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Circular Motion: Newton's Laws in Action

Explore concepts such as uniform circular motion, centripetal force, conical pendulum, and more in this detailed study of motion in circular paths based on Newton's laws.

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Circular Motion: Newton's Laws in Action

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  1. Chapter 6 Circular Motion and Other Applications of Newton’s Laws

  2. Uniform Circular Motion • A force, Fr , is directed toward the center of the circle • This force is associated with an acceleration, ac • Applying Newton’s Second Law along the radial direction gives

  3. Uniform Circular Motion, cont • A force causing a centripetal acceleration acts toward the center of the circle • It causes a change in the direction of the velocity vector • If the force vanishes, the object would move in a straight-line path tangent to the circle

  4. Centripetal Force • The force causing the centripetal acceleration is sometimes called the centripetal force • This is not a new force, it is a new role for a force • It is a force acting in the role of a force that causes a circular motion

  5. Conical Pendulum • The object is in equilibrium in the vertical direction and undergoes uniform circular motion in the horizontal direction • v is independent of m

  6. Motion in a Horizontal Circle • The speed at which the object moves depends on the mass of the object and the tension in the cord • The centripetal force is supplied by the tension

  7. Horizontal (Flat) Curve • The force of static friction supplies the centripetal force • The maximum speed at which the car can negotiate the curve is • Note, this does not depend on the mass of the car

  8. Banked Curve • These are designed with friction equaling zero • There is a component of the normal force that supplies the centripetal force

  9. Loop-the-Loop • This is an example of a vertical circle • At the bottom of the loop (b), the upward force experienced by the object is greater than its weight

  10. Loop-the-Loop, Part 2 • At the top of the circle (c), the force exerted on the object is less than its weight

  11. Non-Uniform Circular Motion • The acceleration and force have tangential components • Fr produces the centripetal acceleration • Ft produces the tangential acceleration • SF = SFr+SFt

  12. Vertical Circle with Non-Uniform Speed • The gravitational force exerts a tangential force on the object • Look at the components of Fg • The tension at any point can be found

  13. Top and Bottom of Circle • The tension at the bottom is a maximum • The tension at the top is a minimum • If Ttop = 0, then

  14. Motion in Accelerated Frames • A fictitious force results from an accelerated frame of reference • A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force

  15. “Centrifugal” Force • From the frame of the passenger (b), a force appears to push her toward the door • From the frame of the Earth, the car applies a leftward force on the passenger • The outward force is often called a centrifugal force • It is a fictitious force due to the acceleration associated with the car’s change in direction

  16. Uniform Circular Motion • Uniform circular motion occurs when an object moves in a circular path with a constant speed • An acceleration exists since the direction of the motion is changing • This change in velocity is related to an acceleration • The velocity vector is always tangent to the path of the object

  17. Changing Velocity in Uniform Circular Motion • The change in the velocity vector is due to the change in direction • The vector diagram shows Dv = vf- vi

  18. Centripetal Acceleration • The acceleration is always perpendicular to the path of the motion • The acceleration always points toward the center of the circle of motion • This acceleration is called the centripetal acceleration

  19. Centripetal Acceleration, cont • The magnitude of the centripetal acceleration vector is given by • The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

  20. Period • The period, T, is the time required for one complete revolution • The speed of the particle would be the circumference of the circle of motion divided by the period • Therefore, the period is

  21. Tangential Acceleration • The magnitude of the velocity could also be changing • In this case, there would be a tangential acceleration

  22. Total Acceleration • The tangential acceleration causes the change in the speed of the particle • The radial acceleration comes from a change in the direction of the velocity vector

  23. Total Acceleration, equations • The tangential acceleration: • The radial acceleration: • The total acceleration: • Magnitude

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