III. Newton’s Laws of Motion A.Kinematics and Mechanics B.Newton’s Laws of Motion C.Common Forces in Nature D.Problem Solving Using Newton’s Laws E.Frictional Forces F.Circular Motion

A. Kinematics and Mechanics Kinematics is concerned with the relationship between position, velocity, and acceleration of an object without reference to the origin of the motion. Mechanics is the study of the relationship between the forces experienced by an object and the motion resulting from these forces.

B. Newton’s Laws of Motion From the time of the Greeks until Galileo, the “natural state” of an object was thought to be rest, so it was assumed that there was a direct connection between the force on an object and the object’s velocity: F  v A nonzero force was thought to imply a nonzero velocity, and vice-versa. Although this idea seems to be just common sense, Galileo and Newton showed that it is wrong!

Galileo Galilei ( ) His genius lie in carrying out “thought experiments”, by which he was able to simplify physical situations and infer the important causal relationships. He realized that a moving object, which seems to slow down as a matter of necessity, would not slow down in the absence of frictional forces.

Isaac Newton ( ) A towering genius in both mathematics and physics, he was the co-inventor of differential calculus as well as the discoverer of the inverse square law of universal gravitation.

First Law: An object moves with constant velocity (or remains at rest) unless acted upon by a net external force. Also called the law of inertia, this principle assumes that the proper connection is between the force on an object and its change in velocity, or its acceleration: F  a Thus, if an object moves with a constant velocity, the net external force acting on it is ZERO!

Second Law: The acceleration of an object is in the direction of the net force it experiences and is inversely proportional to its mass, or inertia.

Third Law: All forces occur in equal and opposite pairs. If object A exerts a force on object B, then object B exerts an equal and opposite force on object A: AB This principle is important in problems involving two or more objects in contact with each other.

An important qualification: Newton’s Laws are only valid in inertial (non- accelerating) frames of reference. Thus, the accelerations and forces that we use to write down these laws must be measured by an observer in an inertial reference frame.

C. Common Forces in Nature Weight is the force with which a massive object (like the earth) attracts other objects. If an object with mass m is near the surface of the earth, it experiences a force directed toward the center of the earth with magnitude W m

C. Common Forces in Nature, contd. Two objects that are touching exert contact forces on each other. The direction of such forces is always normal (perpendicular) to the surface of contact, so they are sometimes called normal forces. F n = force of table on box F n ' = force of box on table

C. Common Forces in Nature, contd. When a 1-D spring (or similar elastic cord) is elongated or compressed from its equilibrium position by a distance  x, it exerts a restoring force (or spring force) F s which is given approximately by xx FsFs x where k is the spring constant. Note that the force is always opposite to the displacement.

D. Problem Solving Using Newton’s Laws We clearly indicate the forces acting on an object by representing the object as a point particle and drawing arrows from the point in the direction of the force. On such a free-body diagram, be sure you only draw arrows for forces! Velocity and acceleration are not forces! m mg FnFn frictionless surface

Problem Solving Steps (p.97): 1.Draw a neat diagram that includes important features. 2.Draw separate free-body diagrams for each object of interest. 3.Choose a convenient coordinate system for each object. 4.Write down Newton’s second law in component form and use Newton’s third law if you have more than one object. 5.Solve the resulting equations for desired unknown(s). 6.Check your answers for units, plausibility, and familiar limiting cases.

Example: Two accelerating blocks Two blocks, with masses m 1 and m 2, are in contact on a frictionless horizontal surface. A force F is applied to m 1, causing both blocks to accelerate along the surface. Find the force of contact between the two blocks. Note carefully the notation for the normal forces and the weights of the blocks. Also note that force F acts only on m 1 and NOT on m 2 ! m1m1 F m2m2 m2gm2g F n2 FcFc m1gm1g F n1 FFcFc

Another example: two blocks connected by a string Two blocks, with masses m 1 and m 2, are connected by a string of negligible mass on a frictionless horizontal surface. A force F is applied to m 1 at an angle  above the horizontal, causing both blocks to accelerate along the surface. Find the acceleration of the two blocks and the tension in the string, in terms of the masses, the angle  and the force F. m1m1 F m2m2  T

An inclined plane problem Two blocks with mass m and 2m are connected by a string of negligible mass over a pulley as shown. The plane is a frictionless surface inclined at an angle  from the horizontal. After the block 2m is released, find the acceleration of the system and the tension in the string, in terms of m and . 2m m 

E. Frictional Forces The two basic cases we need to understand are static friction and sliding friction. Consider a block of mass m on a surface and subject to a horizontal force F, but still at rest: m F mg FnFn F fsfs The symbol f s represents the force of static friction. We know it must be present because the net horizontal force must equal zero if the block is not accelerating.

If the force F is increased, the block will remain at rest until F is large enough to cause it to accelerate from rest. At this point the static friction force is a maximum (f s ) max. Graphically we can represent this: f (f s ) max F fkfk static friction kinetic friction f k = force of kinetic friction

What does the frictional force depend on? Experiment shows the same frictional force in both cases. Thus, f k does not depend on contact area. m F fkfk m F fkfk 2m F 2f k Experiment also shows that as the block mass increases, the frictional force increases proportionally. Thus, f k depends linearly on normal force between block and surface.

Approximate quantitative relationships  k = coefficient of kinetic friction (approximately a constant, independent of velocity)  s = coefficient of static friction This equation only holds when the object is on the verge of sliding (  s >  k )

F. Circular Motion Consider a particle of mass m moving with constant speed in a circular path of radius r. By using a simple vector diagram we can calculate its average acceleration over a time interval  t: Since these triangles are similar, and the magnitudes

To find the acceleration: centripetal acceleration (center-seeking)

Newton’s Second Law in Circular Motion: an example A car of mass m drives over a hill of height h whose shape near the top approximates a circular arc. How fast can the car be moving and stay in contact with the road at all times? h v max = ?

Circular Motion with Frictional Forces: an example An automobile has mass m = 1000 kg, and the coefficient of friction  s between its tires and a dry road surface is approximately 1.0. Find the maximum speed at which it can safely (i.e., without sliding) navigate a circular curve with radius of curvature 25 m. r v m top view of path side view r center of circle v into the screen

Remember: Problems worthy of attack Prove their worth by hitting back --Piet Hein