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Dynamics - Momentum and Circular Motion

Newton’s Laws of Motion
  • First Law: If a body is at rest, it remains at rest or of it is in motion, it moves with a uniform velocity until it is acted on by a resultant force or torque.
  • Second Law: The rate of change of momentum of a body is proportional to the resultant force and occurs in the direction of the force; F= ma
  • Third Law: If a body A exerts a force on a body B, then body B exerts an equal but opposite force on body A, forming an action-reaction pair.
Mass and Weight

  • Mass is a measure of the amount of matter in a body, and is the property of a body which resists a change in motion.
  • Weight is the force of gravitational attraction (exerted by the Earth) on a body.
MOMENTUM
  • Linear momentum is the product of mass and velocity; p = mv
  • Force is the rate of change of momentum; F = (mvmu)/t
  • Impulse is the product of force and the time for which it acts; Ft = mvmu
  • Principle of Conservation of Linear Momentum occurs when bodies in a system interact, the total momentum remains constant provided there is no external force acting on the system;
    mAuA + mBuB = mAvA + mBvB
Elastic Collisions
  • Momentum is conserved.
  • Kinetic energy is conserved
  • For example:
    • Two identical spheres collide elastically. Initially, X is moving with speed v and Y is stationary. What happens after the collision?
    • relative velocity before collision = -(relative velocity after collision)
    • uAuB = vBvA
Inelastic Collisions
  • Relative speed of approach > Relative speed of separation
  • Momentum is conserved.
  • Perfectly Inelastic Collision: In this form of collision, only momentum is conserved, and the particles stick together after the collision, i.e. move with the same velocity.
  • In inelastic collision, energy is conserved but Ek may be converted into other forms, e.g. heat.
Collisions in Two Dimensions

  • The change in momentum (impulse) affecting each sphere acts along the line of impact.
  • The law of conservation of momentum applies along the line of impact.
  • The components of velocities of the spheres along the plane of impact remain unchanged.
CIRCULAR MOTION
  • A body moving in a circle at a constant speed changes its velocity since its direction changes. Thus, it is accelerating and hence experiences a force.
  • Centripetal Force is the resultant force acting on an object moving in a circle, it is always directed towards the center of the circle. F = mv2/r = mrω2
  • Centripetal Acceleration is derived by equating Newton’s second law and centripetal force; a = 2 or a = v2/r
Examples
  1. A horizontal flat plate is free to rotate about a vertical axis through its center. A mass M is placed on the plate, a distance d, 35 cm, from the axis of rotation. The speed of rotation is increased from zero until the mass slides off the plate. The maximum frictional force F between the plate and the mass is given by the expression:

    F = 0.72 W

    Determine the maximum number of revolutions per minute for the mass M to remain on the plate

    • The centripetal force on the particle is the frictional force so the maximum speed is when friction is at maximum:
      • Centripetal Force = Frictional Force
      • Mv2/r = 0.72 W
      • Manipulating the expression by adding ω and removing M: M(ωr)2/r = 0.72 Mg
      • ω2r = 0.72 g
    • Find the angular velocity: ω = √([0.72 × 9.81]/0.35) = 4.49 s-1
    • Find the radians covered in a minute using ratios:
      • 4.49 rad = 1 sec
      • 269.5 rad = 60 secs
    • Divide the radians covered by 2π to find revolutions:
      • 269.5/ = 42.9
      • ∴ 42 revolutions min-1
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