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Impulse and Momentum Chapter 7

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Expectations After chapter 7, students will: be able to define and calculate the impulse of a force. be able to define and calculate the momentum of a moving object. understand and apply the impulse-momentum theorem. understand the conditions in which the momentum of a system is conserved.

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Expectations After chapter 7, students will: distinguish between elastic and inelastic collisions. know what quantities are conserved in what sorts of collisions. be able to calculate the location and velocity of the center of mass of a system of objects.

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Impulse and Momentum In the last chapter, we defined a force-related quantity (work) and a motion-related quantity (kinetic energy), and showed how they were linked: This was the work-energy theorem.

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Impulse and Momentum Now, we define a new force-related quantity: the impulse of a force. The impulse is the product of the net force and the time over which it is exerted. It is a vector quantity, having the dimensions of force · time. SI units: N·s

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Impulse and Momentum The net force F, acting on an object of mass m, produces an acceleration: Substituting that into the definition of impulse: Solve for at:

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Impulse and Momentum Substitute into the kinematic equation: Multiply through by m:

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Impulse and Momentum Again, we have a relationship between a force- related quantity, I, and a motion-related quantity, mv. We define this quantity as linear momentum: SI units: kg·m/s Note that linear momentum is a vector quantity.

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Impulse and Momentum The above relationship is called the impulse- momentum theorem: the impulse of a net force acting on an object is equal to the resulting change in its linear momentum.

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Impulse and Momentum Bottom line: a net force changes an object’s state of motion. The work-energy theorem and the impulse-momentum theorem are two different descriptions of the same change. Which description is more useful to you depends on the situation: what things you know, and what things you want to find out.

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Systems of Objects A system of objects is any collection of objects. You define the system. You can say what objects are part of it, and what objects are outside of it.

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Isolated Systems A system of objects is called isolated if the net external force acting on the system is zero. An external force is a force exerted on the system (any object in it) by an object that is not a part of the system. A force could be external or internal, depending on how the system is defined.

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Conservation of Momentum The conservation of linear momentum principle tells us: If a system is isolated, its total linear momentum does not change. Why not?

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Conservation of Momentum The only force experienced by any part of an isolated system is an internal force – a force exerted on one part of the system by another. Newton’s third law tells us: If objects A and B are both part of the system, then the net force on the system as a whole is

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Conservation of Momentum If the net force is zero, its impulse is zero:... and if the impulse is zero, the change in (total) momentum is zero:... and the total linear momentum is conserved.

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Conservation of Momentum We said the change in total (i.e., system) momentum was zero. Notice that the linear momentum of any particular object within the system will, in general, change due to internal forces. But the sum of the individual momenta of the objects in the system, added as the vectors that they are, does not change – as long as the system is isolated.

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Collisions If we define a system of two or more objects, it is usually because those objects are busy colliding with each other. In every collision among the objects in an isolated system, the total linear momentum of the system is conserved. We established that earlier, with the linear momentum conservation principle.

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Elastic Collisions We now want to consider a special class of collisions: elastic collisions. In an elastic collision between objects in an isolated system, the total kinetic energy of the system is also conserved. Why should that be?

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Elastic Collisions The work-energy theorem tells us that the total or net work due to conservative forces does not change the total energy of the system. The gravitational potential energy of the system does not change unless the system experiences a net (external) gravitational force... in which case, it is no longer an isolated system.

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Elastic Collisions This means that, if only conservative forces do work within the system, its kinetic energy must be constant... since we already know its potential energy is constant. What conservative forces do work within the system? Spring forces (as long as deformations are elastic).

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Elastic Collisions If objects in the system are elastically deformed by the forces they exert on each other, the work done by those forces is stored as elastic potential energy. When the objects’ pre-collision shapes are restored, that elastic potential energy is re- converted into kinetic energy. So, the total kinetic energy of the system after the elastic collision is the same as before the collision.

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Collision Summary If the system is isolated (no net external force acts on the system), momentum is conserved: If only conservative forces do work in the collision, the collision is elastic, and kinetic energy is conserved:

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Collision Summary Collisions are not, in general, either perfectly elastic or perfectly inelastic. Those two conditions are idealizations that can give us insights into real collisions.

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Center of Mass A system of several objects has its total mass distributed among those objects. When we consider the conservation of a system’s total momentum, it will be useful to us to define a single location where the system’s total mass is effectively located: the center of mass.

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Center of Mass Consider a system of two objects, both located on the X axis. If m 1 = m 2 = m,

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Center of Mass Now, we let the objects in our system move (independently):

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Center of Mass The motion takes place in a time t: or, : velocity of the center of mass

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Center of Mass Notice that the center-of-mass velocity is the total momentum of the system, divided by its total mass. If the system is isolated, its total momentum is constant … and so is its total mass. So, the center-of-mass velocity of an isolated system is constant, regardless of intra-system collisions. … for isolated systems!

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