Work and Energy
Scalar (Dot) Product
• When two vectors are multiplied together a
scalar is the result:
Dot Product
ˆ
i
Unit vector iˆhas a
magnitude of 1 and is
in the x direction
Dot Product
ĵ
Unit vector ĵ has a
magnitude of 1 and is
in the y direction
Dot Product
k̂
Unit vector k̂ has a
magnitude of 1 and is
in the y direction
Dot Product
Dot Product
Dot Product
5.1 Work Done by a Constant
Force
• 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
Unit of work: Newton• meter (N • m)
1 N • m = 1 ________(J)
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5.1 Work Done by a Constant
Force
• If there is a force but no displacement: no
work is done.
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5.1 Work Done by a Constant
Force
• If the force is _______________to the
displacement, work is done
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5.1 Work Done by a Constant
Force
• If the force is at an angle to the
displacement, the ________________component
of the force does the work
W  F||d
 F cos d
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5.1 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 ______________.
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5.1 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 _________________.
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5.3 The Work–Energy Theorem: Kinetic
Energy
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5.3 The Work–Energy Theorem: Kinetic
Energy
We can use this
relation to calculate
the work done:
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5.3 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.
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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
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
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