Rotational Motion by 4ZtQyQU


									 Jan    4th   Review Questions
• What is the difference between weight and mass?

• How many centimeters are in 2.35 meters?
Rotational Motion
  motion of a body about an INTERNAL axis.
          Circular Motion
• Motion of an object about an External axis.
• Riding a ferris wheel.
• Any object attached to the object in rotational

Find as many examples of Circular Motion in the
        Rotational Motion
     Axis of rotation           Distance traveled
• Is perpendicular to the   • The angle through
  side of the rotating        which the object
  object.                     travels is required.

                            • All objects no matter
                              the radius go through
                              the same angle.
              The Details
       Variables                      Units
• s = Arc Length           • Unit of length (m)
• r = Radius               • Unit of length (m)
• θ = Angle through        • Degrees or Radians
       which object moves.
  (angular displacement)

                          r       s
•   An angle whose arc length (s) is equal to its radius.
•   USUAL measurement for angular motion.
•   Most commonly communicated as multiples of π.
•   1 radian ~ 57.3°
    Problem Solve:
  Give the value of each radian
        value in degrees.
p = ___         7p = ___
 6                6
p = ___         5p = ___
 4                4
p = ___         4p = ___
 3                3
p = ___         3p = ___
 2                2
2p = ___        5p = ___
  3               3
3p = ___        7p = ___
  4               4

5p = ___       11p = ___
  6               6

  p = ___          0 = ___
           Unit Conversion
•   1 revolution = 360° = 2π radians
•   1 radian = 57.3°
•   2π= 360°
•   π = 180°

Convert 75° to radians:
         Unit Conversion
• Convert π/6 radians to degrees.

• A wheel rotates 4.5 times. How many radians does
  the wheel rotate?
 Jan     5 th   Review Questions
• Describe the difference between rotational and
  circular motion.

• Show how to convert 300 to radians.

• A 250lb man falls into a crevasse that reaches to
  the center of the earth. Assuming he lands intact,
  how much does he weigh?
   Angular Displacement
• Variable representation- θ

• Describes:
  o How much an object rotated.

• Textbook wording:
  o It is the angle through which a point, line, or body is rotated in a specified
    direction about a specified axis.

• Mathematical representation:                                       s
  o Angular displacement in radians equals the change            
    in arc length divided by the radius of the circle.               r
     + or –             Arc Length?
Notice how the values
increase in the unit

Arc Length is + when
rotation occurs

Arc Length is – When
rotation occurs
• While riding on a carousel that is rotating clockwise,
  a child travels an arc length of 11.5m. If the
  carousel has a diameter of 8.00m, through what
  angular displacement does the child travel?
  Answer is both radians and degrees.
• A nail on a wheel with radius 2.2m moves 110
  counterclockwise. How far does the nail move?
   Linear Speed Refresher
• Variable representation- 

• What is it?
   o The time rate of linear displacement.

• Mathematical representation-
   o The average linear velocity is equal to the displacement divided by the
     time required for the displacement to occur.

                                vavg   
                 Angular Speed
• Variable representation-                
• What is it?
   o   1The    time rate of angular displacement.
   o   2It is the rate at which a body rotates about an axis, usually expressed in

       radians per second, rad/s.

• Bottom Line-
   o Angular speed describes how quickly does a rotation occur.

• Mathematical representation-                                            
                                                                avg    
   o Angular speed is equal to the angular displacement
     divided by the amount of time for the angular                        t
     displacement to occur.
• A child at an ice cream parlow spins on a stool. If
  the child turns clockwise through 10 during a 10.0s
  interval, what is the angular speed of the child’s
• Before compact discs, musical recordings were
  commonly sold on vinyl discs that could be played
  on a turntable at 45 rpm or 33.3rpm. Calculate the
  corresponding angular speeds in rad/s.
        Linear Acceleration
• Variable Representation- a

• What is it?
   o The time rate of change of linear speed.
   o The rate at which the velocity changes every second.
      Angular Acceleration
• Variable representation- 

• What is it?
   o The time rate of change of angular velocity expressed in rad/s2
   o It occurs when angular speed changes.

• Mathematical representation-
   o The average angular acceleration is equal to the change in angular
     speed divided by the amount of time it took for the angular speed to

                                                     ( f  i )
            avg                             avg 
                     t                                     t
     Angular Acceleration
• May
  o Change the rate of rotation
  o Change the direction of the axis of rotation,
  o Or do both

• Fact
  o All points on a rotating rigid object have the same angular acceleration
    and angular speed.
• A car’s tire rotates at an angular speed of 21.5
  rad/s. The driver accelerates, and after 3.5 s the
  tire’s angular speed is 28.0 rad/s. What is the tire’s
  average angular acceleration during the 3.5
  second time interval?
 Jan        6th      Review Questions
• A fisherman tries to jump from his unmoving boat to
  the dock, a distance of 0.5m. Instead of landing on
  the dock, he falls into the water. What statement
  best explains what happened?
   o The force of his propelling himself out of the boat caused the boat to
     move backward, increasing the distance between boat and dock.
   o Friction held his foot in place and tripped him.
   o The force of being propelled out of the stationary boat caused him to
     miss the dock and fall.
   o The opposing force attracted the boat to the dock.

• What is the difference between a scalar and a
   Practice Kinematics for
         constant .
• The wheel on an upside-down bicycle rotates with
  a constant angular acceleration of 3.5rad/s2. If the
  initial angular speed of the wheel is 2.0 rad/s,
  through what angular displacement does the wheel
  rotate in 2.0s?
   Practice Kinematics for
         constant .
• A barrel is given a downhill rolling start of 1.5 rad/s
  at the top of a hill. Assume a constant angular
  acceleration of 2.9 rad/s2. If it takes 11.5 s to get to
  the bottom of the hill,
   o What is the final angular speed of the barrel?
   o What angular displacement does the barrel experience?
          Jan.     10th
• A racecar accelerates from 0 meters per second
  (m/s) to 72 m/s in 5 seconds. What is the average
  acceleration of the car in meters per second
  squared (m/s2)?

• What is the relationship between mass,
  acceleration, and force?
            Circular Motion
• What is circular motion?
   Motion of an object about an external axis.

• Why do we care to know the relationship

  • Angular Speed                   • Linear Speed
     o ω                                o ν

  • Angular Acceleration            • Linear Acceleration
     o α                                o a
         Circular Motion
• Brainstorm Reasons why we care about the
  relationships between ω/ν and α/a.
      Example of Purpose
• The most effective method for hitting a golf ball a
  long distance involves swinging the club in an
  approximate circle around the body. If the club
  head undergoes a large angular acceleration ()
  then the linear acceleration (a) causes the club
  head to strike the ball at a high speed and produce
  a significant force (F) on the ball.

 It’s a benefit to us to understand that we can take
   advantage of the TANGENTIAL SPEED of an object in
                      circular motion.
             Tangential Speed
• Tangent:
   o Touching a circle or sphere at only one point.

• Variable Representation:               vt
• Definition:
   o The instantaneous linear speed () of an object directed along the
     tangent line to the object’s circular path.

• Mathematical Representation:                                  vt  m
   o Tangential velocity is equal to the radius
     (center of object in rotational motion to
     the object in circular motion) multiplied    vt  r       rm
     by the angular speed.                                        rad s
          Tangential Speed
• By looking at the equation for tangential speed-
                              vt  r
   o All the velocities have the same angular displacement.
     How do the values of the velocities compare?
          o It can be seen that 3 must have a greater tangential speed to
            travel the same angular displacement () as 1.

      • Why?
          o The radius of 3 is greater.
          Practice Problems
• #1
  o If the radius of a CD is a computer is 0.0600m and the disc turns at a
    constant angular speed of 31.4 rad/s, what is the tangential speed of a
    microbe riding on the disc’s rim?
   Tangential Acceleration
• When does it occur?
   o   When angular acceleration occurs.
        • Angular acceleration occurs- If a rigid rotating object speeds up .

• What is it?
   o   The linear acceleration that is tangent to this angular acceleration.

• Variable Representation-              at
• Defined as:
   o   The instantaneous linear acceleration of an object directed along the tangent to
       the object’s circular path.
                                                                               at  m
• Mathematical Representation:
   o   Tangential acceleration equals the radius to the
                                                        at  r                rm
       object in circular motion and the angular acceleration.
                                                                                 rad
          Practice Problems
• #2
  o On a spinning ride at a carnival, the tangential acceleration of a rider
    who sits 6.5 m from the center is 3.3 m/s2. What is the angular acceleration
    of the ride?
          Practice Problems
• #3
  o A yo-yo has an angular acceleration of 280 rad/s2 when it is released. The
    string is wound around a central shaft of radius 0.35cm. What is the
    tangential acceleration of the yo-yo?
           Practice Problems
• #4
  o A vial is placed in a centrifuge. The centrifuge starts from rest and
    accelerates to 10.4 rad/s in 2.3s. If the tangential acceleration is 0.21m/s2,
    how far from the center is the vial?
Jan       11th         Review Questions
• What is the difference between static friction and
  sliding friction?

• A swimmer is moving through the water. His arms
  complete one stroke as he moves through the
  water and towards the other side of the pool.
  Which statement most correctly describes the
  movement of the swimmer.
   o (F.) The swimmer moves in the same direction as the force he applies.
   o (G.) The swimmer moves in the same direction as the water current he
   o (H.) The swimmer moves in a direction opposite to the force he applies.
   o (I.) The forces are equal, and the swimmer uses the balanced forces to
  Centripetal Acceleration
                “center seeking” acceleration

•What is it?
   o Acceleration directed toward the center of a circular path.

•Variable Representation- ac

•Mathematical Representation
       • Centripetal acceleration is equal to the tangential
         velocity squared divided by the radius.
               ac =                          ac = rw 2
       Linear Acceleration
Acceleration can be
produced by a
change in magnitude
or direction in velocity
since velocity is a

Does a car traveling in
a circle at constant
speed experience
       Centripetal Acceleration
• A car is moving around a
  circular path.
   o   Its speed may be increasing or

• The car has a component                    at
  of centripetal acceleration
  since the car is continually          ac
  changing direction.

• The car has a tangential
  component of
  acceleration, since the
  car’s speed is increasing or
   Total (Net) Acceleration
• at and ac are
  components (at right
  angles) of acceleration
  that the car is             ac
  experiencing.                         atotal

• Use the Pythagorean
  Theorem to find the total
  (net) acceleration.

     a 2
       total   = a +a
   Total (Net) Acceleration
• The direction can be             θ
  found by using         ac
  trigonometry                     atotal
       tanq =
          Practice Problems
• #5
  o A test car moves at a constant speed of 19.7m/s around a circular track. If
    the distance from the car to the center of the track is 48.2 m, what is the
    centripetal acceleration of the car?
          Practice Problems
• #6
  o A young boy swings a yo-yo horizontally above his head at an angular
    speed of 7 rev/s. If the string is 0.50m long, what is the centripetal
    acceleration of the yo-yo at the end of the string?
          Jan.     12th
• An object of unknown mass has an acceleration
  rate of 9m/s2 and a force of 630 N. What is the mass
  of the object in kilograms?

• What is kinetic energy?
Causes of Circular Motion
• A force must be present that pulls/pushes the
  object to the center of the circular path.

   o Why?
      • Because according to Newton’s 1st law objects naturally want to
        continue in a straight line path.

Inertia- the tendency of an object to continue doing
           what it is doing in a straight line path.
            Centripetal Force
• What is it?
   o The force that is directed toward the center of the circle that is needed to
     maintain circular motion.

• Variable Representation: Fc                                     Fc

• What is the unit?
   o Newton! It’s a Force!

• Real Life:
   o The centripetal force enables a car to travel in a circular path. The friction
     between the car’s tires and the road provides the centripetal force.
   o The gravitational force the earth exerts on the moon provides the
     centripetal force needed to keep the moon in orbit.
          Centripetal Force
• Mathematical Formulas:
  o Combining the following formulas….

      Fc = mac                       v   2
                                ac =     t
                                                    ac = rw   2

  o Yields…

         Fc = m(ac )
                                             Fc = m(ac )
                æ vt2 ö
         Fc = m ç ÷                          Fc = mrw   2

         Centripetal Force

• Centripetal force acts at right angles to the motion
  and causes a change in direction of the velocity
         Practice Problem
• What happens to an object in circular motion when
  the centripetal force disappears?
          Practice Problem
• A 70kg pilot is flying a small plane at 30.0 m/s in a
  circular path with a radius of 100.0m. Find the
  magnitude of the force that maintains the circular
  motion of the pilot.
          Practice Problem
• A 8.0kg ball on a string is whirled in a horizontal
  circle. The string is 1.0m long and rotates at an
  angular speed of 4.0 rad/s. What is the force that
  acts on the ball to maintain its circular motion?
         Gravitational Force
• Defined as:
   o The mutual force of attraction between particles of matter.

• When does it occur?
   o It exists between any two masses regardless of size and composition.

• Newton’s Law of Universal Gravitation is…
   o The force of attraction between two objects is directly proportional to the
     product of the masses of the objects and inversely proportional to the
     square of the distance between their centers of mass.

          Gm1 m2                G = Constant of Universal Gravitation
     Fg                        m = masses in kg
           r2                   r = distance between the centers of mass
         Gravitational Force
• Universal Constant of Gravitation
                                  Nm 2
           G  6.6731011                                                  Moon
                                  kg 2
• By Newton’s Third Law                   Earth

       FmE  FEm
• The Universal Law of Gravitation..
   o The universal law of gravitation is an inverse-square law. This means that
     the force varies as the inverse square of the separation, so that the force
     between the two masses decreases as the masses move farther apart.

           F 2
         Practice Problem
• What is the gravitational force of attraction
  between two 11.0kg masses held 3.0m apart?
          Practice Problem
• Determine the force that maintains the circular
  motion of Mercury (3.18 x 1023 kg) around the Sun
  (1.991 x 1030kg). Mercury orbits the Sun at a range of
  5.79 x 1010m.

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