Momentum [!copyright and waiver^SectionEdit.png!]
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Introduction to Linear Momentum
Although Newton is famous for the law F-ma, he actually stated his Second Law in terms of momentum and its change due to impressed forces or impulses (the time integral of the force). He defined Momentum as
The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. - DEFINITiON II, Principia (Motte and Cajori).
Momentum
Mass times velocity, or, equivalently, a quantity whose time rate of change is equal to the net force applied to a system.
Motivation for Concept
Forces are actions which cause a change in the velocity of an object, but a given application of force will have very different results when applied to objects of very different mass. Consider the force imparted by a baseball player swinging a bat. When delivered to a baseball, the change in velocity is dramatic. A 95 mph fasball might be completely reversed and exit the bat moving 110 mph in the other direction. When delivered to a car, however, the change in velocity is miniscule. A car moving 95 mph will not be slowed noticeably by the action of a bat. Thus, although the change in velocity of a system is proportional to the force applied, it is not equal to the force applied. To define a quantity whose rate of change is equal to the force applied, we must include both the mass and velocity of the system subject to the force.
Mathematical Definition
Momentum of a Point Particle
The momentum (p) of a point particle with mass m and velocity v is defined as:
Momentum of a System
For a system composed of N objects which are approximated as point particles with their position specified by the objects' centers of mass, the system momentum is defined as the vector sum of the momentum of the constituents:
This definition is completely equivalent to
where Msys is the total mass of the system and vCM is the velocity of the system's center of mass.
Momentum and Newton's Laws
Momentum and Newton's Second Law
One way of stating Newton's Second Law is that the rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:
Momentum and Newton's Third Law
By Newton's 3rd Law, internal forces cancel from the vector sum above, leaving only the contribution of external forces:
Momentum and Impulse
The integrated change in momentum can be found explicitly by using the net external impulse (Jext):
Conservation of Momentum
Conditions for True Conservation
In the absence of a net external force, the momentum of a system is constant:
This equation is normally broken up to explicitly show the system constituents and the vector components:
Approximate Conservation in Collisions
Because the change in momentum is proportional to the impulse, which involves a time integral, for instantaneous events:
For approximately instantaneous events such as collisions, it is often reasonable to approximate the external impulse as zero by considering a system composed of all the objects involved in the collision. The key to the utility of this assumption is that often during collisions the change in momentum of any individual system constituent being analyzed is dominated by the internal collision forces (the external forces make a negligible contribution to that constituent's change in momentum during the collision).
Before the collision occurs and after the collision is complete, the collision forces will usually drop to zero. Neglecting external impulse can only be justified during the collision. It is also completely incorrect to say that the momentum of each object is conserved. Only the system momentum is (approximately) conserved.
Newton emphasizes the linearity of momentum with both mass and velocity. He specifies our sum over different bodies by further stating that the motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the momentum is double; with twice the velocity also, it is quadruple.
Newton's Second Law is the fundamental law of change for momentum,
Frac
=\Sigma_j\vec
Where the sum goes over all the forces acting on any body in the system.
The mathematical relationship between force and momentum, or, for systems with constant mass, the relationship between force and acceleration.
Motivation for Concept
When you push, kick or use some other means to apply force to an object, its velocity will change. It is of value to be able to quantitatively define the strength of such a push, kick or other force by examining the effects of the force on the object which is the target. Such a quantitative understanding of force is the basis of the science of dynamics.
Statement of the Law
Newton's Statement
"A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed." (The Principia by I. Newton, translated by I.B. Cohen and A. Whitman.)
Modern Statement
The modern form of the Law which is perhaps most consistent with Newton's statement is the integral formulation:
It is more common to express the Law in a differential form:
and, since it is rare to consider a system with changing mass, this form is often reduced to:
Use of the Law in Problem Solving
Form of the Second Law for Multiple Impressed Forces
When more than one force is impressed, the change in momentum is proportional to the vector sum of the forces. Thus, Newton's 2nd Law is usually expanded to state:
"Writing" Newton's Second Law
When a physics teacher or student says they are "writing Newton's 2nd Law" for a system, the form used should be that of the previous subsection, but expressed as several equations separated by vector component:
Directions that clearly have no forces acting are sometimes ignored.
Free Body Diagrams
Free body diagrams are pictorial guides to the specific form of Newton's 2nd Law for a given system. Drawing an accurate free body diagram shows visually what forces should be included in the statement of the law, and also gives information about the sign of the vector components of these forces.
Technically, the concept of linear momentum applies only to collections of point objects. The momentum of a rigid body is simply the sum of the momentum of each of the atoms in the body, which turns out to be the body's mass times its center of mass velocity. The momentum of a system composed of many rigid bodies and point particles is then the sum of their individual momenta, which again can be expressed as the total mass of this system times the velocity of the system's center of mass. This is true, and a valuable concept, especially when the system is composed of disparate objects going in different directions - for example a system of two cars about to have a collision, or after they have just had a collision.
Introduction to the ModelDescription and Assumptions
Learning ObjectivesStudents will be assumed to understand this model who can:
Relevant DefinitionsS.I.M. Structure of the ModelCompatible SystemsThe system must be effectively composed of point particles, though rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid body when this model is used. Relevant InteractionsOnly external forces need be considered, since internal forces do not change the system's momentum. Laws of ChangeMathematical RepresentationDifferential FormIntegral FormDiagrammatic RepresentationsRelevant ExamplesExamples Involving Constant MomentumExamples Involving ImpulseExamples Involving 1-D CollisionsExamples Involving 2-D CollisionsExamples Involving Elastic CollisionsExamples Involving Totally Inelastic CollisionsExamples Involving Continuous Momentum FluxAll Examples Using this Model |
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