Module 7 Rotational Mechanics

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Module 7 Rotational Mechanics

• Serway Faughn Chapters 7 8

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Module Study Objectives

• Circular motion (revisited)
• Law of gravity
• Torque
• Centre of gravity
• Rotational kinetic energy
• Angular momentum

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Angular Measure
4
Angular Speed
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Example

• A helicopter rotor has an angular speed of 320
  rpm. What is this in radians per second?

6
Angular Acceleration
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Rotation Under Constant Angular Acceleration
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Example

• A bicycle wheel experiences angular acceleration
  of 3.5 rad/s2. If the initial angular speed is
  2.0 rad/s, through what angle does the wheel
  rotate in 2.0s and what is its speed?

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Relations Between Angular and Linear Quantities
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Example

• A computer floppy discs rotates from rest to 31.4
  rad/s in 0.892s. How many rotations does it make
  coming up to speed?

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Example

• A CD is read with constant linear speed of 1.3
  m/s. What angular speeds are needed at radii of
  5.0 8.0cm and how long a track is needed for 1
  hour of play?

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Centripetal Acceleration

• In circular motionthe centripetal acceleration
  is directed inward toward the centre of the
  circle and has a magnitude given by either v2/r
  or r?2.

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Example

• A test car moves at 10 m/s around a circular road
  of radius 50m. Find centripetal acceleration and
  angular speed.

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Centripetal Forces

• Tension
• Gravity
• Friction

v
m
F
r
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Problem-solving Strategy Centripetal Forces

• Draw a diagram
• Choose coordinate system
• Find the net force towards the centre
• Solve using Fma

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Example

• A car travels 13.4 m/s on a level circular turn,
  radius 50.0m. What minimum coefficient of static
  friction between tyres and road is needed to
  prevent sliding?

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Rotating Systems

• Centrifugal force is based on an erroneous
  understanding of motion in an accelerated
  reference frame. Objects are merely obeying
  Newtons first law.

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Example

• What speed must a roller coaster car have at the
  bottom of a loop of radius 10m to reach the top?

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Newtons Law of Gravity

• Every particle in the universe attracts every
  other particle with a force that is directly
  proportional to the product of their masses and
  inversely proportional to the square of the
  distance between them

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Features of Gravitation

• Gravity is a field force independent of the
  medium separating bodies.
• Force decreases rapidly with distance.
• Proportional to the product of the masses.
• Force exerted by a spherical mass on an outside
  particle acts as if all the mass is at the
  centre.

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Dark Matter

• Star and galaxy motion indicate much more
  gravitating matter than is visible.

Orbiting galaxy
Massive galaxy dark matter halo
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Example

• Use the law of gravity to estimate the Earths
  mass.

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Gravitational Potential Energy

• Newtons law provides an exact expression for
  potential energy.
• mgh is a good approximation near the earths
  surface.

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Escape Speed

• An object requires an escape speed to leave the
  Earth or some other gravitating body
• This speed is 11 km/s for any object (eg gas
  molecule) to leave the Earth

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Keplers Laws

• All planets move in elliptical orbits with the
  Sun at one of the focal points.
• A line drawn from the Sun to any planet sweeps
  out equal areas in equal time intervals.
• The square of the orbital period of any planet is
  proportional to the cube of the average distance
  from the planet to the Sun.

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Torque

• The tendency of a force to rotate a body about
  some axis is measured by the quantity called the
  torque.

d
hinge
F
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Example

• What is the torque produced by a 300-N force
  applied at 600 to the door?

d2.0m
600
hinge
F
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Torque Equilibrium

• A system is in static equilibrium if
• Resultant external force is zero
• Resultant external torque is zero

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Centre of Gravity

• The centre of gravity is where all the mass of
  the body can be considered to be concentrated.

THIS END UP
CoG
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Problem-solving for Objects in Equilibrium

• Draw a diagram
• Show all the force vectors
• Establish a co-ordinate system
• Apply 2nd equilibrium condition (no net torque)
• Apply 1st condition (no net force) solve
  simultaneous equations

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Example

• Determine where the second mass should be for
  static equilibrium.

500N
350N
x
1.5m
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Force and Torque

• The torque for a mass in circular motion about a
  point leads to an expression similar to that of
  force.

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Moment of Inertia

• Rotational mechanics analogue of mass.
• Need to sum all contributions with respect to a
  specific rotational axis.

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Torque Angular Acceleration

• In general, the total torque on a rigid body
  rotating about a fixed axis is given by moment of
  inertia times angular acceleration.

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Rotational Kinetic Energy

• A body rotating about some axis with an angular
  speed has kinetic energy that depends on its
  moment of inertia

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Angular Momentum

• Product of moment of inertia times angular
  velocity.

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Angular Momentum Torque

• Angular momentum is conserved when the net
  external torque acting is zero.

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Problem-solving for Rotational Motion

• Steps analogous to linear motion strategy
• Analogous equations (eg ?? I? instead of ?Fma)

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Example

• In a supernova explosion, a star collapses from
  radius 106 km to a pulsar of radius 10km
  (such events do occur). Assume mass and angular
  momentum are preserved. If the star rotated about
  once a fortnight (106s), how fast does the pulsar
  spin?

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End of Module

• Any questions?