Chapter 8 Rotational Motion

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Chapter 8 Rotational Motion

• Glencoe Physics Principles and Problems

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(No Transcript)
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• A. Describing Rotational Motion
• 1. Angular Displacement symbol is Greek
  letter theta - ?
• ????s / r
• s arc length, along circumference
• r radius
• ? is the angle measured in RADIANS, a pure
  number with no dimensions
• 1 radian 57.3o 2? radians 360 o

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a. The Radian

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

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b. More About Radians

• (1) Comparing degrees and radians
• (2) Converting from degrees to radians

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c. Measuring Angular Displacement

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

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• 2. Angular velocity symbol is lower case Greek
  omega- ?
• a. avg???? ??????t, measured in rads/sec
• b. ???is angular displacement that occurs
• 3. Angular acceleration symbol is lower case
  Greek alpha - ?
• a. measured in radians/sec2.
• b. avg ? ?? / ?t?

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• 4. ?? v / r where v is tangential velocity
• v t r ? tangential velocity
• a t r ? tangential acceleration
• Angular Frequency
• Number of complete revolutions a rotating object
  makes in 1 second
• Formula ? 2 p f

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• 6. formulas
• ?f ?????????t
• ?????? t ????????t 2
• ?f 2????????????

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• B. Torque
• 1. The product of a force (F) and the lever arm
  (L) over which the force acts.
• a. Direction is specified as CW or CCW.
• b. Lever arm is the perpendicular distance from
  the axis of rotation to the force
• 2. Force vs. Torque
• a. Forces cause accelerations
• b.Torques cause angular accelerations
• c. Force and torque are related

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3. Torque

• a. The door is free to rotate about an axis
  through the hinge at point O
• b. There are three factors that determine the
  effectiveness of the force in opening the door
• (1) The magnitude of the force
• (2) The position of the application of the force
• (3) The angle at which the force is applied

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• c. Torque, t, is the tendency of a force to
  rotate an object about some axis
• t F r sin?
• t is the torque
• F is the force
• symbol is the Greek tau
• r sin? is the length of the lever arm acting
  perpendicular to the force where ? is the angle
  between F and r
• d. SI unit is N.m

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4. Direction of Torque

• a. Torque is a vector quantity
• (1) The direction is perpendicular to the plane
  determined by the position vector and the force
• (2) If the turning tendency of the force is
  counterclockwise, the torque will be positive
• (3) If the turning tendency is clockwise, the
  torque will be negative

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• b. Multiple Torques
• (1) When two or more torques are acting on an
  object, the torques are added as vectors
• (2) If the net torque is zero, the objects rate
  of rotation doesnt change

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• 5. General Definition of Torque
• a.The applied force is not always perpendicular
  to the position vector
• b. The component of the force perpendicular to
  the object will cause it to rotate
• c. Greater the torque - the greater is the
  rotational motion
• d. Second condition for equilibrium is for the
  sum of the torques to be equal to zero.

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• 6. Net Torque
• a. The net torque is the sum of all the torques
  produced by all the forces
• b. Remember to account for the direction of the
  tendency for rotation
• (1) Counterclockwise torques are positive
• (2) Clockwise torques are negative

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• 7. Moment of Inertia
• a. resistance to rotation
• b. symbol is I
• c. units are kg-m2
• d. different formulas used to calculate I
• (1) see page 206 in text
• (2) point mass I mr2
• e. I depends on mass, how it is distributed
  around the axis of rotation, and the axis of
  rotation chosen

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• 8. Newtons 2nd Law for Rotational Motion
• t I a
• Torque as the force like quantity causing
  acceleration
• I acts as the mass of object
• Angular acceleration is rate of change

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• C. Equilibrium
• 1. Center of Mass the point on an object that
  moves the same way a point particle would move
• a. Freely rotating objects rotate about an axis
  through their center of mass
• b. center of mass follows parabolic path of
  2-dimensional motion
• c. center of mass of humans can change with
  movement of limbs and age

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• 2. Center of Mass and Stability
• a. object stable if an external force is required
  to tip it
• b. object stable if center of mass is over
  support then torque from weight tends to keep
  it upright
• c. the higher the center of mass the easier it is
  to tip the vehicle over

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• 3. Center of Gravity
• a. The force of gravity acting on an object must
  be considered
• b. 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
  this is also known as the center of gravity of
  the object

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• c. Center of Gravity of a Uniform Object
• (1) The center of gravity of a homogenous,
  symmetric body must lie on the axis of symmetry.
• (2) Often, the center of gravity of such an
  object is the geometric center of the object.

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

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

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• 7. Rotating Frames of Reference
• a. Newtons Laws of Motion are only valid for
  inertial or non-accelerated frames of motion and
  since Earth rotates can cause effects.
• (1) effects too small to see in classroom or lab
• (2) effects play an important influence on
  climate and weather

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• b. Centrifugal Force a force that appears to
  cause objects to be flung away from the center of
  a rotating object doesnt really exist.
• (1) water in a washing machine going through spin
  cycle
• (2) spinning amusement park rides
• (3) car sliding on ice around a corner
• c. second effect of rotation is Coriolis Effect
• (1) apparent deflection in horizontal motion when
  in rotating frame of reference

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• (2) observer on ground sees straight line motion
• (3) observer on moving frame sees deflection of
  object which is due to the Coriolis force
• d. Winds and Coriolis Effect
• (1) winds go from high pressure to low pressure
  areas
• (2) in Northern hemisphere winds from south go
  the east of low pressure areas

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• (3) winds from the north go to the west of low
  pressure areas
• (4) these deflections cause winds to rotate
  counter-clockwise around low pressure areas in
  the northern hemisphere
• (5) rotate clockwise in southern hemisphere