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PHY2053 Lecture 17 Ch. 8.4, 8.6 (skip 8.5) Rotational Equilibrium,Rotational Formulation of Newton’s Laws
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Linear vs Rotational VariablesTransformation Table
2
Linear Variable → Rotational Variable
distance angle
velocity angular velocity
acceleration angular acceleration
mass [inertia] moment of inertia
momentum p = mv angular momentum, L = Iωforce, F = ma torque, τ = IαKtrans = ½mv2 Krotation = ½Iω2
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
“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
3
∆θ
∆θ
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Reminder: from angle to angular acceleration
• instantaneous angular velocity:
• instantaneous angular acceleration:
4
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Torque• The equivalent of force, for rotational systems
(somewhat hand-waving derivation of equivalence)
5
• not quite so trivial, need to account for the orientation of force wrt “lever arm” of object i, “useful force”
torque = momentum of inertia × angular acceleration
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Lever Arm
6
F
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“lever arm”
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PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Work done by torque• Start from definition of work, for rotational systems:
7
• Hooke’s law, for rotationalsystems (torsion spring):
PHY2053, Lecture 16, Rotational Energy and Inertia
Unusual Atwood’s Machine
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Unusual Atwood’s Machine, via energy conservation
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Unusual Atwood’s Machine, via force, tension and torque
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Unusual Atwood’s Machine, via force, tension and torque 2
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PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Statics, Rotational Equilibrium• Condition 1: Newton’s 1st law: a stationary object
that has no net force acting on it remains stationary
12
• 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
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?
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Springboard
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Springboard 2
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PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Angular Momentum• Momentum-impulse theorem:
16
• by analogy with p = mv, define angular momentum:
• angular momentum-impulse theorem:
PHY2053, Lecture 17 Rotational Form of Newton’s Laws
Example: “Angular Collision”
• A disk of mass m1 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?
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Angular Collision
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PHY2053, Lecture 16, Rotational Energy and Inertia
Discussion: Angular Collision 2
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PHY2053, Lecture 17 Rotational Form of Newton’s Laws
The vector nature of torque and angular momentum
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