Dr. Steve Peterson Steve.peterson@uct.ac.za Physics 1025F Mechanics NEWTON’S LAWS Dr. Steve Peterson Steve.peterson@uct.ac.za

Chapter 4: Newton’s Laws of Motion Dynamics is the description of why objects move and the connection between forces and motion

Why do objects move? What is the “natural state” of an object [Aristotle, ~350 BC]? In Motion or At Rest? What if we remove friction (the idealized case [Galileo, ~1600 AD])?

Newton’s First Law of Motion Galileo reasoned that the natural state of an object (if free of external influences) is uniform motion with a constant velocity. Consider an object with no force acting on it. If it is at rest, it will remain at rest; if it is moving, it will continue to move in a straight line at a constant speed.

Fundamental Forces Four Basic Fundamental Forces Characteristics Strong nuclear force (holds nucleus together) Electromagnetic force Weak nuclear force (radioactive decay) Gravitational force Characteristics All long-range (fields) forces Listed in order of decreasing strength Only gravity and electromagnetic in mechanics MOVE HIGHER UP IN SLIDES

Free Body Diagrams Identify the object of interest The object whose motion you want to study Draw a picture of the situation Show the object of interest and all directly-interacting objects Choose an appropriate coordinate system Name and label all the forces acting on the object of interest Contact and long-range forces If the free body diagram is incorrect, the solution will likely be incorrect

Examples: Free Body Diagrams A block is dragged uphill by a rope. Identify all forces acting on the block. Block A hangs from the ceiling by a rope. Another block B hangs from A. Identify the forces acting on A. A ball, hanging from the ceiling by a string, is pulled back and released. Identify the forces acting on it just after its release.

Newton’s First Law of Motion A body at rest will remain at rest, and a body undergoing uniform motion will remain in uniform motion, unless a net force acts on the body. This is sometimes referred to as the Law of Inertia. Inertia is the tendency of an object to maintain its state of rest or of uniform motion in a straight line.

Newton’s First Law of Motion If no net force is acting on a body it is said to be in “equilibrium”. e.g. book on table

Static Equilibrium An object is in equilibrium when the net force acting on it is zero. In component form, this is Does acceleration = 0? Object at rest or constant v?

Newton’s Second Law of Motion What happens if a net force is exerted on an object? The acceleration of a body is directly proportional to the net force acting on the body, and is inversely proportional to the mass. The direction is in the same direction as the net applied force. i.e.

Newton’s Second Law of Motion Note: The mass is the property of the body which resists acceleration, and it is referred to as “inertial mass” when measured this way. If a 1 kg mass is accelerating at 1 m/s2 it must be subjected to a force of 1 kg m/s2 or 1 N.

Free Body Diagrams Identify the object of interest The object whose motion you want to study Draw a picture of the situation Show the object of interest and all directly-interacting objects Choose an appropriate coordinate system Name and label all the forces acting on the object of interest Contact and long-range forces Draw and label the net force vector Fnet. If the free body diagram is incorrect, the solution will likely be incorrect

Example: Free Body Diagram An elevator, lifted by a cable, is going up at a steady speed. Identify the forces acting on the elevator. Is T greater than, equal to, or less than W? Or is there not enough information to tell?

Dynamics An object is in motion when the net force acting on it is not zero. In component form, this is

Example: Dynamics A 10 kg block is pulled across a frictionless horizontal floor by a rope which makes an angle of 30° to the horizontal. What is the acceleration of the block if the force exerted on it by the rope is 5 N?

Newton’s Third Law of Motion When body A exerts a force on body B, then body B will exert an equal and opposite force on A. i.e. Note: Action and reaction forces ALWAYS act on different bodies.

Example: Newton’s Third Law A book is placed on a table. Are these two forces action and reaction forces? W = Force of Earth on Book N = Force of Table on Book A. YES B. NO

Weight: The Force of Gravity A body undergoing free fall accelerated downwards under gravity with an acceleration g. From Newton’s 2nd law this means that the body must be experiencing a “force of gravity” We refer to the gravitational force as the “weight”, i.e.

Mass Mass is the measure of inertia of an object. In the SI system, mass is measured in kilograms. Mass is not weight: Mass is a property of an object. Weight is the force exerted on that object by gravity. If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same.

Apparent Weight Gravity is not a force that you can feel or sense directly. The sensation of weight (how heavy you feel) is due to the contact forces pressing against you. When you stand on a scale, the contact force is the upward spring force acting on your feet. If you and the scale are in equilibrium, the scale will read your weight. If not, it will read your “apparent weight.”

Apparent Weight Let’s define the apparent weight (Wapp) as the magnitude of the contact force (N) that supports you.

Example: Apparent Weight A 50 kg student gets in a 1000 kg elevator at rest. As the elevator begins to move, she has an apparent weight of 600 N for the first 3 s. How far has the elevator moved, and in which direction, at the end of 3 s?

Normal Force The key is understanding the normal force is that it adjusts to the force applied by the object. The atomic “springs” that make up the surface produce the normal force. The harder the objects bears down on the surface, the more the normal responds, adjusting itself so the object stays on the surface

Example: Normal Force A 10.0 kg box rests on a table. (a) Calculate the normal force. (b) A rope is now tied around the box and a 40.0 N force is applied upward, calculate the normal force. (c) What happens if a 100.0 N force is applied to the rope?

Example: Normal Force A 75 kg skier starts down a 50-m-high, 10° slope on frictionless skis. What is his speed at the bottom?

Friction Consider the application of a small force to a stationary body on a rough horizontal surface. At first the application of small force does not induce any motion. If we apply the force on the other side of the body it still does not move. (not directional dependant) We increase the applied force and the body still does not move. As we increase the applied force the body will eventually begin to move. (There is an upper limit on the frictional force) There is a maximum value beyond which the frictional force cannot increase.

Friction Experiments show that the size of the force needed to just cause movement of the body depends on: the nature of the surface (i.e. smooth / rough) the normal force exerted by the surface on the body where μs is the static coefficient of friction. Static = two surfaces are not moving relative to each other

Friction In general for the stationary situation Once the body begins to move the frictional force usually reduces so where k is the kinetic coefficient of friction and, usually, k < s. Kinetic = two surfaces are moving relative to each other

Friction The static frictional force increases as the applied force increases, until it reaches its maximum. Then the object starts to move, and the kinetic frictional force takes over.

Example: Friction A 2 kg block is pulled across a rough horizontal surface (k = 0.2) by a rope which makes an angle of 30 to the horizontal. What is the acceleration of the block if the force exerted by the rope is 5 N? EXPLAIN WHY PLUG NUMBERS IN AT END

Example: Friction A 2 kg block slides along a smooth horizontal surface at 2.4 m/s. It then encounters a rough section of the surface and travels for a further 1.5 m before it comes to rest. What is the coefficient of friction between block and table?

Interacting Objects Objects in contact will produce Newton’s third law action/reaction pairs. Solving two or more objects interacting via direct contact forces will require applying Newton’s second and third laws. Two objects moving together will experience the same acceleration. There is a maximum value beyond which the frictional force cannot increase.

Example: Interacting Objects A 2 kg and a 3 kg block are placed in contact with each other on a smooth frictionless surface. A 10 N force is then used to push the two blocks across the surface. What is the acceleration of the blocks? What force does the 2 kg block exert on the 3 kg? What would change if swapped around?

Tension When a cord or rope pulls on an object, it is said to be under tension, and the force it exerts is called a tension force. Assumptions: Ignore any frictional effects of the rope Ignore the mass of the rope The magnitude of the force exerted along the rope is called the tension and is the same at all points in the rope

Tension: Connected Objects Apply Newton’s Laws separately to each object. The magnitude of the acceleration of both objects will be the same. The tension is the same in each diagram.

Example: Connected Objects A block with mass m1 = 4.00 kg and a ball with mass m2 = 7.00 kg are connected by a light string that passes over a frictionless pulley. The coefficient of kinetic friction between the block and the surface is 0.300. Find the acceleration of the two objects and the tension in the string.