Four Chapter 5

• An object going in a circular path at the end of a string experiences an acceleration (centripetal acceleration).
• Centripetal acceleration is perpendicular to the velocity vector and points radially inwards.
• Because of Newton's 1st law, if the string is cut, then the object will move off in a straight line.
• Acceleration (centripetal or otherwise) must be caused by a force.
• Centripetal force - The force that causes centripetal acceleration.
  • It is a push or pull.
  • It is produced like other forces.
  • In fact, it is caused by other forces like tension in a string, rods, gravity, electrical forces, magnetic forces etc.
  • They cause changes in the direction of velocity but not the magnitude.
  • They are radial forces. That is, forces directed towards the center of the circular path of the object.
• Newton's 2nd law states:

• Centripetal acceleration is given by:

• R in the above equations is radius of the path, T is period of revolution, f is frequence of the motion, v is the velocity of the object, and a_c is centripetal acceleration.
• These three equations can be substituted into Newton's 2nd law to give three equations for centripetal force (F_c). The variable m is mass.

• It is common to use a fixed frame of reference.
• The motion of an object is measured with respect to (wrt) the frame of reference.
• Galilean relativity principle - Newton's laws of motion hold equally well whether the frame of reference is moving at uniform velocity or at rest.
• Consider a ball dropped from the crows nest of a sailing ship. It lands at the base of the mast regardless of the boat's motion.
• What we observe depends upon what frame of reference is used.
  • We are sometimes fooled into thinking our bus is rolling backwards when another bus beside us moves forward. We used the other bus as our frame of reference and assumed it was stationary. When we looked out a different window, we change our frame of reference and draw different conclusions (The other bus moved forwards.)
  • A person in a bus throws a ball up and catches it. That person sees the ball go straight up and down. Someone on the street sees the ball follow a curved path.
• If there was an absolute reference frame, we could determine how all objects move.
• Absolute reference frames do not exist. Therefore the best we can do is describe how one object moves in relation to another (reference frame).
• Some quantities depend upon the frame of reference used. Examples of this are velocity and position.
• Some quantities are independent of the frame of reference used. Examples of this are force, mass and acceleration. This statement assumes that the reference frame is not accelerating.
• Note: Force, mass and acceleration relate to Newton's laws.
• Inertial frames of reference - Reference frames that move with uniform motion.
  • Newton's laws apply to these frames.
  • These frames are not accelerating.
    • Examples of inertial reference frames include:
      • A bus moving at constant speed.
      • A bus that is stopped.
    • Examples of non inertial reference frames include:
      • A bus which is accelerating.
      • A bus which is turning a corner.
• Non inertial frames of reference - Reference frames that do not move with uniform motion.
  • Objects in these frames appear to have forces on them even though they don't.
    • A person riding on a bus throws a small ball straight up. To the person's surprise, it flies forward and lands in the lap of the person infront. The reason for the unexpected observation is that the bus driver applied the brakes while the ball was in the air. The bus (the reference frame) accelerated. The ball behaved as if a force was applied to it.
  • To explain these apparent forces, we invent fictitious forces.
    • An example of a fictitious force is centrifugal force. We feel our self thrown outwards when a car goes around a corner. We say that we experienced centrifugal force. In fact, you traveled in a straight line obeying Newton's 1st law. As the car went around the corner, you traveled in a straight line wrt the ground. Wrt the car, it appeared as if you were thrown outwards.
• We consider the Earth to be an inertial frame even though it isn't
  • Because the effects of the accelerations we experience on Earth are so small, they can for the most part be ignored.
• Situations where Newton's Laws do not apply are:
  • non inertial frames.
  • frames moving at speed close to 3.00 X 10⁸ m/s (the speed of light).
• Einstein developed his special theory of relativity to explain these effects.
• Einstein's theory reduces to the same things as Newton's theory at slow speeds.
• This reduction characteristic of complicated new theories in physics is common.
• You should now try to do the lab Circular Motion.

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August 6, 2020