Download Lecture 10

Physics 218
Lecture 10
Dr. David Toback
Physics 218, Lecture X
1
Overview: Chapters 6 & 7
Combine Chapter 6 & 7 into four
lectures
Today we’ll cover Work:
• Intuitive understanding
• The math and multiple ways to
calculate work
Next time:
• How much energy does it take to
accomplish a task?
Physics 218, Lecture X
2
Physics 218, Lecture X
3
Why are we learning this stuff?
This is Fundamental to Engineering
• How much work can a machine
do? (today)
• How much energy does it take to
accomplish a task? (next time)
Physics 218, Lecture X
4
Work
• The word “Work” means something
specific in Physics (Kinda like Force)
• The amount of Work we do is the
amount of Forcing we do over some
distance
• Example: If we are accelerating a car
for 1 mile, then there is a force and a
distance  We do Work
Physics 218, Lecture X
5
Calculating the work
• Work is done only if the force
(or some component of it) is in
the same (or opposite) direction
as the displacement
• Work is the force done
Parallel to the displacement
Physics 218, Lecture X
6
Work for Constant Forces
The Math: Work can be complicated.
Start with a simple case.
For constant forces, the work is:
.
W=F d
…(more on this later)
Physics 218, Lecture X
7
1 Dimension Example
Pull a box with a constant force of
30N for 50m where the force and the
displacement are in the same
direction
How much work is done on the box?
W = F.d = 30N . 50m= 1500 N . m
= 1500 Joules
Physics 218, Lecture X
8
What if the Force and
the Displacement
aren’t in the same
direction?
Physics 218, Lecture X
9
2 Dim: Force Parallel to Displacement
W = F||d = F.d = Fdcosq where q is the angle
between the net Force and the net displacement.
You can think of this as the force component in
the direction of the displacement.
Force
Force
Rotate
Displacement
F|| = Fcosq
Physics 218, Lecture X
Displacement
10
Work done and Work experienced
• Something subtle: The amount of
work YOU do on a body may not be
the same as the work done ON a
body
• Only the NET force on the object is
used in the total work calculation
• Add up all the work done on an
object to find the total work done!
Physics 218, Lecture X
11
Examples
• Holding a bag of groceries in place
– Is it heavy?
– Will you get tired holding it?
– Are you doing “Work?”
• Moving a bag of groceries with
constant speed across a room
– Is it heavy?
– Will you get tired doing it?
– Are you doing “Work?”
• Lifting a bag of groceries a height h
with constant speed?
– Work by you?
– Work on the bag? Physics 218, Lecture X
12
Groceries: With the math
• Holding a bag of groceries
– W=F.d = Fdcosq =(0)*(0)*cosq = 0
• Moving a bag of groceries with constant speed
across a room
– Force exerted by you = mg, Net Force on bag = 0
– Work on bag= F.d = Fdcosq =0*dcosq =0
– Work exerted by you =Fdcosq =mgd*cos(900)=0
• Lifting a bag of groceries a height h with constant
speed?
– Work on bag = Fd*cosq = (0)*h*(00) = 0
– Work by you =Fdcosq =(mg)hcos(00)=mgh
Physics 218, Lecture X
13
Work in Two Dimensions
You pull a crate of mass M a distance X along a
horizontal floor with a constant force. Your pull has
magnitude FP, and acts at an angle of Q. The floor is
rough and has coefficient of friction m. Determine:
• The work done by each force
• The net work on the crate
Q
X
Physics 218, Lecture X
14
What if the Force is
changing direction?
What if the Force is
changing magnitude?
Physics 218, Lecture X
15
What if the force or direction isn’t constant?
I exert a force over a distance for awhile, then exert
a different force over a different distance (or
direction) for awhile. Do this a number of times.
How much work did I do?
Need to
add up all
the little
pieces of
work!
Physics 218, Lecture X
16
Find the work: Calculus
To find the total work, we must sum up all the little
pieces of work (i.e., F.d). If the force is continually
changing, then we have to take smaller and smaller
lengths to add. In the limit, this sum becomes an
integral.
b
 
F

d
x

a
Fancy sum
notationIntegral
Physics 218, Lecture X
17
Use an Integral for a Constant Force
Assume a constant Force, F, doing
work in the same direction,
starting at x=0 and continuing for a
distance d. What is the work?
W 
d
0
 
d
x d
F  dx  Fdx  Fx|x 0  Fd  F 0  Fd
0
Region of integration
W=Fd
Physics 218, Lecture X
18
Non-Constant Force: Springs
• Springs are a good example of
the types of problems we
come back to over and over
again!
• Hooke’s Law


F  kx
Some constant
Displacement
• Force is NOT CONSTANT
over a distance
Physics 218, Lecture X
19
Work done to stretch a Spring
How much
work do you
do to stretch
a spring, at
constant
velocity, from
x=0 to x=D?
D
Physics 218, Lecture X
20
This Week
• Next Lecture: More on Work
and Energy
• Finish the reading for Chapter 7
• Recitation on Chapter 5, with
HW5 due Monday
• Get caught up on your
homework
Physics 218, Lecture X
21
Physics 218, Lecture X
22
Examples
• While you are lifting up a bottle with mass m, the
bottle moves a distance d with constant velocity.
As you lift it:
–
–
–
–
What is the force you exert?
What is the work done by you?
What is the work done by gravity?
What is the net work?
• You push a box with Force F on a rough floor
with coefficient of friction m for a distance d, and
the box moves with constant velocity. As it moves:
– What is the work done by you?
– What is the work done by friction?
– What is the net work?
Physics 218, Lecture X
23
Exam 1 Results
• Overall:
• Mean=50/75 (after bonus) or ~66%
• This was a hard exam… We’ll probably
curve it.
• Preliminary curve… will change
• >85% => A
• >75% =>B
• Between 40 and 52 out of 75 => C
• <40/75 in the D or F range
• Remember: Exam 1 only worth 75 points
Physics 218, Lecture X
24
Does the Earth do work on the Moon?
Physics 218, Lecture X
25
Simple Case
Start with our spherical cow:
–Constant Forces in a single
direction
• Work is the force done Parallel to the
displacement
• Work is done only if the force (or
some component of it) is in the same
(or opposite) direction as the
displacement
Physics 218, Lecture X
26
Hiker
A hiker carries a backpack
of mass M with constant
speed up a hill of angle Q
and height h.
Determine:
• The work done by the
hiker
• The work done by gravity
• The work on the backpack
Physics 218, Lecture X
27
Simple Example with Unit Vectors
A woman pulls a
box of mass M
with Force FP in
the Q direction
for a distance d.
Ignore friction
Find the work
using unit vectors
Physics 218, Lecture X
28