Download Lecture 12 - McMaster Physics and Astronomy

Torque
• Torque in 2 and 3 dimensions
Text Sections : 10.6
Physics 1D03 - Lecture 10
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Torque is what causes rotation
An unbalanced force applied
to an object causes it to move
(accelerate).
F
a
Even with no net force, the
object is not necessarily in
equilibrium. A net torque
will cause it to rotate.
-F
F
Physics 1D03 - Lecture 10
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A force directed toward the axis of rotation produces no torque.
You cannot shut a door by pushing directly toward the hinges (F1), or
pulling directly away from the hinges (F2).
hinge
F1
F2
Physics 1D03 - Lecture 10
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Divide the force vector into radial and tangential components.
Only the tangential component produces a torque about the
rotation axis.
Torque = (tangential force)  (distance to rotation axis)
  rFt  rF sin 
( is the angle between r and F )
r
Units: N•m
This is identical to the unit of energy
(joule), but is always called “N•m” when
it refers to torque.
Fr
Ft

F
Physics 1D03 - Lecture 10
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Another way to look at torque...
“effective lever arm”,
r sin 
The shortest distance from the pivot to the
line of action of the force is the “effective
lever arm”. If the force were applied at right
angles to this shorter distance, the same
torque would result.
“line of action” of F
r
Torque = (force)  (effective lever arm)

F
Physics 1D03 - Lecture 10
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Torque as a vector
Vector product or “cross product” of 2 vectors:
C  AB
C  A B sin   A B
B
B

A
Direction: C is perpendicular to A and B (out of the page, towards you).
Right-hand rule: Fingers parallel to A when straight, and along B as you
curl them; then the thumb gives the direction of C.
Not commutative!
( A  B)  (B  A)
Physics 1D03 - Lecture 10
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The cross-product is used to
define torque as a vector.
 
  r F


F
F
Φ
r
 rF sin ( )
The direction of  is along the axis about which the body
would tend to rotate. Right-hand rule: Thumb in
direction of , fingers indicate sense of rotation.
Physics 1D03 - Lecture 10
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Torque depends on:
• The magnitude of the force
• The direction of the force
• The location at which the force is applied
• The point which you assume as the “pivot” or axis
of rotation.
In two-dimensional problems, torque is either “clockwise” or
“counterclockwise” (we can take one sense as positive, and
the other as negative).
In 3-D, torque is a vector.
Physics 1D03 - Lecture 10
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Quiz
A weight W is placed at one end of a light plank. What
weight should be placed at the other end so that the
plank will balance on a pivot placed 1/3 of the way along
the plank?
?
W
a) 3W
b) 2W
c) W
d) ½ W
L/3
2L/3
e) 1/3 W
Physics 1D03 - Lecture 10
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Each force produces a torque, 
(“tau”), which tends to rotate the
plank about the pivot.
F

R
Torque is proportional to the force and to the distance from the pivot:
torque = (length of “lever”) x (force)
  rF
(when the force is perpendicular to the “lever” – more
complicated otherwise)
Physics 1D03 - Lecture 10
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Each force produces a
torque,  (“tau”), which
tends to rotate the plank
about the pivot. The
plank is in equilibrium
when the torques are
equal and opposite.
F1=W
F2=W/2
1

2
L/3
2L/3
Torque is proportional to the force and to the distance
from the pivot:
torque = (force) x (length of “lever”)
Physics 1D03 - Lecture 10
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Example
What is direction of rotation due to the 20N force?
What is direction of rotation due to the 25N force?
20 N
25 N
30 cm
40 cm
pivot
Physics 1D03 - Lecture 10
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Example
What is the torque due to the 20-N force? What is the total torque?
20 N
25 N
30 cm
40 cm
pivot
Physics 1D03 - Lecture 10
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Example
A force F=10N is applied to the end of a wrench (of
length L=0.4m) in order to loosen a nut on a bolt. The
force is applied at an angle of 30° to the wrench.
What is the torque applied to
the nut?
30°
F
L
Physics 1D03 - Lecture 10
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Example
A mass M is hanging from a string wrapped around a
massless wheel of radius R that is pivoted through its
center.
What is the torque applied to the wheel?
Physics 1D03 - Lecture 10
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Example
A mass M1 is hanging from a string wrapped around a
cylinder of radius R1 that is pivoted through its center,
and another mass M2, on a string wrapped in the
opposite direction, from a core section protruding from
the larger drum, at radius R2. They cylinders are
massless.
What is the torque applied to the cylinder?
Physics 1D03 - Lecture 10
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