Chapter 4 The Laws of Motion
Classical Mechanics Describes the relationship between the motion of objects in our everyday world and the forces acting on them Conditions when Classical Mechanics does not apply very tiny objects (< atomic sizes) objects moving near the speed of light
Forces Usually think of a force as a push or pull Vector quantity May be a contact force or a field force Contact forces result from physical contact between two objects Field forces act between disconnected objects Also called action at a distance
Contact and Field Forces
Fundamental Forces Types Strong nuclear force Electromagnetic force Weak nuclear force Gravity Characteristics All field forces Listed in order of decreasing strength Only gravity and electromagnetic in mechanics
External and Internal Forces External force Any force that results from the interaction between the object and its environment Internal forces Forces that originate within the object itself They cannot change the object s velocity
Net Force Net force is the combination of all forces that change an object s state of motion. example: If you pull on a box with 10 N and a friend pulls oppositely with 5 N, the net force is 5N in the direction you are pulling.
Net Force Vector quantity a quantity whose description requires both magnitude (how much) and direction (which way) can be represented by arrows drawn to scale, called vectors length of arrow represents magnitude and arrowhead shows direction examples: force, velocity, acceleration
The Equilibrium Rule The equilibrium rule the vector sum of forces acting on a non-accelerating object equals zero in equation form: F = 0
The Equilibrium Rule example: a string holding up a bag of flour two forces act on the bag of flour: tension force acts upward weight acts downward direction equal in magnitude and opposite in when added, cancel to zero bag of flour remains at rest
Inertia Is the tendency of an object to continue in its original motion
Galileo (1564 1642) Inertia
Galileo s Concept of Inertia Italian scientist Galileo demolished Aristotle s assertions in early 1500s. Galileo s discovery objects of different weight fall to the ground at the same time in the absence of air resistance a moving object needs no force to keep it moving in the absence of friction
Galileo s Concept of Inertia Force is a push or a pull Inertia is a property of matter to resist changes in motion depends on the amount of matter in an object (its mass)
Galileo s Inclinded Plane The use of inclined planes for Galileo s experiments helped him to discover the property called inertia. Intertia is a property of matter.
Mass A Measure of Inertia Mass a measure of the inertia of a material object independent of gravity greater inertia greater mass unit of measurement is the kilogram (kg) Weight the force on an object due to gravity scientific unit of force is the Newton (N) unit is also the pound (lb)
Mass A Measure of Inertia Mass and weight in everyday conversation are interchangeable. Mass, however, is different and more fundamental than weight. Mass versus weight on Moon and Earth weight of an object on Moon is less than on Earth (1/6 th ) mass of an object is the same in both locations
Mass A Measure of Inertia One Kilogram Weighs 9.8 Newtons Relationship between kilograms and pounds 1 kg = 2.2 lb = 9.8 N at Earth s surface 1 lb = 4.45 N
Mass A measure of the resistance of an object to changes in its motion due to a force Scalar quantity SI units are kg
Sir Isaac Newton 1642 1727 Formulated basic concepts and laws of mechanics Universal Gravitation Calculus Light and optics Isaac my man
Newton s First Law An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force The net force is defined as the vector sum of all the external forces exerted on the object
Newton s Second Law The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. F and a are both vectors Can also be applied three-dimensionally
Units of Force SI unit of force is a Newton (N) 1N kg m 1 2 s US Customary unit of force is a pound (lb) 1 N = 0.225 lb
Problem 4.1 A 6.0-kg object undergoes an acceleration of 2.0 m/s 2. (a) What is the magnitude of the resultant force acting on it? (b) If this same force is applied to a 4.0-kg object, what acceleration is produced?
Problem 4.12 The force exerted by the wind on the sails of a sailboat is 390 N north. The water exerts a force of 180 N east. If the boat (including its crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?
Problem 4.19 A 2 000-kg car is slowed down uniformly from 20.0 m/s to 5.00 m/s in 4.00 s. (a) What average force acted on the car during that time, and (b) how far did the car travel during that time?
Gravitational Force Mutual force of attraction between any two objects Expressed by Newton s Law of Universal Gravitation: m1 m Fg G 2 r 2
Weight The magnitude of the gravitational force acting on an object of mass m near the Earth s surface is called the weight w of the object w = m g is a special case of Newton s Second Law g is the acceleration due to gravity g can also be found from the Law of Universal Gravitation
More about weight Weight is not an inherent property of an object mass is an inherent property Weight depends upon location
Newton s Third Law If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1. F F 12 21 Equivalent to saying a single isolated force cannot exist
Newton s Third Law cont. F 12 may be called the action force and F 21 the reaction force Actually, either force can be the action or the reaction force The action and reaction forces act on different objects
Some Action-Reaction Pairs n and n' n is the normal force, the force the table exerts on the TV n is always perpendicular to the surface n' is the reaction the TV on the table n n'
More Action-Reaction pairs F g F g and is the force the Earth exerts on the object ' F g is the force the object exerts on the earth Fg F ' g F ' g
Forces Acting on an Object Newton s Law uses the forces acting on an object n and F are g acting on the object ' n' and Fare g acting on other objects
Applications of Newton s Laws Assumptions Objects behave as particles can ignore rotational motion (for now) Masses of strings or ropes are negligible Interested only in the forces acting on the object can neglect reaction forces
Free Body Diagram Must identify all the forces acting on the object of interest Choose an appropriate coordinate system If the free body diagram is incorrect, the solution will likely be incorrect
Free Body Diagram, Example The force is the tension acting on the box The tension is the same at all points along the rope n and Fg are the forces exerted by the earth and the ground
Free Body Diagram, final Only forces acting directly on the object are included in the free body diagram Reaction forces act on other objects and so are not included The reaction forces do not directly influence the object s motion
Solving Newton s Second Law Problems Read the problem at least once Draw a picture of the system Identify the object of primary interest Indicate forces with arrows Label each force Use labels that bring to mind the physical quantity involved
Solving Newton s Second Law Problems Draw a free body diagram If additional objects are involved, draw separate free body diagrams for each object Choose a convenient coordinate system for each object Apply Newton s Second Law The x- and y-components should be taken from the vector equation and written separately Solve for the unknown(s)
Equilibrium An object either at rest or moving with a constant velocity is said to be in equilibrium The net force acting on the object is zero (since the acceleration is zero) F 0
Equilibrium cont. Easier to work with the equation in terms of its components: F 0 and F 0 x This could be extended to three dimensions y
Equilibrium Example Free Body Diagrams
Problem 4.45 (a) What is the resultant force exerted by the two cables supporting the traffic light in Figure P4.45? (b) What is the weight of the light?
Problem 4.26 (a) An elevator of mass m moving upward has two forces acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, which is greater, T or w? (b) When the elevator is moving at a constant velocity upward, which is greater, T or w? (c) When the elevator is moving upward, but the acceleration is downward, which is greater, T or w? (d) Let the elevator have a mass of 1 500 kg and an upward acceleration of 2.5 m/s 2. Find T. Is your answer consistent with the answer to part (a)? (e) The elevator of part (d) now moves with a constant upward velocity of 10 m/s. Find T. Is your answer consistent with your answer to part (b)? (f) Having initially moved upward with a constant velocity, the elevator begins to accelerate downward at 1.50 m/s 2. Find T. Is your answer consistent with your answer to part (c)?
Inclined Planes Choose the coordinate system with x along the incline and y perpendicular to the incline Replace the force of gravity with its components
Multiple Objects Example When you have more than one object, the problem-solving strategy is applied to each object Draw free body diagrams for each object Apply Newton s Laws to each object Solve the equations
Multiple Objects Example, cont.
Problem 4.20 Two packing crates of masses 10.0 kg and 5.00 kg are connected by a light string that passes over a frictionless pulley as in Figure P4.20. The 5.00-kg crate lies on a smooth incline of angle 40.0. Find the acceleration of the 5.00-kg crate and the tension in the string.
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 Solve the simultaneous equations
More About Connected Objects Treating the system as one object allows an alternative method or a check Use only external forces Not the tension it s internal The mass is the mass of the system Doesn t tell you anything about any internal forces
Problem 4.52 Three objects are connected by light strings as shown in Figure P4.52. The string connecting the 4.00-kg object and the 5.00-kg object passes over a light frictionless pulley. Determine (a) the acceleration of each object and (b) the tension in the two strings.
Forces of Friction When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion This is due to the interactions between the object and its environment This is resistance is called friction
More About Friction Friction is proportional to the normal force The force of static friction is generally greater than the force of kinetic friction The coefficient of friction (µ) depends on the surfaces in contact The direction of the frictional force is opposite the direction of motion The coefficients of friction are nearly independent of the area of contact
Static Friction, ƒ s Static friction acts to keep the object from moving If F increases, so does ƒ s If F decreases, so does ƒ s ƒ s µ n
Kinetic Friction, ƒ k The force of kinetic friction acts when the object is in motion ƒ k = µ n Variations of the coefficient with speed will be ignored
Block on a Ramp, Example Axes are rotated as usual on an incline The direction of impending motion would be down the plane Friction acts up the plane Opposes the motion Apply Newton s Laws and solve equations
Problem 4.41 Find the acceleration reached by each of the two objects shown in Figure P4.41 if the coefficient of kinetic friction between the 7.00-kg object and the plane is 0.250.
Problem 4.48 (a) What is the minimum force of friction required to hold the system of Figure P4.48 in equilibrium? (b) What coefficient of static friction between the 100-N block and the table ensures equilibrium? (c) If the coefficient of kinetic friction between the 100-N block and the table is 0.250, what hanging weight should replace the 50.0-N weight to allow the system to move at a constant speed once it is set in motion?
Friction of Air When acceleration of fall is less than g non-free fall occurs when air resistance is non-negligible depends on two things: speed and frontal surface area
Friction of Air example: A skydiver jumps from plane. Weight is the only force until air resistance acts. As falling speed increases, air resistance on diver builds up, net force is reduced, and acceleration becomes less. When air resistance equals the diver s weight, net force is zero and acceleration terminates. Diver reaches terminal velocity, then continues the fall at constant speed.
Friction of Air Terminal speed occurs when acceleration terminates (when air resistance equals weight and net force is zero) Terminal velocity same as terminal speed, with direction implied or specified
Newton s Second Law of Motion CHECK YOUR NEIGHBOR When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is A. less than g. B. more than g. C. g. D. terminated.
Newton s Second Law of Motion CHECK YOUR ANSWER When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is A. less than g. B. more than g. C. g. D. terminated. Comment: Acceleration of a nonfree-fall is always less than g. Acceleration will actually be (20 N 5 N)/2 kg = 7.5 m/s 2.
Newton s Second Law of Motion CHECK YOUR NEIGHBOR If a 50-N person is to fall at terminal speed, the air resistance needed is A. less than 50 N. B. 50 N. C. more than 50 N. D. none of the above
Newton s Second Law of Motion CHECK YOUR ANSWER If a 50-N person is to fall at terminal speed, the air resistance needed is A. less than 50 N. B. 50 N. C. more than 50 N. D. none of the above Explanation: Then, F = 0 and acceleration = 0.
Newton s Second Law of Motion CHECK YOUR NEIGHBOR As the skydiver falls faster and faster through the air, air resistance A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion CHECK YOUR ANSWER As the skydiver falls faster and faster through the air, air resistance A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion CHECK YOUR NEIGHBOR As the skydiver continues to fall faster and faster through the air, net force A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion CHECK YOUR ANSWER As the skydiver continues to fall faster and faster through the air, net force A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion CHECK YOUR NEIGHBOR As the skydiver continues to fall faster and faster through the air, her acceleration A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion CHECK YOUR ANSWER As the skydiver continues to fall faster and faster through the air, her acceleration A. increases. B. decreases. C. remains the same. D. not enough information
Newton s Second Law of Motion A situation to ponder Consider a heavy and light person jumping together with the samesize parachutes from the same altitude.
A situation to ponder CHECK YOUR NEIGHBOR Who will reach the ground first? A. the light person B. the heavy person C. both will reach at the same time D. not enough information
A situation to ponder CHECK YOUR ANSWER Who will reach the ground first? A. the light person B. the heavy person C. both will reach at the same time D. not enough information Explanation: The heavier person has a greater terminal velocity. Do you know why?
Newton s Second Law of Motion Coin and feather fall feather reaches terminal velocity very quickly and falls slowly at constant speed, reaching the bottom after the coin does coin falls very quickly and air resistance doesn t build up to its weight over short-falling distances, which is why the coin hits the bottom much sooner than the falling feather
Air Resistance How to be a flying squirrel: Jeb Corliss "Grinding the Crack"
Air Resistance Felix Baumgartner: Record high sky dive 24 miles high 725 mph
Backup
Problem 4.42 A 2.00-kg block is held in equilibrium on an incline of angle θ = 60.0 by a horizontal force applied in the direction shown in Figure P4.42. If the coefficient of static friction between block and incline is μ s = 0.300, determine (a) the minimum value of and (b) the normal force exerted by the incline on the block.
Forces on an Elevator Tension T Force of gravity Force of elevator acceleration T ma T mg mg ma