Torque and Rotational Inertia in Rigid Bodies
L-10(M-9) torque and rotational inertia
•
We consider the rotation of
rigid bodies
. A rigid
body is an extended object in which the mass is
distributed spatially.
•
Where should a force be applied to make it
rotate (spin)? The same force applied at
different
locations produces
different
results.
1
AXLE
TORQUE – Greek letter tau
•
To make an object rotate, a force must be
applied in the right place.
•
the combination of force and point of application
is called
TORQUE
•
The
lever arm L
is the distance from the axis of
rotation to the point where the force is applied
•
If the line of action of F passes through the axis
of rotation,
it produces no torque
.
Force, F
lever arm: L
Axis
2
•
force F in Newtons, N
•
lever arm L in meters, m
•
Torque
in units of N m
Torque:
(Greek tau)
Torque = force (F) x lever arm (L)
F L
3
Torque example
F
L
What is the torque on a bolt
applied with a wrench that
has a lever arm: L= 20 cm
with a force: F = 30 N?
Solution:
F
L
= 30 N
(1/5) m
= 6 N m
For the same force, you get more torque
with a bigger wrench
the job is easier!
4
Torque wrench
5
•
A torque wrench is a wrench that applies
a
calibrated torque
to a bolt.
•
It prevents a bolt from being over-tightened
and possibly breaking.
Homer attempts to straighten out
the leaning tower of Pisa
fulcrum
lever
6
Net Force = 0 , Net Torque
≠ 0
10 N
10 N
•
> The net force = 0, since the forces are applied in
opposite directions so it will not accelerate.
•
> However, together these forces will make the rod
rotate in the clockwise direction.
7
Net torque = 0, net force
≠ 0
The rod will accelerate upward under these
two forces, but will not rotate.
8
Balancing torques
10 N
20 N
1 m
0.5 m
Left torque = 10 N x 1 m = 10 n m
Right torque = 20 N x 0.5 m = 10 N m
9
Equilibrium
•
To ensure that an
object does not
accelerate or rotate
two conditions must
be met:
–
net force = 0
–
net torque = 0
•
this results in the
practical 4-1 “ladder
rule”
10
When is an object stable?
•
If you can tip it over a
bit and it doesn’t fall
•
The object may
wobble a bit but it
eventually stops and
settles down to its
upright position.
A thinner object is
easier to topple
An object that is thicker
at its base is more stable
11
Why do tall objects tend to fall over
•
Every object has a special point called the
center of gravity
(
CG
). The CG is usually
in the center of the object.
•
if the center of gravity is supported, the
object will not fall over
.
•
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12
Condition for stability
If the CG is above
the edge of the
table, the object
will not fall off.
13
Why makes an object tip over?
•
For the wide object, the dashed line extending from the CG
down is to the left of the point of contact; the torque due to
the weight tends to rotate the object counterclockwise
•
For the narrow object, the dashed line extending from the
CG down is to the right of the point of contact, the torque
due to the weight tends to rotate the object clockwise.
14
STABLE
UNSTABLE
D
D
CG
CG
Stable structures
Structures are
wider at their
base to lower their
center of gravity
15
If the center of gravity
is supported, the blocks
do not fall over
Playing with blocks
CG
16
Coin Stack
17
As more stuff is loaded into a semi, its center
of gravity moves upward, making it
susceptible to tipping over in high winds.
High Profile Vehicles
wind
18
Rotational Inertia
(moment of inertia) symbol I
•
A rigid body is characterized by a parameter
called its rotational inertia
•
The rotational inertia of a RB depends on how its
mass is distributed relative to the axis of rotation
•
The rotational inertia of a RB is the parameter
that is analogous to inertia (mass) for a non-
extended object
•
For a RB, the rotational inertia determines how
much torque is needed to produce a certain
amount of rotational acceleration (spin).
19
rotational inertia examples
Rods of equal mass m and length L
axis through center
axis through end
20
The rod with the axis through the end
requires more torque to get it rotating.
How fast does it spin?
•
For spinning or rotational motion, the
rotational inertia of an object plays the
same role as ordinary mass for simple
motion
•
For a given amount of torque applied to an
object, its rotational inertia determines its
rotational acceleration
the smaller the
rotational inertia, the bigger the rotational
acceleration
21
Big rotational
inertia
Small rotational
inertia
Same torque, different rotational inertia
s
p
i
n
s
s
l
o
w
s
p
i
n
s
f
a
s
t
22
L
=
R
F
=
m
g
F
L
=
m
g
R
In the realm of rigid bodies, the application of force at specific points is crucial for inducing rotation. Torque, denoted by the Greek letter tau, is the product of the force and the lever arm length, determining the rotational effect. By grasping the concept of torque and its applications, one can comprehend how forces create rotational motion in objects. Examples, such as torque wrenches and bolt tightening scenarios, illustrate the relevance and practicality of torque in everyday contexts.
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Presentation Transcript
L-10(M-9) torque and rotational inertia We consider the rotation of rigid bodies. A rigid body is an extended object in which the mass is distributed spatially. Where should a force be applied to make it rotate (spin)? The same force applied at different locations produces different results. AXLE 1
TORQUE Greek letter tau To make an object rotate, a force must be applied in the right place. the combination of force and point of application is called TORQUE The lever arm L is the distance from the axis of rotation to the point where the force is applied If the line of action of F passes through the axis of rotation, it produces no torque. lever arm: L Axis Force, F 2
Torque: (Greek tau) Torque = force (F) x lever arm (L) = F L force F in Newtons, N lever arm L in meters, m Torque in units of N m 3
Torque example What is the torque on a bolt applied with a wrench that has a lever arm: L= 20 cm with a force: F = 30 N? F Solution: = F L = 30 N (1/5) m = 6 N m L For the same force, you get more torque with a bigger wrench the job is easier! 4
Torque wrench A torque wrench is a wrench that applies a calibrated torque to a bolt. It prevents a bolt from being over-tightened and possibly breaking. 5
Homer attempts to straighten out the leaning tower of Pisa lever fulcrum 6
Net Force = 0 , Net Torque 0 10 N 10 N > The net force = 0, since the forces are applied in opposite directions so it will not accelerate. > However, together these forces will make the rod rotate in the clockwise direction. 7
Net torque = 0, net force 0 The rod will accelerate upward under these two forces, but will not rotate. 8
Balancing torques 20 N 10 N 0.5 m 1 m Left torque = 10 N x 1 m = 10 n m Right torque = 20 N x 0.5 m = 10 N m 9
Equilibrium To ensure that an object does not accelerate or rotate two conditions must be met: net force = 0 net torque = 0 this results in the practical 4-1 ladder rule 10
When is an object stable? If you can tip it over a bit and it doesn t fall The object may wobble a bit but it eventually stops and settles down to its upright position. A thinner object is easier to topple An object that is thicker at its base is more stable 11
Why do tall objects tend to fall over Every object has a special point called the center of gravity (CG). The CG is usually in the center of the object. if the center of gravity is supported, the object will not fall over. The lower the CG the more stable an object is. stable not easy to knock over! 12
Condition for stability If the CG is above the edge of the table, the object will not fall off. CG 13
Why makes an object tip over? CG CG STABLE UNSTABLE D D For the wide object, the dashed line extending from the CG down is to the left of the point of contact; the torque due to the weight tends to rotate the object counterclockwise For the narrow object, the dashed line extending from the CG down is to the right of the point of contact, the torque due to the weight tends to rotate the object clockwise. 14
Stable structures Structures are wider at their base to lower their center of gravity 15
Playing with blocks CG If the center of gravity is supported, the blocks do not fall over 16
Coin Stack 17
High Profile Vehicles wind As more stuff is loaded into a semi, its center of gravity moves upward, making it susceptible to tipping over in high winds. 18
Rotational Inertia (moment of inertia) symbol I A rigid body is characterized by a parameter called its rotational inertia The rotational inertia of a RB depends on how its mass is distributed relative to the axis of rotation The rotational inertia of a RB is the parameter that is analogous to inertia (mass) for a non- extended object For a RB, the rotational inertia determines how much torque is needed to produce a certain amount of rotational acceleration (spin). 19
rotational inertia examples Rods of equal mass m and length L axis through end axis through center I = 4 I end center The rod with the axis through the end requires more torque to get it rotating. 20
How fast does it spin? For spinning or rotational motion, the rotational inertia of an object plays the same role as ordinary mass for simple motion For a given amount of torque applied to an object, its rotational inertia determines its rotational acceleration the smaller the rotational inertia, the bigger the rotational acceleration 21
Same torque, different rotational inertia = =F L = mgR L = R Small rotational inertia Big rotational inertia spins slow spins fast F = mg 22
