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 actionreaction 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_{A}u_{A} + m_{B}u_{B} = m_{A}v_{A} + m_{B}v_{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}
 Two identical spheres collide elastically. Initially, X is moving with speed v and Y is stationary. What happens after the collision?
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} = mrω^{2}
 Centripetal Acceleration is derived by equating Newton’s second law and centripetal force; a = rω^{2} or a = ^{v2}/_{r}
Examples
 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
 ω^{2}r = 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}
 The centripetal force on the particle is the frictional force so the maximum speed is when friction is at maximum: