Physics 221, March 1
Key Concepts:
•Rotations
•Kinematics and dynamics
•Energy and angular momentum
•Equilibrium
Please do not use social media during class.
Rotations
Extended object can have translational and rotational motion.
How do we describe rotational motion?
Angular speed:
ω = ∆θ/∆t
Angular velocity:
ω = ∆θ/∆t n,
n = direction indicator
Angular acceleration:
α = ∆ω/∆t
Right-hand rule:
direction of angular
velocity
The direction of the angular acceleration is the
direction of the change of the angular velocity.
A pocket watch and Big Ben are both keeping perfect time.
Which minute hand has the larger angular velocity?
1. Big Ben’s
2. The pocket watch’s
3. They both have the
same ω.
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Kinematics and dynamics
Motion with constant angular acceleration
Kinematic equations: ωf = ωi + α(tf - ti)
θf = θi + ωi(tf - ti)+ ½ α(tf - ti)2
ωavg = (ωi + ωf )/2.
ωf 2 = ωi 2 + 2 α(θf - θ i).
A fan rotating with an initial angular velocity of
1000 rev/min is switched off. In 2 seconds, the
angular velocity decreases to 200 rev/min.
Assuming the angular acceleration is constant,
how many revolutions does the blade undergo
during this time?
Think about it!
How can you approach this problem?
A fan rotating with an initial angular velocity of 1000 rev/min is
switched off. In 2 seconds, the angular velocity decreases to
200 rev/min. Assuming the angular acceleration is constant,
how many revolutions does the blade undergo during this time?
Given:
wi = 1000 rev/min wf = 200 rev/min, Dw = -800 rev/min
a = constant, Dt = 2s
Asked:
Dq = ?
Possible approach to the solution:
Find wavg = (wi + wf)/2, then use Dq = wavgDt.
Another approach to the solution:
Find a = (wf - wi)/ Dt, then use Dq = ωiDt + ½αDt2.
Extra credit:
A fan rotating with an initial angular velocity of 1000 rev/min is
switched off. In 2 seconds, the angular velocity decreases to 200
rev/min. Assuming the angular acceleration is constant, how
many revolutions does the blade undergo during this time?
1.
2.
3.
4.
5.
10
20
125
100
1200
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Torque
A net torque causes angular acceleration. α = τ /I
torque = lever arm × force
τ=r×F
Torque is a vector. It is the cross product of r and F.
The cross product of two vectors A and B is defined as the vector C = A×B.
The magnitude of C is C = ABsinθ, where θ is the smallest angle between the
directions of the vectors A and B.
C is perpendicular to both A and B.
The direction of C can be found by using the right-hand rule.
Let the fingers of your right hand point in the direction of A.
Orient the palm of your hand so that, as you curl your fingers, you can sweep
them over to point in the direction of B.
Your thumb points in the direction of C = A×B.
Direction of the torque:
Let the fingers of your right hand point from the axis of rotation to the point where the
force is applied. Curl them into the direction of F. Your thumb points in the direction of the
torque vector.
F1, F2 and F3 have the same magnitude. Which statement is correct?
1.
2.
3.
4.
All forces produce a torque
pointing out of the page.
The torque produced by F1 and
F3 point in opposite directions.
F2 produces no torque, and the
torques produced by F1 and F3
have equal magnitudes,
F1 produces the torque with the
largest magnitude.
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Extra credit:
Consider the earth spinning from
west to east on its axis. It is
slowing down in its rotation due to
friction with the air. Using the
right hand rule, in what direction
is the frictional torque?
1. S pole to N pole
2. N pole to S pole
3. Counterclockwise along the
equator
4. Clockwise along the
equator
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You are staring at a non-rotating
wind turbine and all of a sudden
the wind comes up and makes
the turbine start rotating
clockwise. The net force that
causes translational motion is
______ and the torque causing
the turbine to rotate is ________.
1.
2.
3.
4.
Positive; towards you
Positive; towards the right
zero; to the left.
zero; away from you.
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Moment of inertia
The moment of inertia is defined with respect to an axis of
rotation.
The farther the bulk of the mass is from the axis of rotation, the
Greater is the moment of inertia of the object.
A cheerleader is given two batons:
one gold and one plastic. He is told
that both batons have the same mass.
He tries spinning each one and finds
that the plastic one is much easier to
start and stop spinning. He correctly
concludes that the plastic baton, in
comparison to the gold baton
1. has has more of its mass
distributed towards the center.
2. has more of its mass distributed
towards the ends.
3. must weigh more.
4. must weigh less.
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Both the disk and the ring have the same mass, and the same radius.
Which has the larger moment of inertia?
1.
2.
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Energy and angular momentum
Energy of an object with translational and rotational motion:
translational kinetic energy = ½mvCM2
rotational kinetic energy = ½Iω2
total kinetic energy = ½mvCM2 + ½Iω2
Rolling: vCM = rω
total kinetic energy = ½ (m + I/r2)vCM2
Ratio Etrans/Erot = mr2/I
Angular momentum
L = Iω, ΔL/Δt = Iα = τ,
ΔL = τΔt
If no external torque acts on a system of interacting objects,
then their total angular momentum is constant.
Assume a disk and a ring with the same radius roll down an incline of height h
and angle theta. If they both start from rest at t = 0, which one will reach the
bottom first?
1. The disk
2. The ring
3. They both will arrive
at the same time.
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Extra credit:
Suppose the earth’s polar ice caps
melted, sending water towards the
equator and increasing the moment
of inertia of the earth to 1.1 times its
present value. The angular velocity
of the earth would be
1.
2.
3.
4.
0.82 times its present value.
0.909 times its present value.
Unchanged.
1.1 times its present value.
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Equilibrium
Mechanical equilibrium:
the net external force and the net external torque are zero.
Ftot = 0, τtot = 0.
Mechanical equilibrium can be stable or unstable.
When an object in stable equilibrium is disturbed be a small
amount, its total potential energy increases.
You can balance a broom on your hand if you’re careful. To do this
trick well, you must look at
1. your hand.
2. the top of the
broom.
3. the broom’s
center of gravity.
4. the middle of the
broom.
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You can balance a broom on your hand if you’re careful.
To do this trick well, you must look at
(3) the broom’s center of gravity.
The center of gravity has to stay above the support.
You track the center of gravity with your eyes and
move your hand accordingly.