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Angular Mechanics
Chapter 8/9
Similarities
Linear Angular
Mass Moment of Inertia
Force Torque
Momentum Angular Momentum
Center of Mass
• The center of mass of an object is the average position of mass.
• Objects tend to rotate about their center of mass.
• Examples: • Meter stick• Rotating Hammer• Rolling Double-Cone
Stability• For stability center of gravity must be over area
of support.
• Examples: • Tower of Pisa• Touching toes with back to wall• Meter stick over the edge
Otherwise we will get a rotation!
I = Rotational Inertia • An object rotating about an axis tends to
remain rotating unless interfered with by some external influence.
• This influence is called torque.
• Rotation adds stability to linear motion.– Examples:
• spinning football• bicycle tires• Frisbee
Moment of Inertia
• Defined as resistance to rotation – depends on mass– depends on
distance from axis of rotation
I = mr2
Moments For Various ObjectsObject Location of
axis Diagram Moment of
InertiaThin Hoop Center
Solid Cylinder Center
Uniform Sphere
Center
Uniform Rod Length L
Center
Uniform Rod Length L
Through End
Thin Plate Length L Width
W
Center
mr2
1
2mr2
2
5mr2
1
12ml2
1
3ml2
1
12m(l2 + w2 )
• The greater the distance between the bulk of an object's mass and its axis of rotation, the greater the rotational inertia.
• Examples: – Tightrope walker– Metronome
Ways to Measure Rotation
• Degrees: 1/360th of a revolution
• Radians: of a
revolution
1 revolution = 2 radians
1
2π
π
Angular Displacement
• Found by change in θ.
• Distance around a pivot is found by
• d = r θ
– Where the angle is measured in radians and r is the radius of the arc.
– Measured in meters
Angular Velocity
• The rate of revolution around an axis.– Measured in rads/sec
• Velocity around an axis is found by
v = rωWhere r is the radius and ω is angular velocity and is
measured in m/s.
ω =ΔθΔt
How Fast Does the Earth Spin?
• 1rev/24 hrs
• 2π radians/revolution
• Radius or earth = 6.38x106m
ω e =1rev
24hrs=
2π rad
86400s= 7.27x10−5 rad
sec
v =rω =6.38x106 m(7.27x10−5 rads
) =464ms
Angular Acceleration
• The change in angular velocity per unit of time.– Measured in rads/sec2
α =ΔωΔt
Acceleration of an object is found by
a = rα And is measured in m/s2.
Linear and Angular MeasuresQuantity Linear Angular Relationship
Displacement d (m) θ (rad) d = r θ
Velocity v (m/s) ω (rad/s) v = r ω
Acceleration a (m/s2) α (rad/s2) a = r α
Direction ofMotion
Centrifugal Force
CentripetalForce
Centripetal or Centrifugal?
No Matter What Faith Hill Says,IT’S NOT CENTRIFICAL MOTION!
Centripetal Force• …is applied by some object.
• Centripetal means "center seeking".
Centrifugal Force
• …results from a natural tendency.
• Centrifugal means "center fleeing".
• This is a fictitious force for us. Why?
Centripetal motion
ac =v2
r
Fc =mv2
r
Practice Wall and Wall
Pg 234, 6-6, Pg 243, 6-12
Examples
• water in bucket
• moon’s orbit
• car on circular path
• coin on a hanger
• jogging in a space station
Centripetal Force
• Bucket
• Earth’s gravity
• Road Friction
• Hanger
• Space Station Floor
Centrifugal Force
• Nature
• Nature
• Nature
• Nature
• Nature
Conservation of Angular Momentum
• angular momentum = rotational inertia rotational velocity
• L = I ω
• Newton's first law for rotating systems: – “A body will maintain its state of angular momentum
unless acted upon by an unbalanced external torque.”
• Examples: –1. ice skater spin–2. cat dropped on back–3. Diving–4. Collapsing Stars (neutron stars)
Torque
• Force directed on an object that has a fixed point is found by
– Where τ is torque, F is force in N, r is distance from the axis in m, and θ is measured IN DEGREES.
– (use sin θ only if force is not || to motion)
τ =Fr sinθ
Levers
• The lever arm is the distance from the axis along a θ to the direction of applied force.
• Torque here is force times the lever arm.
Lever arm (r sin θ)
rθ
τ =Fr sinθ
A Balancing Act
• Static equilibrium occurs when the sum of the torques add to equal zero.
NOW BUILD YOUR OWN!
• Must involve at least 3 different axes of rotation.
• Must hang at least 8 objects.
• No 2 objects can have the same mass.• No two hangers can have the same length.• No fulcrum can be in the middle of a hanger.