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Circular Motion Terms The point or line that is the center of the circle is the axis of rotation. If the axis of rotation is inside the object, the object is rotating (spinning). If the axis of rotation is outside the object, the object is revolving.

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Linear/Tangential Velocity Objects moving in a circle still have a linear velocity = distance/time. This is often called tangential velocity, since the direction of the linear velocity is tangent to the circle. v

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Rotational/Angular Velocity Objects moving in a circle also have a rotational or angular velocity, which is the rate angular position changes. Rotational velocity is measured in degrees/second, rotations/minute (rpm), etc. Common symbol, (Greek letter omega)

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Rotational/Angular Velocity Rotational velocity = Change in angle time

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Rotational & Linear Velocity If an object is rotating: All points on the object have the same rotational (angular) velocity. All points on the object do not have the same linear (tangential) velocity.

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Rotational & Linear Velocity Linear velocity of a point depends on: The rotational velocity of the point. More rotational velocity means more linear velocity. The distance from the point to the axis of rotation. More distance from the axis means more linear velocity.

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Rotational & Linear Velocity In symbols: v = r v r

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Acceleration As an object moves around a circle, its direction of motion is constantly changing. Therefore its velocity is changing. Therefore an object moving in a circle is constantly accelerating.

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Centripetal Acceleration The acceleration of an object moving in a circle points toward the center of the circle. This is called a centripetal (center pointing) acceleration. a

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Centripetal Acceleration The centripetal acceleration depends on: The speed of the object. The radius of the circle. A cent = v2v2 r

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Centripetal Force Newton’s Second Law says that if an object is accelerating, there must be a net force on it. For an object moving in a circle, this is called the centripetal force. The centripetal force points toward the center of the circle.

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Centripetal Force In order to make an object revolve about an axis, the net force on the object must pull it toward the center of the circle. This force is called a centripetal (center seeking) force. F net

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Centripetal Force Centripetal force on an object depends on: The object’s mass - more mass means more force. The object’s speed - more speed means more force. And…

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Centripetal Force The centripetal force on an object also depends on: The object’s distance from the axis (radius). If linear velocity is held constant, more distance requires less force. If rotational velocity is held constant, more distance requires more force.

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Centripetal Force In symbols: F cent = mv 2 r = mr 2

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Work Done by the Centripetal Force Since the centripetal force on an object is always perpendicular to the object’s velocity, the centripetal force never does work on the object - no energy is transformed. v F cent

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“Centrifugal Force” “Centrifugal force” is a fictitious force - it is not an interaction between 2 objects, and therefore not a real force. Nothing pulls an object away from the center of the circle.

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“Centrifugal Force” What is erroneously attributed to “centrifugal force” is actually the action of the object’s inertia - whatever velocity it has (speed + direction) it wants to keep.

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Rotational Motion of an Object

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Rotational Motion All Spinning Objects Axis of Rotation The line about which everything rotates. Speed of Rotation Period of rotation The time of a single complete rotation (T) Frequency of rotation The number of cycles completed in a given time (f = hertz) Period = 1/Frequency or Frequency = 1/Period T= 1/ff=1/T

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Rotational Motion

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Spinning Angular speed = s/r ( is measured in radians) For 360 degreed, s = 2 r 360 0 = 2 radians Angular speed = 2 f

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Spinning Angular speed = s/r ( is measured in radians) For 360 degreed, s = 2 r 360 0 = 2 radians Angular speed = 2 f

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Torque

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Angular Momentum Moment of inertia In general, the farther away a mass is from the axis the greater its moment of inertia is. I = kmr 2

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Angular Momentum (Cont.) Momentum of inertia times angular speed L = I Conservation of Angular momentum Direction of Rotation The right hand rule

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Moment of Inertia Momentum of inertia equals the resistance to motion I = mr 2 Moment of Inertia = mass times the distance from the axis squared

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Angular acceleration

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Center of Gravity Center of Mass "All of science is nothing more than the refinement of everyday thinking." -- Albert Einstein

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Center of Gravity Point of an object located at the average position of weight.

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Center of Gravity Point of an object located at the average position of weight.

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Center of Gravity Point of an object located at the average position of weight.

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Center of Mass The Average position of matter

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Center of Mass The Average position of matter

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Center of Mass The Average position of matter

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Center of Mass The Average position of matter

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Toppling Toppling occurs when the center of gravity extends beyond the support base.

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Stability Unstable – CG is lowered with displacement Stable – work must be done to raise the CG Neutral – displacement neither raises or lowers the CG

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Coriolis “force” An apparent force that seems to deflect a moving object from its path Only observed in rotating references Related to Centifrugal “force”

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Coriolis “force” An apparent force that seems to deflect a moving object from its path Only observed in rotating references Related to Centifrugal “force”

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Coriolis “force” An apparent force that seems to deflect a moving object from its path Only observed in rotating references Related to Centifrugal “force”

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Objects rotate around their center of gravity.

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Center of Gravity

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Throw a ball through the air and it travels a smooth parabolic path. Throw a bat through the air and it wobbles all over the place (class example: marker). However if you watch the path of the bat, the middle of it follows the same path that the ball followed. The bat is a sum of two motions. A spin around the center point A movement through the air as if all the weight were concentrated at this point.

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The Center of Gravity for an object is the point located at the object’s average position of weight. For a symmetrical object this point would be located at the center For an irregular shaped object, the center of gravity is toward the heavier end ( i.e. bat)

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Center of Mass Center of Gravity is often called Center of Mass, which is the average positions of all the particles of mass that make up an object. The Center of mass or center of gravity can lie outside of the object (i.e. Donut, tire, banana, chair Finding center of mass for a 1-D situation. We can use the equation: X cm = (m 1 x 1 + m 2 x 2 + …) / (m 1 + m 2 + …) 2-D is easy to follow the same trend, but use Y cm as well.

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Locating the Center of Gravity Using a plumb line and bob, you can suspend the object from some other point and constructing a second vertical line. The Center of Gravity is where the two lines intersect.

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Toppling The rule for Toppling: If the center of gravity of an object is above the area of support, the object will remain upright. If the Center of Gravity extends outside the area of support, the object will topple. Example: When a male tries to push a penny with his nose on the floor. The center of gravity extends beyond the supports and he will fall over. The Leaning Tower of Pisa does not topple over, WHY??

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Stability We say that an object balanced so that any displacement lowers its center of gravity is in Unstable Equilibrium. An example would be a cone that was point down, if it is moved, it’s center of gravity would lower and it would then topple. We say an object that is balanced so that any displacement raises its center of gravity is in Stable Equilibrium. An example would be a cone that was point up. Any movement would cause the center of gravity to rise up. So that would need to be overcome before toppling can happen.

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Place the cone on its side and its center of gravity is neither raised nor lowered with displacement. This is called Neutral Equilibrium. A book that is standing is at stable Equilibrium and so is a book laying flat. Which one is more stable and why? Why does a tightrope walker use a long poll that bends downward?

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Center of Gravity of People What happens when we touch our toes? Isn’t it true that we push our but back to touch our toes? Why? When we stand our center of gravity is generally a few cm’s below our navel. Women are typically lower than men. What happens when we stand against the wall and then try to lean forward and touch our toes?

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The End

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Chapter 7 Rotational Motion and the Law of Gravity.

Chapter 7 Rotational Motion and the Law of Gravity.

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