Download Torque & Rotation

It doesn’t seem stable why doesn’t it fall?
Center of Mass
Balanced Rock
Arches National Park
Moab, Utah
When a ball is spun in the air it spins around its center
& so does a block of wood
That is to say the axis of rotation is through the center of the object
A hammer
behaves a little
differently
An object naturally spins around its:
center of mass
which is very similar to (we will use these interchangeably)
center of gravity
Center of Gravity or Center of Mass
This is a location of the average position
of mass in an object
CM
If an object is symmetrical and
uniform (made of the same material throughout)
the center of mass is just the center
CM
If an object has more mass at one end,
the center of mass if shifted that way
CM
Lead
Plastic
Where is the center of mass?
Where is the center of mass?
Where would the CM of a large pot be?
But you can’t just say that there is the same
amount of mass to the left and to the right
Where
At what
do point
you think
would
this
there
set of
be masses
the same
connected
mass to
bythe
a string
left & would
right (assume
rotate about
the string
if swung
is massless)
in the air
Actual CM
1 kg
Equal mass L/R & U/D
2 kg
Geometric center shown at line
Multiply the mass times its distance from the
center of mass
Actual CM
1 kg
1m
2 m * 1 kg
1m
1m
2 kg
1 m * 2 kg
A massive sun and 3 small planets.
Where is the center of mass?
Sun
CM
The solar system does NOT rotate around
the center of the sun.
Sun
The sun actually wobbles a bit because…
Sun
Does the center of mass of our solar system
change or is it constant?
Sun
Compare the center of mass after the
explosion?
If there are no outside forces, the center of
mass is the same based on conservation of
momentum
What if the fragments are different sizes?
Why does a washing machine vibrate during the
spin cycle sometimes but not others?
Where is the center of mass
If a puck is pushed across a frictionless table
It moves in a straight line at a constant speed
A motion graph for this is…
If a wrench is slid across a frictionless table
The wrench below spins about is CM
And its CM moves in a straight line at a constant speed
If a baseball is thrown into the air it follows a
parabolic path
If a bat is tossed it looks a little more chaotic but….
But really just 2 things happen
1.) the object rotates around its CM
2.) the CM follows a parabolic path
Complex motion like the hammer can be simplified
1.) The CM moves
(translates) in a straight
line or parabola
2.) The object rotates
around the CM
APPLET
If ball is thrown into the air, it follows a parabolic
path (sorry can’t control the speed any better)
The hammer does 2 things when tossed
1.) rotates around the CM
2.) the CM follows the typical path
Now at the same time.
A hammer thrown into the air
The center of mass / center of gravity
is special for several reasons.
FOR EXAMPLE:
Gravity pulls down on an entire meter stick (each atom)
Fg
BUT this would make calculations a bit tiresome
FOR EXAMPLE:
Fg = mg
BUT you can PRETEND like the Force of gravity
is only pulling down on the CENTER OF MASS
FN = mg
Fg = mg
When a fulcrum is placed under the CM,
It balances!!!
(the net force is zero)
This is an easy way to find the center of mass
of an object
JUST BALANCE IT ON YOUR FINGER
If a weight is hung at one end, what happens to the CM
A Force causes acceleration
A Torque causes angular acceleration (rotation)
Torque
Force
Anytime you rotate an object you apply a torque
When you open door, it rotates. Where is
the pivot point / axis of rotation?
To open a heavy door where would you
push on the door to make it easiest?
A
B
C
The distance from the pivot point to the
location of the force is called the
LEVER ARM (distance)
A
B
C
The longer the lever arm the greater the torque
This force has no lever arm and produces no
torque or rotation
The direction of the force matters too, how well will
these two forces work for opening the door?
B
A
To Rotate an object the force must be applied perpendicular
to the lever arm (the door in this case)
B
A
A is perpendicular to the lever
TORQUE
t=Fd
Force perpendicular
to lever arm (N)
lever arm length (m)
NOT distance traveled
Where should you push the wrench in order to
loosen a stubborn bolt?
t=Fd
Torque Force
MORE Torque
Force
A wrench is all about TORQUE
You create the most torque by holding the
wrench at the end and by exerting a large
force.
If you can’t loosen the bolt with all your
strength, what do you do?
Force
Force
Get a longer wrench!!
lever arm
Even with the same force, twice the
lever arm gives twice the TORQUE
The greater the force,the greater the torque
t=Fd
Torque
Force
Force
MORE Torque
The more perpendicular the force the
greater the torque
t=Fd
Torque
Force
only the blue
part counts
MORE Torque
Force
Any force which points through the axis
of rotation creates no torque
Force
t=Fd
Torque = 0
Torque requirement on your tires lug nuts is 190 Nm.
If you have a wrench which is .25 m, how hard do you
have to push?
.25 m
A force of 70 N is exerted as below on a .25 m wrench.
What torque is generated?
70 N
.25 m
35o
Consider the two wrenches below, compare torques
If the applied force is the same
F
F
The lever arm distance is really still the same
F
F
Another way of looking at torque, this is easier
in some situtations.
1.) Draw a line through the force
2.) Bring a line perpendicular to the previous line
through the axis of rotation
3.) The lever arm is the distance that line
to the center of rotation
lever arm distance
4.) Torque = Total force x lever arm
1.) Draw a line through the force
2.) The lever arm is the distance that line
to the center of rotation
3.) Torque = Total force x lever arm
Lever arm distance (d)
F
1.) Draw a line through the force
2.) The lever arm is the distance that line
to the center of rotation
3.) Torque = Total force x lever arm
F
Lever arm distance (d)
A force of 100 N is exerted as below on a rope attached
to a .25 m wrench. What torque is generated?
100 N
.2 m
F
Using the 2nd method to find the torque
A force of 70 N is exerted as below on a .25 m wrench.
What torque is generated?
70 N
.25 m
35o
35o
Torque Feeler DEMO
TORQUE is also a vector.
There are two ways of looking at finding the direction of torque.
1.) Look at the angular velocity the torque
would give the object. Use the right hand
rule
Where does the torque vector point?
F
A mass on a string exerts a torque on a pulley,
what is the direction of the torque vector?
20.0 N
4.0 m from fulcrum
30 N
2.0 m from fulcrum
What is the net torque on the see saw
(ignoring the mass of the seesaw itself)?
4.0 m from fulcrum
20 N
What is the net torque on the see saw
(ignoring the mass of the seesaw itself)?
2.0 m from fulcrum
30 N
What is the net torque on the wheel and axle below?
.25 m
.10 m
40 N
20 N
DEMONSTRATION: Will the object rotate
.10 m
500 g
.25 m
500 g
Does a thicker wrench give you
more torque?
Does a thicker screwdriver?
TORQUE FEELER DEMO
Torque WS 1
What Force(s) are acting on a rolling ball?
Is the force of gravity causing a torque?
Fg
What else is acting on the ball
FN
Fg
Will the normal force cause a torque?
What else is acting on the ball
Friction is the only force
causing a torque
FF
FN
Fg
When a football spins in the air where is
the axis of rotation?
Through the center of mass!
When a force is applied through the CM.
Will it accelerate? YES
Will it rotate?
Is there torque?
What about here-Will it accelerate? YES
Will it rotaterotate? Is there torque YES
Will it accelerate? YES
Will it rotate?
Balanced Torques
Two 400 N children balance on a seesaw
The same force on each side, right
400 N
400 N
1m
1m
What if one child scoots in?
Forces are still equal on both sides
400 N
400 N
1m
0.5 m
All forces would cancel (adding the fulcrum)
BUT THE TORQUES DO NOT
400 N
400 N
1m
t = 400 Nm (CCW)
0.5 m
t = 200 Nm (CW)
Net t = 200 Nm (CCW)
If an object is balanced there can be
NO NET TORQUE
in other “words”
Stcw = Stccw
Can a big kid and a little kid balance on a seesaw?
100 N
400 N
Can a big kid and a little kid balance on a seesaw?
400 N
100 N
?
1m
What would happen if the 100
N person leaned forward?
A 250 N child sits 1.0 meter away from the
pivot point. Where should a 325 N adult sit
for the see saw to balance?
325 N
?
250 N
1.0 m
How much force needs to be placed at the 9 m
mark to balance the board?
9
?N
8
7
6
5
100 N
4
3
2
1
0
500 N
A 150 kg beam which 8.0 m long and is
supported by a fulcrum 2.5 m from one end.
How much force must be applied to one of
the ends to balance the object?
A 100 kg beam which 10.0 m long and is
supported by a rope 4.0 m from one end. An
additional force of 2,000 N is applied 2 meters
from the same end pulling down. Where should
a 800 N downward force be place to cancel?
800 N
2,000 N
where to place
Torque problems
Balanced Torque Lab
In order for an object to be perfectly still
or static,
2 Conditions MUST BE MET
#1 All FORCES must cancel
#2 All TORQUES must cancel
Will this object accelerate or rotate?
Consider the CM its center of rotation
100 N
100 N
Will this object accelerate or rotate?
100 N
100 N
Will this object accelerate or rotate?
100 N
100 N
100 N
Will this object accelerate or rotate?
100 N
200 N
100 N
Consider an object in space with only 2 forces on it.
Will the object accelerate or rotate?
100 N
100 N
Will this object accelerate or rotate?
100 N
50 N
Will this object accelerate or rotate?
100 N
50 N
Will this object accelerate or rotate?
100 N
50 N
Will this object accelerate or rotate?
100 N
150 N
50 N
Statics- Getting things not to move
If an object is completely stationary it is said to be static
It must not ROTATE OR ACCELERATE
Neither of these are good for a building!!!
Rotation and Acceleration is bad!!
(although some flex is necessary)
Many factors need to be considered
when designing a structure such as:
Wind / Earthquake resistance
Moving loads (like semi-trucks)
Strength of materials
But the first is just supporting its own weight
and anticipated loads
TWO criteria must be met
1st
all forces cancel
S FX =0
lefts = rights
S FY =0
ups = downs
But just having the FORCES cancel,
isn’t enough.
Consider the bridge below, a 5,000 N car on top of a 90,000 N bridge
is supported by 1 column. Forces balance. But UH OH
95,000 N
5,000 N
90,000 N
TWO criteria must be met
2nd all Torques cancel
S t =0
or
tcw = tccw
In order for the paint stick to be “static”
Forces and Torques must cancel
In order for the paint stick to be “static”
Forces and Torques must cancel
A
B
D
C
A+B+C+D=0
C & D are negative
In order for the paint stick to be “static”
Forces and Torques must cancel
dA
A
B
D
C
dD
dC
Still keeping C
& D negative
AdA + CdC + DdD = 0
A massless meter stick is supported by 2 spring scales.
A 1.0 kg mass is hung at the 25 cm mark. What is the force
exerted by each scale if the meterstick is static?
1st - sum of all forces = 0
UPS = DOWNS
FA
A
FB
25 cm
C
100 cm
B
9.8 N
FA + FB = 9.8 N
2nd - sum of all torques = 0
What should be do about center of rotation for lever arm
Should we use A, B, C, or D?
FA
A
FB
25 cm
C
9.8 N
D
B
TRICK #1You can use ANY POINT as the center of
rotation. Because a static object will
not rotate about ANY POINT.
BUT BE CONSISTENT!!!!!!!!!!!!!!!!!!
Pick a convenient location- such as one with
a unkown force on it. This way it has no
lever arm and therefore no torque.
This simplifies your equation.
2nd - sum of all torques = 0
CW = CCW
Lets use point A.
FA
FB
B
C
A
9.8 N
tC = tB
9.8 N .25 m = 1.0 m FB
DO the same thing using point C
UPS = DOWNS & LEFTS = RIGHTS
CW = CCW
FA
FB
25 cm
C
A
9.8 N
100 cm
B
A 65 kg person stands on a 6.0 m long
plank (ignore it weight). Where should
they stand so that the tension in the left
cable is 180 N.
A 25 kg cantilevered pole is supported by
2 posts. Find the force exerted by each on the
pole.
5m
A
B
8m
C
16 m
D
TRICK #2You may not know the direction of a force.
Just pick a direction. If you answer ends up
negative, it just means it goes opposite the
way you have drawn.
5m
A
FA
B
FB
7m
C
14 m
D
Fg = 25 kg * 9.8 m/s2
5m
A
FA
B
FB
7m
C
Fg = 245 N
15 m
D
a 7 kg beam 3 m long is held by a
hinge and cable. What are the forces
at pts A & B. The tension in the cable?
36o
A
3.0 m
B
a 7 kg beam supporting a 11 kg mass
36o
A
B
.5 m
2.5 m
11 kg
A 5 m ladder with a mass of 12 kg is
supported at the floor and frictionless wall.
What are the forces on the ladder?
5m
4m
3m
Problems:
Honors Physics p 267-268
handout prob
optional
19, 23, 25, 26
542 N, 409 N, -59 N
A 25 kg board (8 m long)
leans against a wall.
Indicate all forces involved.
(245, 71, 71)
60o