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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. 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
• 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