Unit 3: Rotational and Orbital Motion€¦ · Mr. Cali Unit 3: Rotational and Orbital Motion Honors...

Unit 3: Rotational and Orbital Motion

Mr. Cali

Honors Physics

Mr. Cali Unit 3: Rotational and Orbital Motion Honors Physics 1 / 26

Overview

1 Angular MotionAngular KinematicsTorqueAngular Kinetic EnergyAngular Momentum

2 Uniform Circular Motion

3 Orbital MotionUniversal GravitationKepler’s Laws


Angular Motion Angular Kinematics

Kinematics: Translation vs Rotation

Translation Rotation



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



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◦





360◦


2π


360

2π

degrees

radians=

180

π

degrees

radians≈ 57.3◦





360◦


2π


360

2π

degrees

radians=

180

π

degrees

radians≈ 57.3◦





360◦


2π


360

2π

degrees

radians=

180

π

degrees

radians≈ 57.3◦



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



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



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



• The rate of change ofdisplacement per unit time:

x = rθ ⇒ ∆x = r∆θ ⇒∆x

∆t= r

∆θ

∆t⇒ v = rω




• The rate of change ofvelocity per unit time:

v = rω ⇒ ∆v = r∆ω ⇒∆v

∆t= r

∆ω

∆t⇒ a = rα



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?



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



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


“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





v 2f = v 2i + 2a∆θ ⇒ r 2ω2f = r 2ω2

i + 2r 2a∆θ ⇒

ω2f = ω2

i + 2a∆θ

∆x = vit +1


1

2rαt2 ⇒

∆θ = ωit +1

2αt2





v 2f = v 2i + 2a∆θ ⇒ r 2ω2f = r 2ω2

i + 2r 2a∆θ ⇒

ω2f = ω2

i + 2a∆θ

∆x = vit +1


1

2rαt2 ⇒

∆θ = ωit +1

2αt2


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



Torque


• τ = rF sin(φ)




Torque


• τ = rF sin(φ)




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?



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?



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).



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.



Moment of Inertia


Angular Motion Angular Kinetic Energy

Kinetic Energy: Translation vs Rotation

Translational

KE =1

2mv 2

Units: J

Rotational

KE =1

2Iω2

Units: J



Energy of Rolling Objects

• Objects that roll have both types of kinetic energy

KE =1

2mv 2 +

1

2Iω2



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?


Angular Motion Angular Momentum

Momentum: Translation vs Rotation

things3


Uniform Circular Motion

Rotation vs Circles

things4


Orbital Motion Universal Gravitation

Newton and Gravity

things5


Orbital Motion Kepler’s Laws

Kepler and Orbits

things6


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Unit 3: Rotational and Orbital Motion€¦ · Mr. Cali Unit 3: Rotational and Orbital Motion Honors...

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