PHY2053 Lecture 17
Ch. 8.4, 8.6 (skip 8.5) Rotational Equilibrium,
Rotational Formulation of Newton’s Laws
Linear vs Rotational Variables
Transformation Table
Linear Variable
→ Rotational Variable
distance
angle
velocity
angular velocity
acceleration
angular acceleration
mass [inertia]
moment of inertia
momentum p = mv
angular momentum, L = Iω
force, F = ma
torque, τ = Iα
Ktrans =
2
½mv
Krotation = ½Iω2
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
2
“Right Hand Rule” for
rotational systems
• change of angle (∆θ) is a
•
vector along the axis of
rotation, and its direction is
determined by the right hand
rule:
orient your right hand so that
the fingers point in the
direction in which the angle is
changing; your thumb points in
the direction of the rotational
vector
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
∆θ
∆θ
3
Reminder: from angle to
angular acceleration
• instantaneous angular velocity:
• instantaneous angular acceleration:
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
4
Torque
• The equivalent of force, for rotational systems
(somewhat hand-waving derivation of equivalence)
torque = momentum of inertia × angular acceleration
• not quite so trivial, need to account for the orientation
of force wrt “lever arm” of object i, “useful force”
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
5
F⊥= F sinθ
θ
“Perpendicular force”
θ
F
F
r⊥ = r sinθ
r
r
“lever arm”
θ
• for the same force, the torque is directly proportional
to the lever arm (projection of r perpendicular to F)
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
6
Work done by torque
• Start from definition of work, for rotational systems:
F
m
∆s
• Hooke’s law, for rotational
systems (torsion spring):
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
r
∆θ
7
Unusual Atwood’s Machine
In the machine below, the small weight (mass m) is connected via
an ideal pulley to the massless horizontal wheel of radius r. Ignore
the moments of inertia of the connecting rods; the two masses M
are both R away from the axis of rotation. What is the velocity of
mass m as it falls h from its initial position, where it was at rest?
R
M
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r
m
h
PHY2053, Lecture 16, Rotational Energy and Inertia
8
Discussion: Unusual Atwood’s Machine,
via energy conservation
PHY2053, Lecture 16, Rotational Energy and Inertia
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via force, tension and torque
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PHY2053, Lecture 16, Rotational Energy and Inertia
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Discussion: Unusual Atwood’s Machine,
via force, tension and torque 2
11
Statics, Rotational Equilibrium
• Condition 1: Newton’s 1st law: a stationary object
that has no net force acting on it remains stationary
• Condition 2: For the object not to start rotating about
an axis, the net torque on the object must be zero
• Important: Conditions of static equilibrium, once
established for one axis, apply to any arbitrary axis
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
12
Example: Springboard
• A diver weighing 80 kg is standing on the end of a 2
m long springboard which weighs 240 kg. The
springboard is supported by two springs on the other
end. One spring is at the very end of the board, and
the other is 0.5 m toward the center of the board.
Compute the forces generated by the springs if the
system is in static equilibrium. Are the springs
compressed or stretched?
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
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PHY2053, Lecture 16, Rotational Energy and Inertia
O
Discussion: Springboard 2
Angular Momentum
• Momentum-impulse theorem:
• by analogy with p = mv, define angular momentum:
• angular momentum-impulse theorem:
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
16
Example: “Angular Collision”
• A disk of mass m
and radius R is spinning with an
angular frequency ωi . A second disk, of mass 3m1 is
dropped onto disk 1 so that they share the same axis
of rotation. After a brief period of friction, the two
disks are rotating with the same angular frequency
ωf.
What is the ratio ωf/ωi ?
What fraction of the initial kinetic energy was wasted
through friction?
1
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PHY2053, Lecture 17 Rotational Form of Newton’s Laws
17
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Discussion: Angular Collision
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PHY2053, Lecture 16, Rotational Energy and Inertia
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Discussion: Angular Collision 2
19
The vector nature of torque
and angular momentum
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
20
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