Part 2a: Newton and His Laws (1642-1727) This topic will be taught largely by example problems. Initially we must define the concept of “force”: A force is that which causes an object to accelerate.

NEWTON I: (Newton’s First Law of Motion) In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion at constant velocity (in a straight line). NEWTON II: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F=ma). NEWTON III: If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force F21 exerted by object 2 on object 1 (F12 = - F21)

Applications of Newton’s Laws EXAMPLE 1 An American traffic-light weighing 125 N hangs from a cable tied to two other cables fastened to a support. The upper cables make angles of 37o and 53o with the horizontal. Find the tension in the three cables. 37o 53o T1 T2 T3 Fg x y Equilibrium !!! Resolving we get: SFx=0= -T1cos37 + T2cos53 SFy=0=T1sin37 + T2sin53 – T3 Solving yields T1=75.1N, T2=99.9N and T3=125N. Solution: We need to consider the forces acting on the two bodies involved: the traffic light and the intersection of the cables (knot). T3 = Fg since the light is held up by the tension in the rope above it.

Crate on a frictionless incline A crate of mass m is placed on a frictionless inclined plane of angle q. Determine (a) the acceleration of the crate after it is released from rest and (b) how long does it take the front edge of the crate to travel a distance d down the plane? a d θ mg mg cos θ mg sin θ n y

Solution: It is convenient to choose the surface of the plane as the x-axis. The forces acting on the crate can be seen in the figure: Thus we can see that the acceleration ax is independent of the mass: We can take this further by: The crate starts from rest so: Finally we can find the final velocity:

Friction: Static and Dynamic When a body is moving either on a surface or in a viscous medium (eg. a liquid), there is resistance to its motion due to the interaction of the body with its surroundings. This is the force of friction. If a book is at rest on a desk and we apply a force, F, then if the book does not move, then another force of equal magnitude must be acting against the applied force. This is called the force of static friction, fs. If the force is gradually increased, eventually the book will move and then the retarding force against the motion is called the force of kinetic friction, fk. If the applied force is then removed, then fk acts on the book to decelerate it; if the applied force is increased above fk, then the book will accelerate. Experimentally, these frictional forces are approximately proportional to the normal force: F, applied force f, frictional force Static Kinetic

Experimental determination of coefficients of friction We gradually increase the angle of incline until the block moves: STATIC CASE (just before the point where the block moves): (1) (2) From (2), so, substituting into (1) gives: (3) When the incline is at the critical angle θc, KINETIC CASE: When the block is accelerating, it is possible to reduce θc’ (< θc) so that it moves down the incline at constant velocity. In this case: θ mg cos θ mg sin θ mg n f y x

Coefficients of Friction ms mk Steel on steel 0.74 0.57 Aluminium on steel 0.61 0.47 Rubber on concrete 1.00 0.80 Wood on wood 0.25-0.50 0.20 Glass on glass 0.94 0.40 Waxed wood on wet snow 0.14 0.10 Ice on ice 0.10 0.03 Teflon on teflon 0.04 0.04 Synovial joints in humans 0.01 0.003