vector product combines the relative position and relative velocity of a point with respect to the center of mass to express angular momentum. This mathematical operation produces a vector perpendicular to both input vectors, capturing the rotational characteristics of the body at that point.
Q5: How do you calculate the total moment of a rigid body system?
The total moment about the center of mass is calculated by summing the moments from all points within the rigid body. Using relative acceleration definitions and vector distributive laws, you combine individual moments to determine the system's overall rotational response to applied forces.
Q6: Why is the center of mass significant in rigid body equations of motion?
The center of mass, point G, is the reference point for both translational and rotational equations of motion. Translational motion follows Newton's Second Law at this point, while all moments and angular momentum are calculated relative to it, simplifying the analysis of complex rigid body dynamics.
Q7: How do translational and rotational motion equations work together?
Translational motion describes how the center of mass moves under applied forces, while rotational motion describes how the body spins about that center. Together, these coupled equations fully characterize rigid body motion, with angular momentum and principle axes of inertia determining rotational behavior.