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Concept Summary Belton High School Physics

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

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Uniform Circular Motion The motion of an object in a circle traveling with uniform or constant speed. Velocity is constantly changing. In circular motion this velocity becomes a little more complex. We now have 2 types of velocity… Tangential velocityTangential velocity Angular velocityAngular velocity

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

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2 pod racers complete a turn and remain neck and neck. Which had the greater Linear Velocity? A.The one closer to the point of rotation. B.The one farther from the point of rotation. C.Both were equal.

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

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Rotational/Angular Velocity Rotational velocity = Change in angle time Notice that we don’t have to use angles! We could just as easily use Radians!

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How would you describe the angular velocity of the 2 pod racers in the previous question? A.Greater for the one closer to the point of rotation. B.Greater for the one farther from the point of rotation. C.Both the same.

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Rotational Vs. Tangential 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 & Tangential Velocity We now see that…. –Tangential (linear) velocity of a point depends on: The rotational velocity of the point. –More rotational velocity can mean more linear velocity. –The distance from the axis of rotation. More distance from the axis means more linear velocity

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Analogies Between Linear and Rotational Motion

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Relationship Between Angular and Linear Quantities DisplacementsDisplacements SpeedsSpeeds AccelerationsAccelerations Every point on the rotating object has the same angular motionEvery point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motionEvery point on the rotating object does not have the same linear motion

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Acceleration As an object moves around a circle, its direction of motion is constantly changing.As an object moves around a circle, its direction of motion is constantly changing. Therefore its velocity is changing.Therefore its velocity is changing. Therefore an object moving in a circle is constantly accelerating.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 Acceleration and Angular Velocity The angular velocity and the linear velocity are related (v = ωr)The angular velocity and the linear velocity are related (v = ωr) The centripetal acceleration can also be related to the angular velocityThe centripetal acceleration can also be related to the angular velocity

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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. Newton’s Second Law also states that this net force must point in the same direction as the acceleration. 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, increasing the radius requires decreases this force. If rotational velocity is held constant, increasing the radius increases the force.

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Centripetal Force In symbols: F= ma so…… Since a c = v 2 /r And v= ωr Then a c = ω 2 r So…. F c = mv 2 /r F c = mω 2 r

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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.“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.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.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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A note on the Vector Nature of Angular Quantities Angular displacement, velocity and acceleration are all vector quantities Direction can be more completely defined by using the right hand rule –Grasp the axis of rotation with your right hand –Wrap your fingers in the direction of rotation –Your thumb points in the direction of ω

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In example (a) the direction of angular velocity ω points into the page. A.True B.False

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If the rotational speed of a space station increases then the apparent weight of its inhabitants will increase. A.True B.False

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An astronaut decides to go jogging in his relatively small rotating space station. If wants to burn the most calories he should A.Jog in the opposite direction that the station is rotating. B.Jog in the same direction that the station is rotating. C.Run in place at the very center of the station.

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Circular motion Problem Solving Strategy Draw a free body diagram, showing and labeling all the forces acting on the object(s)Draw a free body diagram, showing and labeling all the forces acting on the object(s) Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, F C ) Apply Newton’s 2 nd. Solve for unknowns.

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Common situations involving Centripetal Acceleration Many specific situations will use forces that cause centripetal acceleration –Level curves –Banked curves –Horizontal circles –Vertical circles Note that F c, v or a c may not be constant

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Level Curves Friction is the force that produces the centripetal acceleration Can find the frictional force, µ, or v

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Banked Curves A component of the normal force adds to the frictional force to allow higher speeds

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Vertical Circle Look at the forces at the top of the circle The minimum speed at the top of the circle can be found

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The End

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Tangential and Centripetal Accelerations

Tangential and Centripetal Accelerations

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