Causes of Motion
The influences which cause changes in the motion of objects are forces and torques. The effects of forces on objects are described by Newton's Laws. A force may be defined as any influence which tends to change the motion of an object. The relationship between force, mass, and acceleration is given by Newton's Second Law:
Newton's First Law states that an object will continue at rest or in motion in a straight line at constant velocity unless acted upon by an external force. Newton's Third Law states that all forces in nature occur in pairs of forces which are equal in magnitude and opposite in direction.
Newton’s Second Law for an Extended Object
The utility of Newton's second law for the description of the motion of extended objects is the key to its general practical usefulness. The motion of any real object may involve rotations as well as linear motion, but the motion of the center of mass of the object can be described by an application of Newton's second law in the following form:
If you know the net force exerted upon an object, you can predict the motion of its center of mass, even though it may be executing rotation or other internal motions.
The standard visualization examples for this application are:
- A ballistic trajectory of a rotating object - the center of mass of an extended object that is rotating will follow the same trajectory that a point mass would follow after it is launched.
- A set of masses connected by springs will follow a path such that its center of mass moves along the same path that a point mass of the same total mass would follow under the influence of the same net force.
- Combining the features of the two examples above, a pair of masses connected by a spring could be launched into a ballistic trajectory with both rotation and oscillation with respect to the center of mass. Even with this complicated internal motion, the center of mass will follow the same simple ballistic trajectory that a point mass would follow with the same launch velocity.
- If an artillery shell in a ballistic trajectory explodes in mid-flight, then in the absence of air friction, the center of mass of the fragments would hit at the same point that the intact shell was targeted.
Center of Mass
The terms "center of mass" and "center of gravity" are used synonymously in a uniform gravity field to represent the unique point in an object or system which can be used to describe the system's response to external forces and torques. The concept of the center of mass is that of an average of the masses factored by their distances from a reference point. In one plane, that is like the balancing of a seesaw about a pivot point with respect to the torques produced.
If you are making measurements from the center of mass point for a two-mass system then the center of mass condition can be expressed as
where r1 and r2 locate the masses. The center of mass lies on the line connecting the two masses.
Reduced Mass
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The relative motion of two objects that are acted upon by a central force can be described by Newton's 2nd Law as if they were a single mass with a value called the "reduced mass". |
From Newton's 3rd Law : |
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The relative acceleration of the two masses is |
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Since a2 is negative, this can be rewritten in terms of the magnitudes of the quantities:
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