- Slides: 34
Newton’s Laws of Motion • three laws of motion: fundamental laws of mechanics • describe the motion of all macroscopic objects (i. e. , everyday size objects) moving at ordinary speeds (i. e. , much less than the speed of light)
Among other accomplishments, Sir Isaac Newton (1642 -1727) invented calculus, developed the laws of motion, and developed the law of gravitational attraction.
Newton's First Law of Motion (The Law of Inertia) • objects tend to remain either at rest or in uniform straight line motion (i. e. , motion with constant velocity) until acted upon by an unbalanced force • inertia: concept introduced by Galileo – an object’s tendency to resist changes in its motion – mass of an object: a measure of the amount of inertia the object has • an object with a larger mass has more inertia (i. e. , more resistance to a change in its motion) • an object with a small mass has less inertia
Top view of a person standing in the aisle of a bus. (A) The bus is at rest, and then starts to move forward. Inertia causes the person to remain in the original position, appearing to fall backward.
(B) The bus turns to the right, but inertia causes the person to retain the original straight line motion until forced in a new direction by the side of the bus.
Newton's Second Law of Motion • A net force acting on an object produces an acceleration (a change in the motion of the object) F = ma m = mass of the object a = acceleration F = net force acting on the object – the acceleration is: • directly proportional to the net force acting on the object • inversely proportional to the mass of the object
If the force of tire friction (F 1) and the force of air resistance (F 2) have a vector sum that equals the applied force (Fa), the net force is zero. Therefore, the acceleration is zero (i. e. , velocity is constant)
More mass results in less acceleration when the same force is applied. With the same force applied, the riders and the bike with twice as much mass will have half the acceleration (with all other factors constant). Note that the second rider is not pedaling.
More about: F = ma • the unit of force in the metric system is: Newton (N) 1 N = 1 kg m/s 2 • the unit of force in the English system is: pound (lb) 1 lb = 1 slug x 1 ft/s 2 (slug is the unit of mass in the English system) • The Weight of an object: – the downward pulling force of the Earth on that object (the force of gravity on the object) – is equal to the mass of an object (m) times the acceleration due to gravity (g) W = mg
A parallel between the mass and the weight of an object mass • The amount of substance (matter) contained in an object • a scalar quantity (no direction) • metric unit: kg • the same everywhere in the universe – ex. : mass of an object on the Moon is the same as on the Earth weight • The force of gravity on an object • a vector ( direction: vertically down) • metric unit: N (English unit: lb) • calculated as: W = mg (g = gravitational acceleration) • changes with location (with change in g) – on the Moon: g. Moon = 1. 6 m/s 2 – weight of an object on the Moon is about six times less than on the Earth
Newton's Third Law of Motion Whenever two objects interact, the force exerted by the first object on second is equal in size and opposite in direction to the force exerted by the second object on the first. F 1 = F 2 F 1
Circular Motion and the Centripetal Force
Centripetal force • the force that pulls an object into a circular path (centripetal means “center-seeking”) • produces an acceleration: centripetal acceleration ac = v 2/r m = mass v = velocity r = radius of the circle – Substitute ac into the force equation F = ma to get an equation for the centripetal force Fc = mv 2/r
A ball is swung on the end of a string in a horizontal circle. The pulling force of the string on the ball acts as centripetal force and causes the ball to change direction continuously, or accelerate into a circular path. Without the unbalanced force acting on it, the ball would continue in a straight line.
Newton's Law of Gravitation
Universal Law of Gravitation – Every object in the universe is attracted to every other object in the universe by a force that is directly proportional to the product of their masses and inversely proportional to the square of the distances between them. F = G(m 1 m 2)/d 2 G = 6. 67 x 10 -11 Nm 2/kg 2 (proportionality constant) – forces of attraction between everyday objects: • unnoticeably small • overshadowed by the large gravitational force of the Earth
F = G (m 1 m 2) / d 2 The force of attraction (F) is proportional to the product of the masses (m 1, m 2) and inversely proportional to the square of the distance (d) between the centers of the two masses.
Deriving the Weight of an object from the Universal Law of Gravitation • Weight: force of attraction by the Earth on an object: W = F = G (m. Em) / d 2 m. E = mass of the earth m = mass of the object d = distance from the center of the earth to the object • but the weight of an object is: W = mg • therefore: mg = G (m. Em) / d 2 which gives: g = G m. E / d 2 • Therefore: the gravitational acceleration (g) does not depend on the mass of the object
The force of gravitational attraction decreases inversely with the square of the distance from the center of the earth. Note the weight of a 70. 0 kg person at various distances above the surface of the earth.
Gravitational attraction acts as a centripetal force that keeps the Moon from following the straight-line path shown by the dashed line to position A. It was pulled to position B by gravity and thus "fell" toward Earth the distance from the dashed line to B, resulting in a somewhat circular path.
Work • the result of the force applied to an object and the distance the object moves due to the force • The work done on an object: the magnitude of the applied force multiplied by the parallel distance through which the force acts. Work = force x distance Work = Fd – Something must move when work is done. – The movement must be in the same direction as the applied force.
Examples of work Remember: work is done on an object by a force that moves the object
Work is done against gravity when lifting an object. Work is measured in joules or foot-pounds.
The force on the book moves it through the vertical distance from the second shelf to the fifth shelf, and the work done is Work = Fd.
Motion, Position, and Energy
Energy: the ability to do work Mechanical energy is of two types: • potential energy • kinetic energy Potential Energy (PE) • Gravitational potential energy is the energy that an object has due to its position in the gravitational field of the earth • PE = weight x height • Since weight is w= mg , PE = mgh where h = height
Kinetic Energy – Kinetic energy is the energy that an object has due to its motion. kinetic energy = 1/2 (mass) (velocity)2 KE = (1/2) mv 2 – Units of Kinetic energy • In the metric system unit of KE = kg (m/s)2 = kg m 2/s 2 = = (kg m/s 2) m =Nm = J unit of KE = J • In the English system unit of KE = ft-lb
The Connection between Work and Energy • Work is transfer of energy • Whenever work is done on an object by an external force, the energy of the object changes • Whenever an object does work its energy changes • Every time the change in the energy of the object is equal to the work done
(A) Work is done on the bowling ball as a force (FB) moves it through a distance. (B) This gives the ball a kinetic energy equal to the amount of work done on it. (C) The ball does work on the pins and has enough remaining energy to crash into the wall behind the pins.
• Conservation of mechanical energy – The total amount of mechanical energy of an object remains constant (in the absence of external forces) • Law of Conservation of Energy - universal • Energy is never created or destroyed. Energy can be converted from one form to another, but the total energy remains constant.
Example of energy conversion and conservation During a free fall an object : – Loses PE (the height decreases) – Gains KE (its velocity increases) PE lost = KE gained mg(Δh) = (1/2)mv 2 ; (initial velocity of the object is zero) Δh= distance through which the object falls Solving for v : v = [2 g(Δh)]1/2 This allows you to calculate the velocity of a falling object using energy conservation principles.
v=0 v=(2 g Δh)1/2 = (2 x 9. 8 m/s 2 x 5 m)1/2 = 9. 9 m/s v=(2 g Δh)1/2 =(2 x 9. 8 m/s 2 x 10 m)1/2 = 14 m/s The ball (mass = 1 kg) trades potential energy for kinetic energy as it falls. Notice that the ball had 98 J of potential energy when dropped and has a kinetic energy of 98 J just as it hits the ground (in absence of air resistance)
This pendulum bob loses potential energy (PE) and gains an equal amount of kinetic energy (KE) as it falls through as distance h. The process reverses as the bob moves up the other side of its swing.