Rotational Kinematics

1
Rotational Kinematics
2
Position

• In translational motion, position is represented
  by a point, such as x.

x
0
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linear

• In rotational motion, position is represented by
  an angle, such as q, and a radius, r.

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Displacement

• Linear displacement is represented by the vector
  Dx.

x
0
5
linear

• Angular displacement is represented by Dq, which
  is not a vector, but behaves like one for small
  values. The right hand rule determines direction.

4
Tangential and angular displacement

• A particle that rotates through an angle Dq also
  translates through a distance s, which is the
  length of the arc defining its path.

r

• This distance s is related to the angular
  displacement Dq by the equation s rDq.

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Speed and velocity

• The instantaneous velocity has magnitude vT
  ds/dt and is tangent to the circle.

r

• The same particle rotates with an angular
  velocity w dq/dt.
• The direction of the angular velocity is given by
  the right hand rule.
• Tangential and angular speeds are related by the
  equation v r w.

w is outward according to RHR
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Acceleration
w is out of page (z in this diagram) according
to RHR

• Tangential acceleration is given by aT dvT/dt.
• This acceleration is parallel or anti-parallel to
  the velocity.
• Angular acceleration of this particle is given by
  a dw/dt.
• Angular acceleration is parallel or anti-parallel
  to the angular velocity.
• Tangential and angular accelerations are related
  by the equation a r a.

vT
r
Dq
vT
Dont forget centripetal acceleration.
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Problem Assume the particle is speeding up.

• What is the direction of the instantaneous
  velocity, v?
• What is the direction of the angular velocity, w?
• What is the direction of the tangential
  acceleration, aT?
• What is the direction of the angular acceleration
  a?
• What is the direction of the centripetal
  acceleration, ac?
• What is the direction of the overall
  acceleration, a, of the particle?

What changes if the particle is slowing down?
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First Kinematic Equation

• v vo at (linear form)
• Substitute angular velocity for velocity.
• Substitute angular acceleration for acceleration.
• ? ?o ?t (angular form)

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Second Kinematic Equation

• x xo vot ½ at2 (linear form)
• Substitute angle for position.
• Substitute angular velocity for velocity.
• Substitute angular acceleration for acceleration.
• q qo ?ot ½ ?t2 (angular form)

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Third Kinematic Equation

• v2 vo2 2a(x - xo)
• Substitute angle for position.
• Substitute angular velocity for velocity.
• Substitute angular acceleration for acceleration.
• ?2 ?o2 2?(q - qo)

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Practice problem

• The Beatles White Album is spinning at 33 1/3
  rpm when the power is turned off. If it takes 1/2
  minute for the albums rotation to stop, what is
  the angular acceleration of the phonograph album?

12
Rotational Energy
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Practice problem

• The angular velocity of a flywheel is described
  by the equation w (8.00 2.00 t 2). Determine
  the angular displacement when the flywheel
  reverses its direction.

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Inertia and Rotational Inertia

• In linear motion, inertia is equivalent to mass.
• Rotating systems have rotational inertia.
• I ?mr2 (for a system of particles)
• I rotational inertia (kg m2)
• m mass (kg)
• r radius of rotation (m)
• Solid objects are more complicated well get to
  those later. See page 278 for a cheat sheet.

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Sample Problem

• A 2.0-kg mass and a 3.0-kg mass are mounted on
  opposite ends a 2.0-m long rod of negligible
  mass. What is the rotational inertia about the
  center of the rod and about each mass, assuming
  the axes of rotation are perpendicular to the rod?

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Kinetic Energy

• Bodies moving in a straight line have
  translational kinetic energy
• Ktrans ½ m v2.
• Bodies that are rotating have rotational kinetic
  energy
• Krot ½ I w2
• It is possible to have both forms at once.
• Ktot ½ m v2 ½ I ?2

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Practice problem

• A 3.0 m long lightweight rod has a 1.0 kg mass
  attached to one end, and a 1.5 kg mass attached
  to the other. If the rod is spinning at 20 rpm
  about its midpoint around an axis that is
  perpendicular to the rod, what is the resulting
  rotational kinetic energy? Ignore the mass of the
  rod.

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Rotational Inertia
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Rotational Inertia Calculations

• I ?mr2 (for a system of particles)
• I ? dm r2 (for a solid object)
• I Icm m h2 (parallel axis theorem)
• I rotational inertia about center of mass
• m mass of body
• h distance between axis in question and axis
  through center of mass

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Practice problem

• A solid ball of mass 300 grams and diameter 80 cm
  is thrown at 28 m/s. As it travels through the
  air, it spins with an angular speed of 110
  rad/second. What is its
• translational kinetic energy?
• rotational kinetic energy?
• total kinetic energy?

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Practice Problem

• Derive the rotational inertia of a long thin rod
  of length L and mass M about a point 1/3 from one
  end
• using integration of I ? r2 dm
• using the parallel axis theorem and the
  rotational inertia of a rod about the center.

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Practice Problem

• Derive the rotational inertia of a ring of mass M
  and radius R about the center using the formula I
  ? r2 dm.

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Torque and Angular Acceleration I
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Equilibrium

• Equilibrium occurs when there is no net force and
  no net torque on a system.
• Static equilibrium occurs when nothing in the
  system is moving or rotating in your reference
  frame.
• Dynamic equilibrium occurs when the system is
  translating at constant velocity and/or rotating
  at constant rotational velocity.
• Conditions for equilibrium
• St 0
• SF 0

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Torque
Torque is the rotational analog of force that
causes rotation to begin.
Consider a force F on the beam that is applied a
distance r from the hinge on a beam. (Define r as
a vector having its tail on the hinge and its
head at the point of application of the force.)
A rotation occurs due to the combination of r and
F. In this case, the direction is clockwise.
What do you think is the direction of the torque?
Direction of torque is INTO THE SCREEN.
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Calculating Torque

• The magnitude of the torque is proportional to
  that of the force and moment arm, and torque is
  at right angles to plane established by the force
  and moment arm vectors. What does that sound
  like?
• ? r ? F
• ? torque
• r moment arm (from point of rotation to point
  of application of force)
• F force

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Practice Problem

• What must F be to achieve equilibrium? Assume
  there is no friction on the pulley axle.

F
3 cm
2 cm
2 kg
10 kg
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Torque and Newtons 2nd Law

• Rewrite SF ma for rotating systems
• Substitute torque for force.
• Substitute rotational inertia for mass.
• Substitute angular acceleration for acceleration.
• S? I ?
• ? torque
• I rotational inertia
• ? angular acceleration

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Practice Problem

• A 1.0-kg wheel of 25-cm radius is at rest on a
  fixed axis. A force of 0.45 N is applied tangent
  to the rim of the wheel for 5 seconds.
• After this time, what is the angular velocity of
  the wheel?
• Through what angle does the wheel rotate during
  this 5 second period?

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Sample problem

• Derive an expression for the acceleration of a
  flat disk of mass M and radius R that rolls
  without slipping down a ramp of angle q.

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Practice problem
Calculate initial angular acceleration of rod of
mass M and length L. Calculate initial
acceleration of end of rod.
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Sample problem
Calculate acceleration. Assume pulley has mass M,
radius R, and is a uniform disk.
m2
m1
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Rotational Dynamics Lab
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Work and Power in Rotating Systems
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Practice Problem

• What is the acceleration of this system, and the
  magnitude of tensions T1 and T2? Assume the
  surface is frictionless, and pulley has the
  rotational inertia of a uniform disk.

T1
mpulley 0.45 kg rpulley 0.25 m
T2
m1 2.0 kg
m2 1.5 kg
30o
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Work in rotating systems

• W F Dr (translational systems)
• Substitute torque for force
• Substitute angular displacement for displacement
• Wrot t Dq
• Wrot work done in rotation
• ? torque
• Dq angular displacement
• Remember that different kinds of work change
  different kinds of energy.
• Wnet DK Wc -DU Wnc DE

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Power in rotating systems

• P dW/dt (in translating or rotating systems)
• P F v (translating systems)
• Substitute torque for force.
• Substitute angular velocity for velocity.
• Prot t w (rotating systems)
• Prot power expended
• ? torque
• w angular velocity

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Conservation of Energy

• Etot U K Constant
• (rotating or linear system)
• For gravitational systems, use the center of mass
  of the object for calculating U
• Use rotational and/or translational kinetic
  energy where necessary.

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Practice Problem

• A rotating flywheel provides power to a
  machine. The flywheel is originally rotating at
  of 2,500 rpm. The flywheel is a solid cylinder of
  mass 1,250 kg and diameter of 0.75 m. If the
  machine requires an average power of 12 kW, for
  how long can the flywheel provide power?

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Practice Problem

• A uniform rod of mass M and length L rotates
  around a pin through one end. It is released from
  rest at the horizontal position. What is the
  angular speed when it reaches the lowest point?
  What is the linear speed of the lowest point of
  the rod at this position?

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Rolling without Slipping
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Rolling without slipping

• Total kinetic energy of a body is the sum of the
  translational and rotational kinetic energies.
• K ½ Mvcm2 ½ I ?2
• When a body is rolling without slipping, another
  equation holds true
• vcm ? r
• Therefore, this equation can be combined with the
  first one to create the two following equations
• K ½ M vcm2 ½ Icm v2/R2
• K ½ m ?2R2 ½ Icm ?2

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Sample Problem

• A solid sphere of mass M and radius R rolls from
  rest down a ramp of length L and angle q. Use
  Conservation of Energy to find the linear
  acceleration and the speed at the bottom of the
  ramp.

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Sample Problem

• A solid sphere of mass M and radius R rolls from
  rest down a ramp of length L and angle q. Use
  Rotational Dynamics to find the linear
  acceleration and the speed at the bottom of the
  ramp.

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Angular Momentum of Particles
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Sample Problem

• A solid sphere of mass M and radius R rolls from
  rest down a ramp of length L and angle q. Use
  Conservation of Energy to find the linear
  acceleration and the speed at the bottom of the
  ramp.

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Sample Problem

• A solid sphere of mass M and radius R rolls from
  rest down a ramp of length L and angle q. Use
  Rotational Dynamics to find the linear
  acceleration and the speed at the bottom of the
  ramp.

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Practice Problem

• A hollow sphere (mass M, radius R) rolls without
  slipping down a ramp of length L and angle q. At
  the bottom of the ramp
• what is its translational speed?
• what is its angular speed?

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Angular Momentum

• Angular momentum is a quantity that tells us how
  hard it is to change the rotational motion of a
  particular spinning body.
• Objects with lots of angular momentum are hard to
  stop spinning, or to turn.
• Objects with lots of angular momentum have great
  orientational stability.

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Angular Momentum of a particle

• For a single particle with known momentum, the
  angular momentum can be calculated with this
  relationship
• L r ? p
• L angular momentum for a single particle
• r distance from particle to point of rotation
• p linear momentum

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Practice Problem

• Determine the angular momentum for the
  revolution of
• the earth about the sun.
• the moon about the earth.

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Practice Problem

• Determine the angular momentum for the 2 kg
  particle shown
• about the origin.
• about x 2.0.

y (m)
5.0
5.0
x (m)
-5.0
v 3.0 m/s
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Angular Momentum of Solid Objectsand
Conservation of Angular Momentum
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Angular Momentum - solid object

• For a solid object, angular momentum is analogous
  to linear momentum of a solid object.
• P mv (linear momentum)
• Replace momentum with angular momentum.
• Replace mass with rotational inertia.
• Replace velocity with angular velocity.
• L I ? (angular momentum)
• L angular momentum
• I rotational inertia
• w angular velocity

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Practice Problem

• Set up the calculation of the angular
  momentum for the rotation of the earth on its
  axis.

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Law of Conservation of Angular Momentum

• The Law of Conservation of Momentum states that
  the momentum of a system will not change unless
  an external force is applied. How would you
  change this statement to create the Law of
  Conservation of Angular Momentum?
• Angular momentum of a system will not change
  unless an external torque is applied to the
  system.
• LB LA (momentum before momentum after)

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Practice Problem

• A figure skater is spinning at angular velocity
  wo. He brings his arms and legs closer to his
  body and reduces his rotational inertia to ½ its
  original value. What happens to his angular
  velocity?

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Practice Problem

• A planet of mass m revolves around a star of mass
  M in a highly elliptical orbit. At point A, the
  planet is 3 times farther away from the star than
  it is at point B. How does the speed v of the
  planet at point A compare to the speed at point B?

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Demonstrations

• Bicycle wheel demonstrations
• Gyroscope demonstrations
• Top demonstration

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Precession
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Practice Problem

• A 50.0 kg child runs toward a 150-kg
  merry-go-round of radius 1.5 m, and jumps aboard
  such that the childs velocity prior to landing
  is 3.0 m/s directed tangent to the circumference
  of the merry-go-round. What will be the angular
  velocity of the merry-go-round if the child lands
  right on its edge?

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Angular momentum and torque

• In translational systems, remember that Newtons
  2nd Law can be written in terms of momentum.
• F dP/dt
• Substitute force for torque.
• Substitute angular momentum for momentum.
• t dL/dt
• t torque
• L angular momentum
• t time

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So how does torque affect angular momentum?

• If t dL/dt, then torque changes L with respect
  to time.
• Torque increases angular momentum when the two
  vectors are parallel.
• Torque decreases angular momentum when the two
  vectors are anti-parallel.
• Torque changes the direction of the angular
  momentum vector in all other situations. This
  results in what is called the precession of
  spinning tops.

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If torque and angular momentum are parallel
Consider a disk rotating as shown. In what
direction is the angular momentum?
The torque vector is parallel to the angular
momentum vector. Since t dL/dt, L will increase
with time as the rotation speeds.
L is out
t is out
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If torque and angular momentum are anti-parallel
Consider a disk rotating as shown. In what
direction is the angular momentum?
The torque vector is anti-parallel to the angular
momentum vector. Since t dL/dt, L will decrease
with time as the rotation slows.
L is in
t is out
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If the torque and angular momentum are not
aligned

• For this spinning top, angular momentum and
  torque interact in a more complex way.
• Torque changes the direction of the angular
  momentum.
• This causes the characteristic precession of a
  spinning top.

t dL/dt
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Rotation Review
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Practice Problem

• A pilot is flying a propeller plane and the
  propeller appears to be rotating clockwise as the
  pilot looks at it. The pilot makes a left turn.
  Does the plane nose up or nose down as the
  plane turns left?