Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

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Chapter 6,9,10

• Circular Motion, Gravitation, Rotation, Bodies in
  Equilibrium

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Circular Motion

• Ball at the end of a string revolving
• Planets around Sun
• Moon around Earth

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The Radian

• The radian is a unit of angular measure
• The radian can be defined as the arc length s
  along a circle divided by the radius r

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More About Radians

• Comparing degrees and radians
• Converting from degrees to radians

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Angular Displacement

• Axis of rotation is the center of the disk
• Need a fixed reference line
• During time t, the reference line moves through
  angle ?

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Angular Displacement, cont.

• The angular displacement is defined as the angle
  the object rotates through during some time
  interval
• The unit of angular displacement is the radian
• Each point on the object undergoes the same
  angular displacement

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Average Angular Speed

• The average angular speed, ?, of a rotating rigid
  object is the ratio of the angular displacement
  to the time interval

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Angular Speed, cont.

• The instantaneous angular speed
• Units of angular speed are radians/sec
• rad/s
• Speed will be positive if ? is increasing
  (counterclockwise)
• Speed will be negative if ? is decreasing
  (clockwise)

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Average Angular Acceleration

• The average angular acceleration of an object
  is defined as the ratio of the change in the
  angular speed to the time it takes for the object
  to undergo the change

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Angular Acceleration, cont

• Units of angular acceleration are rad/s²
• Positive angular accelerations are in the
  counterclockwise direction and negative
  accelerations are in the clockwise direction
• When a rigid object rotates about a fixed axis,
  every portion of the object has the same angular
  speed and the same angular acceleration

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Angular Acceleration, final

• The sign of the acceleration does not have to be
  the same as the sign of the angular speed
• The instantaneous angular acceleration

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Analogies Between Linear and Rotational Motion
Linear Motion with constant acc. (x,v,a)
Rotational Motion with fixed axis and constant
a (q,?,a)

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Examples

• 78 rev/min?
• A fan turns at a rate of 900 rpm
• Tangential speed of tips of 20cm long blades?
• Now the fan is uniformly accelerated to 1200 rpm
  in 20 s

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Relationship Between Angular and Linear Quantities

• Displacements
• Speeds
• Accelerations

• Every point on the rotating object has the same
  angular motion
• Every point on the rotating object does not have
  the same linear motion

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Centripetal Acceleration

• An object traveling in a circle, even though it
  moves with a constant speed, will have an
  acceleration
• The centripetal acceleration is due to the change
  in the direction of the velocity

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Centripetal Acceleration, cont.

• Centripetal refers to center-seeking
• The direction of the velocity changes
• The acceleration is directed toward the center of
  the circle of motion

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Centripetal Acceleration, final

• The magnitude of the centripetal acceleration is
  given by
• This direction is toward the center of the circle

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Centripetal Acceleration and Angular Velocity

• The angular velocity and the linear velocity are
  related (v ?R)
• The centripetal acceleration can also be related
  to the angular velocity

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Forces Causing Centripetal Acceleration

• Newtons Second Law says that the centripetal
  acceleration is accompanied by a force
• F ma ?
• F stands for any force that keeps an object
  following a circular path
• Tension in a string
• Gravity
• Force of friction

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Examples

• Ball at the end of revolving string
• Fast car rounding a curve

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More on circular Motion

• Length of circumference 2?R
• Period T (time for one complete circle)

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Example

• 200 grams mass revolving in uniform circular
  motion on an horizontal frictionless surface at 2
  revolutions/s. What is the force on the mass by
  the string (R20cm)?

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Newtons Law of Universal Gravitation

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

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Universal Gravitation, 2

• G is the constant of universal gravitational
• G 6.673 x 10-11 N m² /kg²
• This is an example of an inverse square law

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Universal Gravitation, 3

• The force that mass 1 exerts on mass 2 is equal
  and opposite to the force mass 2 exerts on mass 1
• The forces form a Newtons third law
  action-reaction

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Universal Gravitation, 4

• The gravitational force exerted by a uniform
  sphere on a particle outside the sphere is the
  same as the force exerted if the entire mass of
  the sphere were concentrated on its center

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Gravitation Constant

• Determined experimentally
• Henry Cavendish
• 1798
• The light beam and mirror serve to amplify the
  motion

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Applications of Universal Gravitation

• Weighing the Earth

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Applications of Universal Gravitation

• g will vary with altitude

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Escape Speed

• The escape speed is the speed needed for an
  object to soar off into space and not return
• For the earth, vesc is about 11.2 km/s
• Note, v is independent of the mass of the object

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Various Escape Speeds

• The escape speeds for various members of the
  solar system
• Escape speed is one factor that determines a
  planets atmosphere

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Motion of Satellites

• Consider only circular orbit
• Radius of orbit r
• Gravitational force is the centripetal force.

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Motion of Satellites

• Period ?

Keplers 3rd Law
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Communications Satellite

• A geosynchronous orbit
• Remains above the same place on the earth
• The period of the satellite will be 24 hr
• r h RE
• Still independent of the mass of the satellite

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Satellites and Weightlessness

• weighting an object in an elevator
• Elevator at rest mg
• Elevator accelerates upward m(ga)
• Elevator accelerates downward m(ga) with alt0
• Satellite a-g!!

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Force vs. Torque

• Forces cause accelerations
• Torques cause angular accelerations
• Force and torque are related

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Torque

• The door is free to rotate about an axis through
  O
• There are three factors that determine the
  effectiveness of the force in opening the door
• The magnitude of the force
• The position of the application of the force
• The angle at which the force is applied

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Torque, cont

• Torque, t, is the tendency of a force to rotate
  an object about some axis
• t is the torque
• F is the force
• symbol is the Greek tau
• l is the length of lever arm
• SI unit is N.m
• Work done by torque W??

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Direction of Torque

• If the turning tendency of the force is
  counterclockwise, the torque will be positive
• If the turning tendency is clockwise, the torque
  will be negative

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Multiple Torques

• When two or more torques are acting on an object,
  the torques are added
• If the net torque is zero, the objects rate of
  rotation doesnt change

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Torque and Equilibrium

• First Condition of Equilibrium
• The net external force must be zero
• This is a necessary, but not sufficient,
  condition to ensure that an object is in complete
  mechanical equilibrium
• This is a statement of translational equilibrium

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Torque and Equilibrium, cont

• To ensure mechanical equilibrium, you need to
  ensure rotational equilibrium as well as
  translational
• The Second Condition of Equilibrium states
• The net external torque must be zero

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Equilibrium Example

• The woman, mass m, sits on the left end of the
  see-saw
• The man, mass M, sits where the see-saw will be
  balanced
• Apply the Second Condition of Equilibrium and
  solve for the unknown distance, x

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Moment of Inertia

• The angular acceleration is inversely
  proportional to the analogy of the mass in a
  rotating system
• This mass analog is called the moment of inertia,
  I, of the object
• SI units are kg m2

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Newtons Second Law for a Rotating Object

• The angular acceleration is directly proportional
  to the net torque
• The angular acceleration is inversely
  proportional to the moment of inertia of the
  object

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More About Moment of Inertia

• There is a major difference between moment of
  inertia and mass the moment of inertia depends
  on the quantity of matter and its distribution in
  the rigid object.
• The moment of inertia also depends upon the
  location of the axis of rotation

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Moment of Inertia of a Uniform Ring

• Image the hoop is divided into a number of small
  segments, m1
• These segments are equidistant from the axis

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Other Moments of Inertia
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Example

• Wheel of radius R20 cm and I30kgm². Force
  F40N acts along the edge of the wheel.
• Angular acceleration?
• Angular speed 4s after starting from rest?
• Number of revolutions for the 4s?
• Work done on the wheel?

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Rotational Kinetic Energy

• An object rotating about some axis with an
  angular speed, ?, has rotational kinetic energy
  KEr½I?2
• Energy concepts can be useful for simplifying the
  analysis of rotational motion
• Units (rad/s)!!

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Total Energy of a System

• Conservation of Mechanical Energy
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be
  present
• Potential energies of any other conservative
  forces could be added

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Rolling down incline

• Energy conservation
• Linear velocity and angular speed are related
  vR?
• Smaller I, bigger v, faster!!

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Work-Energy in a Rotating System

• In the case where there are dissipative forces
  such as friction, use the generalized Work-Energy
  Theorem instead of Conservation of Energy
• (KEtKERPE)iW(KEtKERPE)f

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Angular Momentum

• Similarly to the relationship between force and
  momentum in a linear system, we can show the
  relationship between torque and angular momentum
• Angular momentum is defined as
• L I ?
• and

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Angular Momentum, cont

• If the net torque is zero, the angular momentum
  remains constant
• Conservation of Angular Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when

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Conservation Rules, Summary

• In an isolated system, the following quantities
  are conserved
• Mechanical energy
• Linear momentum
• Angular momentum

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Conservation of Angular Momentum, Example

• With hands and feet drawn closer to the body, the
  skaters angular speed increases
• L is conserved, I decreases, w increases

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Example

• A 500 grams uniform sphere of 7.0 cm radius spins
  at 30 rev/s on an axis through its center.
• Moment of inertia
• Rotational kinetic energy
• Angular momentum

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Example

• Find work done to open 30? a 1m wide door with a
  steady force of 0.9N at right angle to the
  surface of the door.

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Example

• A turntable is a uniform disk of metal of mass
  1.5 kg and radius 13 cm. What torque is required
  to drive the turntable so that it accelerates at
  a constant rate from 0 to 33.3 rpm in 2 seconds?

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Center of Gravity

• The force of gravity acting on an object must be
  considered
• In finding the torque produced by the force of
  gravity, all of the weight of the object can be
  considered to be concentrated at a single point

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Calculating the Center of Gravity

• The object is divided up into a large number of
  very small particles of weight (mg)
• Each particle will have a set of coordinates
  indicating its location (x,y)

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Calculating the Center of Gravity, cont.

• We wish to locate the point of application of the
  single force whose magnitude is equal to the
  weight of the object, and whose effect on the
  rotation is the same as all the individual
  particles.
• This point is called the center of gravity of the
  object

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Coordinates of the Center of Gravity

• The coordinates of the center of gravity can be
  found

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Center of Gravity of a Uniform Object

• The center of gravity of a homogenous, symmetric
  body must lie on the axis of symmetry.
• Often, the center of gravity of such an object is
  the geometric center of the object.

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Example

• Find the center of mass (gravity) of these
  masses 3kg (0,1), 2kg (0,0)
• And 1kg (2,0)

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Example

• Find the center of mass (gravity) of the
  dumbbell, 4 kg and 2 kg with a 4m long 3kg rod.

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Torque, review

• t is the torque
• F is the force
• symbol is the Greek tau
• l is the length of lever arm
• SI unit is N.m

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Direction of Torque

• If the turning tendency of the force is
  counterclockwise, the torque will be positive
• If the turning tendency is clockwise, the torque
  will be negative

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Multiple Torques

• When two or more torques are acting on an object,
  the torques are added
• If the net torque is zero, the objects rate of
  rotation doesnt change

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Example

• A 2 m by 2 m square metal plate rotates about its
  center. Calculate the torque of all five forces
  each with magnitude 50N.

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Torque and Equilibrium

• First Condition of Equilibrium
• The net external force must be zero

• The Second Condition of Equilibrium states
• The net external torque must be zero

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Example

• The system is in equilibrium. Calculate W and
  find the tension in the rope (T).

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Example

• A 160 N boy stands on a 600 N concrete beam in
  equilibrium with two end supports. If he stands
  one quarter the length from one support, what are
  the forces exerted on the beam by the two
  supports?

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