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Unit 3: Rotational and Orbital Motion
Mr. Cali
Honors Physics
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Overview
1 Angular MotionAngular KinematicsTorqueAngular Kinetic EnergyAngular Momentum
2 Uniform Circular Motion
3 Orbital MotionUniversal GravitationKepler’s Laws
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Angular Motion Angular Kinematics
Kinematics: Translation vs Rotation
Translation Rotation
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Angular Motion Angular Kinematics
Radians
• It is convenient todescribe the amount ofrotation of an objectthrough anglemeasures.
• The SI unit of angulardisplacement is theradian.
• One radian is the anglethat subtends an arcwith an arclength ofone radius
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Angular Motion Angular Kinematics
How Many Degrees are in a Radian?
• How many degrees are in a circle?
360◦
• How many radians are in a circle?
2π
• How many degrees per radian?
360
2π
degrees
radians=
180
π
degrees
radians≈ 57.3◦
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 5 / 26
Angular Motion Angular Kinematics
How Many Degrees are in a Radian?
• How many degrees are in a circle?
360◦
• How many radians are in a circle?
2π
• How many degrees per radian?
360
2π
degrees
radians=
180
π
degrees
radians≈ 57.3◦
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 5 / 26
Angular Motion Angular Kinematics
How Many Degrees are in a Radian?
• How many degrees are in a circle?
360◦
• How many radians are in a circle?
2π
• How many degrees per radian?
360
2π
degrees
radians=
180
π
degrees
radians≈ 57.3◦
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 5 / 26
Angular Motion Angular Kinematics
How Many Degrees are in a Radian?
• How many degrees are in a circle?
360◦
• How many radians are in a circle?
2π
• How many degrees per radian?
360
2π
degrees
radians=
180
π
degrees
radians≈ 57.3◦
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 5 / 26
Angular Motion Angular Kinematics
Revolutions
• A common unit of measure in the US is revolutionsper minute, rpms.
• How do we convert rpms to radians per second,units used by physicists?
• Again, dimensional analysis!
n
1
revolutions
minute× 2π
1
radians
revolution× 1
60
minute
seconds
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 6 / 26
Angular Motion Angular Kinematics
Revolutions
• A common unit of measure in the US is revolutionsper minute, rpms.
• How do we convert rpms to radians per second,units used by physicists?
• Again, dimensional analysis!
n
1
revolutions
minute× 2π
1
radians
revolution× 1
60
minute
seconds
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 6 / 26
Angular Motion Angular Kinematics
Relating Translational and AngularQuantities
• The arclength formulaconverts linear displacementto angular displacement:
s = rθ
• But we will change thisslightly to make it look morefamiliar to us.
x = rθMr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 7 / 26
Angular Motion Angular Kinematics
Relating Translational and AngularQuantities
• The rate of change ofdisplacement per unit time:
x = rθ ⇒ ∆x = r∆θ ⇒∆x
∆t= r
∆θ
∆t⇒ v = rω
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 8 / 26
Angular Motion Angular Kinematics
Relating Translational and AngularQuantities
• The rate of change ofvelocity per unit time:
v = rω ⇒ ∆v = r∆ω ⇒∆v
∆t= r
∆ω
∆t⇒ a = rα
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Angular Motion Angular Kinematics
Example: Merry-go-round
Your cousin wants to ride ona merry-go-round. When thetwo of your sit down, you sitfarther away. If themerry-go-round has aconstant angular velocity of2 rad/s, your cousin issitting 2 m from the center,and you are sitting 3 m fromthe center, what is the lineardistance covered, the linearvelocity, and the linearacceleration of your cousinin one revolution? Of you?
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Angular Motion Angular Kinematics
Centripetal Acceleration
• Even when the translationalvelocity has a constant speedits direction is changing.
• Centripetal acceleration: aninward acceleration thatkeeps an object rotating.
•ac =
v 2
r
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Angular Motion Angular Kinematics
Frequency and Period of Rotation
• Frequency: How many revolutions per second.
f =ω
2π
(rads
radrevolution
=revolution
s= Hz
)• Period: How long does it take to makeone complete
revolution.
T =2π
ω
(rad
revolutionrads
=s
revolution= s
)• The relationship between frequency and period:
f =1
TMr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 12 / 26
Angular Motion Angular Kinematics
“New” Kinematic Equations
vf = vi + at ⇒ rωf = rωi + rαt ⇒ωf = ωi + αt
v 2f = v 2i + 2a∆θ ⇒ r 2ω2f = r 2ω2
i + 2r 2a∆θ ⇒
ω2f = ω2
i + 2a∆θ
∆x = vit +1
2at2 ⇒ r∆θ = rωit +
1
2rαt2 ⇒
∆θ = ωit +1
2αt2
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Angular Motion Angular Kinematics
“New” Kinematic Equations
vf = vi + at ⇒ rωf = rωi + rαt ⇒ωf = ωi + αt
v 2f = v 2i + 2a∆θ ⇒ r 2ω2f = r 2ω2
i + 2r 2a∆θ ⇒
ω2f = ω2
i + 2a∆θ
∆x = vit +1
2at2 ⇒ r∆θ = rωit +
1
2rαt2 ⇒
∆θ = ωit +1
2αt2
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 13 / 26
Angular Motion Angular Kinematics
“New” Kinematic Equations
vf = vi + at ⇒ rωf = rωi + rαt ⇒ωf = ωi + αt
v 2f = v 2i + 2a∆θ ⇒ r 2ω2f = r 2ω2
i + 2r 2a∆θ ⇒
ω2f = ω2
i + 2a∆θ
∆x = vit +1
2at2 ⇒ r∆θ = rωit +
1
2rαt2 ⇒
∆θ = ωit +1
2αt2
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 13 / 26
Angular Motion Torque
Torque
• The rotational analogue for Force is Torque, τ .
• τ = rF sin(φ)
• Στ = r1F1 sin(φ1) + r2F2 sin(φ2) + · · ·• Units: N·m
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 14 / 26
Angular Motion Torque
Torque
• The rotational analogue for Force is Torque, τ .
• τ = rF sin(φ)
• Στ = r1F1 sin(φ1) + r2F2 sin(φ2) + · · ·• Units: N·m
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 14 / 26
Angular Motion Torque
Torque
• The rotational analogue for Force is Torque, τ .
• τ = rF sin(φ)
• Στ = r1F1 sin(φ1) + r2F2 sin(φ2) + · · ·• Units: N·m
Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 14 / 26
Angular Motion Torque
Example: Opening Doors
A force, F1 = 100 N, is appliedperpendicular to the door adistance r1 = 0.5 m from thehinge. Separately, a force,F2 = 75 N, is appliedperpendicular to the door adistance r2 = 0.8 m from thehinge at an angle of θ = 45◦.Which force applies a greatertorque on the door?
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Angular Motion Torque
Example: Bicep torque
A person is holding a 3.17-kgbowling ball. The bowling ball isheld a distance of 35 cm from thepoint of rotation (the elbow). Ifthe bicep attaches to the forearma distance of 4 cm from theelbow, with what force does itpull vertically upward to hold thebowling ball in place?
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Angular Motion Torque
Example: Why We Don’t Lift With OurBacks
Let the same person try to deadlift 100 kg lifting withtheir back horizontal and with their back angled. Thedistance from the person’s point of rotation and theirshoulders is r = 0.65 m. With the angled back, the anglebetween the spine and the hanging arms is θ = 45◦.What is the mimimum force required by the lower backto lift the load if the muscle attaches to the spine adistance of r = 18 cm from the point of rotation?(Assume the back always pulls perpendicular to thespine).
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Angular Motion Torque
Moment of Inertia
• Moment of inertia, I , is the rotational analogue ofmass.
F = ma ⇒ F = mrα ⇒ rF = rmrα ⇒τ = mr 2α ⇒ τ = Iα
• The above relation I = mr 2 assumes that the massis a point particle at a distance r from the center ofrotation.
• I is different for all shapes that may rotate.
• An equation sheet will always be provided.
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Angular Motion Torque
Moment of Inertia
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Angular Motion Angular Kinetic Energy
Kinetic Energy: Translation vs Rotation
Translational
KE =1
2mv 2
Units: J
Rotational
KE =1
2Iω2
Units: J
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Angular Motion Angular Kinetic Energy
Energy of Rolling Objects
• Objects that roll have both types of kinetic energy
KE =1
2mv 2 +
1
2Iω2
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Angular Motion Angular Kinetic Energy
Example: Tire Rolling Down a Ramp
A 1.3-kg bicycle tire with a radius of 70 cm starts fromrest and rolls down a ramp. If the ramp has a length of15 m and is inclined at 20◦,
a) what is the total kinetic energy of the wheel at thebottom of the ramp?
b) what is the translational velocity at the bottom ofthe ramp?
c) what is the angular velocity at the bottom of theramp?
d) what is the ratio of angular kinetic energy totranslational kinetic energy?
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Angular Motion Angular Momentum
Momentum: Translation vs Rotation
things3
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Uniform Circular Motion
Rotation vs Circles
things4
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Orbital Motion Universal Gravitation
Newton and Gravity
things5
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Orbital Motion Kepler’s Laws
Kepler and Orbits
things6
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