Work and Energy

Scalar (Dot) Product When two vectors are multiplied together a scalar is the result:

Dot Product Unit vector has a magnitude of 1 and is in the x direction

Dot Product Unit vector has a magnitude of 1 and is in the y direction

Dot Product Unit vector has a magnitude of 1 and is in the y direction

Dot Product

9 Work: The _______done by a constant force acting on an object is equal to the product of the magnitudes of the displacement and the component of the force __________to that displacement 5.1 Work Done by a Constant Force Unit of work: Newton meter (N m) 1 N m = 1 ________(J)

Work Done by a Constant Force If there is a force but no displacement: no work is done.

Work Done by a Constant Force If the force is _______________to the displacement, work is done

Work Done by a Constant Force If the force is at an angle to the displacement, the ________________component of the force does the work

Work Done by a Constant Force If the force (or a component) is in the direction of motion, the work done is _____________. If the force (or a component) is opposite to the direction of motion, the work done is ______________.

Work Done by a Constant Force If there is more than one force acting on an object, it is useful to define the _________work: The total, or net, work is defined as the work done by all the forces acting on the object, or the scalar sum of all those quantities of _________________.

The Work–Energy Theorem: Kinetic Energy

The Work–Energy Theorem: Kinetic Energy We can use this relation to calculate the work done:

The Work–Energy Theorem: Kinetic Energy Kinetic energy is therefore defined: (Kinetic Energy) Unit: Joule (J) The net work on an object changes its kinetic energy.

Work Done by Varying Force

Work by a Varying Force We allow the size of the Δx’s to approach zero The work done by the varying force is: and

Work Done by a Varying Force In vector form:

Work Done by a Spring Force exerted by a spring: Where k is the spring constant in N/m and x is the displacement The neg sign indicates that the force is always opposite the displacement from equilibrium

Work done by a spring

Work Done by a Spring The work done by a spring:

Work done by an applied force on a spring

Potential Energy

From a previous slide

Conservative Forces Gravity and the Spring Force are conservative The work done by conservative forces:

Conservative Forces Two properties for a force to be conservative 1.The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle 2.The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one for which the beginning point and the endpoints are identical) Do not change the mechanical energy of a system

Conservative & Nonconservative Forces Some conservative forces: – – – Nonconservative Forces – – –

Conservative Forces We can define the potential energy function, U

Potential Energy Function Suppose we had this equation: The integral would be:

Potential Energy Function If we have this equation: We can do the reverse of what we did on the previous slide: The x component of a conservative force acting on an object within a system equals the negative derivative of the PE of the system w/ respect to x

Check of Previous Equation

Energy Diagrams & Equilibrium of a System   