Knowledge-11 CM

• An object is in uniform circular motion if it is moving in a circle at a constant speed.
• Period T = time taken to complete one rotation in seconds.
• Frequency f = number of revolutions per second in hertz.
• Period and frequency are linked by the equation:

  • f = 1/t
• Radius r = distance from the object to the centre of circle in metres.
• Distance travelled in one revolution in meters is:

  • 2pie x radius
• linear speed v = distance travelled in per second in ms^-1. :

  • v = circumference / time period
  • v = 2 pie r / t
• The direction of velocity is at any instant is tangent to the circle.
• Angular speed w = angle turned through per second in radians per second.
• Angular speed equation:

  • w = angle / t = 2 pie/T = 2pie f
• magnitude of the angular speed is related to the linear speed by:

  • w = v/r
• An object in uniform circular motion is always accelerating even though it is travelling at a steady speed because it is constantly changing direction so its constantly changing velocity.
• The acceleration in uniform circular motion is called the centripetal acceleration:

  • a = v^2 / r = angular speed^2 x r
• if an object is accelerating in circular motion it must have a resultant force acting on it. this is called the centripetal force and always acts towards the centre of a circle:

  • F = mv^2 / r = m x angular speed^2 x r
• Centripetal force is not a type of force but the resultant of the forces acting on an object in circular motion.
• For a vehicle going around a bend:

  the centripetal force is friction between the cars tires and the surface.
• For a vehicle going over a hill or curved bridge:

  the centripetal force is the resultant of the weight mg and the support force from the road S:

  • mv^2 / r = mg-S
  • if a vehicle is going too fast, it will lose contact with the road at which mv^2 / r = mg
  • to find maximum speed at which a vehicle is still in contact with surface you use : v = root ( gr )
• For a vehicle on a banked track:

  the centripetal force is horizontal component of the normal reaction to the surface:

  • mv^2 / r = Nh = Nsin()
  • since mg= Ncos() it can be shown that for a vehicle on a banked track : v^2 = gr tan()
• For an object suspended on a string:

  at the top of a circle:

  • centripetal force mv^2 / r = T +mg = T=mv^2/r -mg
• For an object suspended on a string:

  at the bottom of a circle:

  • centripetal force mv^2 / r = T - mg = T=mv^2/r + mg
• For an object supported by a surface:

  for example a person on a rollercoaster going around a vertical loop.

  At the top of a lop:

  • centripetal force mv^2 / r = N +mg
  • N = mv^2 / - mg
• For an object supported by a surface:

  for example a person on a rollercoaster going around a vertical loop.

  At the bottom of a loop:

  • centripetal force mv^2 / r = N - mg
  • N = mv^2 / + mg