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Circular Motion

r. F c , a c. v. Circular Motion. Circular motion is very similar to linear motion in many ways. Linear. Angular. Analogies Between Linear and Rotational Motion. There are two types of circular motion. An axis is the straight line around which rotation takes place.

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Circular Motion

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  1. r Fc, ac v Circular Motion

  2. Circular motion is very similar to linear motion in many ways. Linear Angular

  3. Analogies Between Linear and Rotational Motion

  4. There are two types of circular motion An axis is the straight line around which rotation takes place. • When an object turns about an internal axis—that is, an axis located within the body of the object—the motion is called rotation, or spin. • When an object turns about an external axis, the motion is called revolution.

  5. Centripetal acceleration – acceleration of an object in circular motion. It is directed toward the center of the circular path. ac = centripetal acceleration, m/s2 v = tangential speed, m/s r = radius, m

  6. Centripetal Force – the net inward force that maintains the circular motion of an object. It is directed toward the center. Fc = centripetal force, N m = mass, kg ac = centripetal acceleration, m/s2 v = tangential speed, m/s r = radius, m

  7. Two types of speed • Linear speed is the distance traveled per unit of time. • A point on the outer edge of the turntable travels a greater distance in one rotation than a point near the center. • The linear speed is greater on the outer edge of a rotating object than it is closer to the axis. • The speed of something moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle.

  8. Tangential Speed (linear) Tangential speed depends on two things 1. rotational speed 2. the distance from the axis of rotation.

  9. Faster? Which part of the turntable moves faster—the outer part where the ladybug sits or a part near the orange center? ***It depends on whether you are talking about linear speed or rotational speed.***

  10. Rotational speed (sometimes called angular speed) is the number of rotations per unit of time. • All parts of the rigid turntable rotate about the axis in the same amount of time. • All parts have the same rate of rotation, or the same number of rotations per unit of time. • It is common to express rotational speed in revolutions per minute (RPM).

  11. Rotational Speed • All parts of the turntable rotate at the same rotational speed. • A point farther away from the center travels a longer path in the same time and therefore has a greater tangential speed. (Linear Speed)

  12. Rotational Speed Remember All parts of the turntable rotate at the same rotational speed. A point farther away from the center travels a longer path in the same time and therefore has a greater tangential speed.(linear speed) Therefore, a ladybug sitting twice as far from the center moves twice as fast.

  13. Question # 1 At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed? Answer: Her rotational speed is also 4 RPM, and her linear speed is 3 m/s.

  14. Calculating Average Speed • An object moving in uniform circular motion would cover the same linear distance in each second of time. • When moving in a circle, an object travels a distance around the perimeter of the circle. • The distance of one complete cycle around the perimeter of a circle is known as the circumference. The circumference of any circle is Circumference = 2*pi*Radius

  15. v = tangential speed, m/s d = distance, m t = time, s r = radius, m T = period, s (time for 1 rev.) For a constant tangential speed: If rpm (revolutions per minute) is given, convert to m/s using these conversion factors: and 1 min. = 60 sec. Or you can find the period by taking the inverse of the frequency. T = period, s – time for one revolution F = frequency, rev/s – number of revolutions per time Note: Period and frequency are inverses.

  16. Constant Speed, but is there constant Velocity? • Remember speed is a scalar quantity and velocity is a vector quantity. • The direction of the velocity vector is directed in the same direction that the object moves. Since an object is moving in a circle, its direction is continuously changing. • The best word that can be used to describe the direction of the velocity vector is the word tangential.

  17. Centripetal force keeps an object in circular motion. Big IDEA….

  18. Centripetal Force The force exerted on a whirling can is toward the center. NO outward force acts on the can.

  19. Sincecentripetal force is a net force, there must be a force causing it. Some examples are • A car going around a curve on a flat road: Fc = Ff (friction force)

  20. Creates a curved path • Centripetal force holds a car in a curved path. • For the car to go around a curve, there must be sufficient friction to provide the required centripetal force. • If the force of friction is not great enough, skidding occurs.

  21. Sincecentripetal force is a net force, there must be a force causing it. Some examples are • A car going around a curve on a flat road: Fc = Ff (friction force) • Orbital motion, such as a satellite: Fc = Fg (weight or force of gravity)

  22. Sincecentripetal force is a net force, there must be a force causing it. Some examples are • A car going around a curve on a flat road: Fc = Ff (friction force) • Orbital motion, such as a satellite: Fc = Fg (weight or force of gravity) • A person going around in a spinning circular room: Fc = FN (normal force)

  23. Sincecentripetal force is a net force, there must be a force causing it. Some examples are • A car going around a curve on a flat road: Fc = Ff (friction force) • Orbital motion, such as a satellite: Fc = Fg (weight or force of gravity) • A person going around in a spinning circular room: Fc = FN (normal force) • A mass on a string (horizontal circle, i.e.. parallel to the ground): Fc = T (tension in the string)

  24. For a mass on a string moving in a vertical circle, the centripetal force is due to different forces in different locations. • At the top of the circle, Fc = T + Fg(tension plus weight or gravity) • At the bottom of the circle, Fc = T - Fg(tension minus weight or gravity) • On the outermost side, Fc = T • Anywhere other than above, you would need to find the component of gravity parallel to the tension and either add or subtract from tension depending on the location on the circular path

  25. v v v + v Bottom mg T T T mg T + T T v v Top of Path Left Side + Top Right Top Right Tension is minimum as weight helps Fc force Weight has no effect on T Maximum tension T, W opposes Fc Weight causes small decrease in tension T Weight has no effect on T Bottom + + + mg mg mg mg Example :Motion in a Vertical Circle Consider the forces on a ball attached to a string as it moves in a vertical loop. Note changes as you click the mouse to show new positions. The velocity of the object is constantly changes depending on which direction gravity is pointing compared to velocity. The tension required to keep this object moving in a circle changes while it is in it motion as well.

  26. Calculating Centripetal Forces Greater speed and greater mass require greater centripetal force. Traveling in a circular path with a smaller radius of curvature requires a greater centripetal force. Centripetal force, Fc, is measured in newtons when m is expressed in kilograms, v in meters/second, and r in meters.

  27. Adding Force Vectors • A conical pendulum is a bob held in a circular path by a string attached above. • This arrangement is called a conical pendulum because the string sweeps out a cone. • Only two forces act on the bob: mg, the force due to gravity, and T, tension in the string. • Both are vectors.

  28. Remember… • The vector T can be resolved into two perpendicular components, Tx(horizontal), and Ty(vertical). • Therefore Tymust be equal and opposite to mg. • Tx is the net force on the bob–the centripetal force. Its magnitude is mv2/r, where r is the radius of the circular path.

  29. Centripetal Force Centripetal force keeps the vehicle in a circular path as it rounds a banked curve.

  30. Centrifugal Forces – MISCONCEPTION!! • When an object moves in a circular motion there MUST be an outward force. • NO!!! • This apparent outward force on a rotating or revolving body is called centrifugal force. Centrifugal means “center-fleeing,” or “away from the center.” • If there was an outward force, we would see something completely different than what actually happens.

  31. Gravity Near the Earth’s Surface The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

  32. Satellites and “Weightlessness” Satellites are routinely put into orbit around the Earth. The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it escapes Earth’s gravity altogether.

  33. Satellites and “Weightlessness” The satellite is kept in orbit by its speed—it is continually falling, but the Earth curves from underneath it.

  34. Satellites and “Weightlessness” Objects in orbit are said to experience weightlessness. They do have a gravitational force acting on them, though! The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness.

  35. Satellites and “Weightlessness” More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:

  36. 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 Fc, v or ac may not be constant

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

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

  39. 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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