Download lecture 21 torque

LECTURE 21
TORQUE
Instructor: Kazumi Tolich
Lecture 21
2
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Reading chapter 11-1 to 11-2
¤ Torque
¤ Newton’s
2nd law for rotation
Torque about an axis
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Torque (𝜏) is a measure of twisting, and the magnitude of
torque is defined as
𝐅⃗
𝜏 = 𝑟𝐹 sin 𝜃 = 𝐹) 𝑟 = 𝐹𝑟*
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𝑟* is called moment arm, or lever arm of 𝐅⃗.
The SI unit for torque is N·∙m, the same as the unit of
work. But torque and work are different physical
quantities.
𝜃
𝐅⃗)
𝐫⃗
Axis
𝑟*
Line of action
Newton’s 2nd law for rotation
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Newton’s 2nd law for rotation is given by
. 𝜏 = 𝐼𝛼
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The sign of torque is the same as the sign of angular acceleration it causes if
it were the only torque acting in the system.
If two or more torques act on a rigid object, the net torque is the sum of the
torques with correct sign assigned to each torque.
This is analogous to Newton’s 2nd law for linear motion: ∑ 𝐅⃗ = 𝑚𝐚
Quiz: 1
5
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Two forces produce the same torque. Does it follow
that the magnitudes of the forces are the same?
A.
B.
Yes
No
Quiz: 21-1 answer
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No
Because torque is 𝜏 = 𝑟𝐹 sin 𝜃 = 𝐹) 𝑟 = 𝐹𝑟* , two different forces
that act at different distances or angles could still give the same
torque.
For example, the diagrams below show the applications of
various forces to various perpendicular level arms. The
magnitudes of the torques about the dark blue dots are the same
in all cases.
L
a
2L
F
b
0.5F
F1
F2
0.5L
c
2F
Applications/Demo: 1
7
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Door knobs: why are all door knobs located farthest from the door
hinges?
Bicycle pedals: why is it hard to get going if you try to start your bike
with the pedal at the highest point?
Wheelchairs: why are the handrims for wheelchairs for racing and
basketball different?
Demo: torque bar
Quiz: 2
8
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In case 1, one end of a horizontal massless rod of length 𝐿 is attached
to a vertical wall by a hinge, and the other end holds a ball of mass
𝑀. In case 2 the massless rod is twice as long and makes an angle of
30° with the wall as shown. In which case is the total torque about an
axis through the hinge bigger?
A.
B.
C.
Case 1
Case 2
same
Quiz: 21-2 answer
9
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Same
The weight of the balls are the same in both cases.
The perpendicular lever arms are the same in both cases.
𝜏 = 𝐹𝑟*
𝐹 = 𝑀𝑔
𝐹 = 𝑀𝑔
𝑟* = 𝐿
𝑟* = 2𝐿 sin 30° = 𝐿
Quiz: 3
10
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Suppose a ladder on a rough floor is leaning against
a frictionless wall as shown, and you are trying to
calculate the net torque on the ladder about the axis
through where the ladder touches the floor. What is
the lever arm of the normal force by the wall?
A.
B.
C.
2𝑟
𝑟
ℎ
𝐍=D
h
𝐖=@
D.
ℎF
E.
ℎF + 𝑟 F
+ 2𝑟
F
𝐍=>
𝐟⃗B,=>
r
r
Quiz: 21-3 answer
11
ℎ
¨ The lever arm is the shortest
distance between the line of
action and the axis.
¨
Line of action
𝐍=D
𝑟* = ℎ
h
𝐖=@
𝐍=>
𝐟⃗B,=>
r
r
Quiz: 4
12
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Forces 𝐹⃗H and 𝐹⃗F have the same magnitude and are applied to identical disks on a frictionless
horizontal table as shown. Which of the following statements is/are correct? Choose all that
apply.
A.
Disk 1 has angular acceleration.
B.
Disk 2 has angular acceleration.
Disk 1 has linear acceleration.
Disk 2 has linear acceleration.
C.
D.
Disk 1
Disk 2
Quiz: 21-4 answer
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Disk 2 has angular acceleration.
The net force on both disks are zero as magnitudes of 𝐹⃗H and 𝐹⃗F are the same, and they point in the
opposite directions.
The net torque on Disk 1 is zero since the lever arms of 𝐹⃗H and 𝐹⃗F on Disk 1 about its center of mass is
zero.
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On the other hand, the lever arms of 𝐹⃗H and 𝐹⃗F on Disk 2 about its center of mass are non-zero.
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𝐹⃗H and 𝐹⃗F both would cause Disk 2 to rotate counterclockwise, having a positive angular acceleration.
𝑟* 𝑟*
Disk 1
Disk 2
Demo: 2
14
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Hinged Stick and Ball
¤ When
the bar is just about to become horizontal, the acceleration of the
free-end of the bar is greater than 𝑔.
¤ 𝛼
¤ 𝑎
I
J
M
KLN
= =O
P
KQN
= 𝐿𝛼 =
RL
F
=
RL
FQ
>𝑔
Example: 1
15
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Figure shows the massive shield door at a neutron
facility at Lawrence Livermore National
Laboratory; this is the world’s heaviest hinged
door. The door has a mass of 𝑀 = 44,000 kg, a
rotational inertia about an axis through its hinges
of 𝐼 = 8.7 × 104 kgŸm2, and a front face width
of 𝑤 = 2.4 m. Neglecting friction, what steady
force, applied at its outer edge and
perpendicular to the plane of the door, can move
it from rest through an angle of 𝜃 = 90º in
𝑡 = 30 s?
Example: 2
16
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A wheel on a game show is given an initial
angular speed of 𝜔X = 1.22 rad/s. It
comes to rest after rotating through
∆𝜃 = 0.75 of a turn.
a)
b)
Find the average torque exerted on the
wheel given that it is a disk of radius
𝑟 = 0.71 m and mass 𝑚 = 6.4 kg.
If the mass of the wheel is doubled, and its
radius is halved, will the angle through
which it rotates before coming to rest
increase, decrease, or stay the same
assuming that the average torque exerted
on the wheel is unchanged?