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Energy and Work
Energy - the ability of a body or system of bodies to perform work.
A body is given energy when a force does work on it.
But What is Work?
A force does work on a body (and changes its energy) when it causes a displacement.
If a force causes no displacement, it does no work.
Riddle Me This
If a man holds a 50 kg box at arms length for 2 hours as he stands still, how
much work does he do on the box?
Nada Zip
Zilch
NONEZERO
Counterintuitive Results There is no work done by a force
if it causes no displacement. Forces perpendicular to
displacement, such as the normal force, can do no work.
Likewise, centripetal forces never do work.
Calculating WorkWork is the dot product of
force and displacement.
Work is a scalar resulting from the interaction of two vectors.
Vector Multiplication
There are three ways to multiply vectors:
•Scalar Multiplication
•Dot Product
•Cross Product
Scalar Multiplication
•Magnitude of vector changes.•Direction of vector does not change.
amF
a = 10 m·s-1
F = 50 N
If m = 5 kg
Dot Product
BAW
Note that the dot
product of two vectors gives a scalar .
. and between angle theis BA
cosABBA
θ
A
B
Dot Product
Geometrically, the dot product is the projection of one vector on a second
vector multiplied by the magnitude of the second vector.
θ
A
B
cosA
Calculating WorkcosFssFW
dxxFW )(
θ
F
s
cosF
Which does more work?
θ
F1
F2
Two forces are acting on the box shown causing it to move across the floor. Which force does more work?
Vectors and Work
F
Vectors and Work
Fs
W = F • sW = F s cos 0o
W = F sMaximum positive work
Vectors and Work
F
Vectors and Work
s
W = F • sW = F s cos Only the component of force aligned with displacement does work. Work is less.
F
Vectors and Work
F
Vectors and Work
F s
W = F • sW = F s cos 180o
W = - F sMaximum negative work.
Gravity often does negative work.
mg
F
When the load goes up, gravity does negative work and the crane does positive work.
When the load goes down, gravity does positive work and the crane does negative work.
Positive, zero, or negative work?
A box is being moved with a velocity v by a force P (parallel to v) along a level floor. The normal force is FN, the frictional force is fk, and the weight of the box is mg.
Decide which forces do positive, zero, or negative work.
Positive, zero, or negative work?
v
mg
P
FN
fk
s
Units of Work
cosFssFW
J = N·m
J = kg·m2·s-2
Work and variable force
The area under the curve of a graph of force vs displacement gives the work done by the force.
F(x)
xxa xb
W = F(x) dxxa
xb
