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 = ^{(mv – mu)}/_t
• Impulse is the product of force and the time for which it acts; Ft = mv – mu
• 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;
  m_Au_A + m_Bu_B = m_Av_A + m_Bv_B

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)
  • u_A – u_B = v_B – v_A

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 E_k 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 = rω² or a = ^v2/_r

Examples

1. 
2. 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
    • M^v2/_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/_2π = 42.9
    • ∴ 42 revolutions min^-1