* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 5 - UCF College of Sciences
Survey
Document related concepts
Hunting oscillation wikipedia , lookup
Center of mass wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Jerk (physics) wikipedia , lookup
Coriolis force wikipedia , lookup
Equations of motion wikipedia , lookup
Inertial frame of reference wikipedia , lookup
Fundamental interaction wikipedia , lookup
Seismometer wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Classical mechanics wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Fictitious force wikipedia , lookup
Centrifugal force wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Classical central-force problem wikipedia , lookup
Transcript
Chapter 5 – Force and Motion I I. Newton’s first law. II. Newton’s second law. III. Particular forces: - Gravitational - Weight - Normal - Friction - Tension IV. Newton’s third law. Force • Forces are what cause any change in the velocity of an object – A force is that which causes an acceleration • The net force is the vector sum of all the forces acting on an object – Also called total force, resultant force, or unbalanced force Zero Net Force • When the net force is equal to zero: – The acceleration is equal to zero – The velocity is constant • Equilibrium occurs when the net force is equal to zero – The object, if at rest, will remain at rest – If the object is moving, it will continue to move at a constant velocity Classes of Forces • Contact forces involve physical contact between two objects • Field forces act through empty space – No physical contact is required Fundamental Forces • Gravitational force – Between two objects • Electromagnetic forces – Between two charges • Nuclear (strong) force – Between subatomic particles • Weak forces – Arise in certain radioactive decay processes • A spring can be used to calibrate the magnitude of a force • Forces are vectors, so you must use the rules for vector addition to find the net force acting on an object Newton mechanics laws cannot be applied when: 1) The speed of the interacting bodies are a fraction of the speed of light Einstein’s special theory of relativity. 2) The interacting bodies are on the scale of the atomic structure Quantum mechanics I. Newton’s first law: If no net force acts on a body, then the body’s velocity cannot change; the body cannot accelerate v = constant in magnitude and direction. Principle of superposition: when two or more forces act on a body, the net force can be obtained by adding the individual forces as vectors. Newton’s First Law • If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration – This is also called the law of inertia – It defines a special set of reference frames called inertial frames, • We call this an inertial frame of reference Inertial reference frame: where Newton’s laws hold. Inertial Frames • Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame • A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame – We can consider the Earth to be such an inertial frame although it has a small centripetal acceleration associated with its motion Newton’s First Law – Alternative Statement • In the absence of external forces, when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with a constant velocity – Newton’s First Law describes what happens in the absence of a force – Also tells us that when no force acts on an object, the acceleration of the object is zero Inertia and Mass • The tendency of an object to resist any attempt to change its velocity is called inertia • Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity Mass and Weight The weight of an object is the force of gravity on the object and may be defined as the mass times the acceleration of gravity, W = mg. Since the weight is a force, its SI unit is the newton. Density is mass/volume. Mass and Weight The mass of an object is a fundamental property of the object; a numerical measure of its inertia; a fundamental measure of the amount of matter in the object. Definitions of mass often seem circular because it is such a fundamental quantity that it is hard to define in terms of something else. All mechanical quantities can be defined in terms of mass, length, and time. The usual symbol for mass is m and its SI unit is the kilogram. While the mass is normally considered to be an unchanging property of an object, at speeds approaching the speed of light one must consider the increase in the relativistic mass. • Mass is an inherent property of an object • Mass is independent of the object’s surroundings • Mass is independent of the method used to measure it • Mass is a scalar quantity • The SI unit of mass is kg Mass vs. Weight • Mass and weight are two different quantities • Weight is equal to the magnitude of the gravitational force exerted on the object – Weight will vary with location Newton’s Second Law • When viewed from an inertial frame, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass – Force is the cause of change in motion, as measured by the acceleration • Algebraically, SF = m a II. Newton’s second law: The net force on a body is equal to the product of the body’s mass and its acceleration. Fnet ma Fnet , x max , Fnet , y may , Fnet , z maz The acceleration component along a given axis is caused only by the sum of the force components along the same axis, and not by force components along any other axis. Units of Force System: collection of bodies. External force: any force on the bodies inside the system. III. Particular forces: Gravitational: pull directed towards a second body, normally the Earth Fg mg Weight: magnitude of the upward force needed to balance the gravitational force on the body due to an astronomical body W mg Normal force: perpendicular force on a body from a surface against which the body presses. N mg Frictional force: force on a body when the body attempts to slide along a surface. It is parallel to the surface and opposite to the motion. Tension: pull on a body directed away from the body along a massless cord. Newton’s Third Law • 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 – Note on notation: FAB is the force exerted by A on B Newton’s Third Law, Alternative Statements • Forces always occur in pairs • A single isolated force cannot exist • The action force is equal in magnitude to the reaction force and opposite in direction – One of the forces is the action force, the other is the reaction force – It doesn’t matter which is considered the action and which the reaction – The action and reaction forces must act on different objects and be of the same type Action-Reaction Examples • The force F12 exerted by object 1 on object 2 is equal in magnitude and opposite in direction to F21 exerted by object 2 on object 1 • F12 = - F21 Newton’s third law: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. FBC FCB Q. A body suspended by a rope has a weigh W of 75N. Is T equal to, greater than, or less than 75N when the body is moving downward at (a) increasing speed and (b) decreasing speed? Fg 5. There are two forces on the 2 kg box in the overhead view of the figure below but only one is shown. The figure also shows the acceleration of the box. Find the second force (a) in unitvector notation and as (b) magnitude and (c) direction. F2 Rules to solve Dynamic problems - Select a reference system. - Make a drawing of the particle system. - Isolate the particles within the system. - Draw the forces that act on each of the isolated bodies. - Find the components of the forces present. - Apply Newton’s second law (F=ma) to each isolated particle. 23. An electron with a speed of 1.2x107m/s moves horizontally into a region where a constant vertical force of 4.5x10-16N acts on it. The mass of the electron is m=9.11x10-31kg. Determine the vertical distance the electron is deflected during the time it has moved 30 mm horizontally. F Fg dy v0 dx=0.03m 13. In the figure below, mblock=8.5kg and θ=30º, and the surface of incline is frictionless. Find (a) Tension in the cord. (b) Normal force acting on the block. (c) If the cord is cut, find the magnitude of the block’s acceleration. N T Fg Friction • Related to microscopic interactions of surfaces • Friction is related to force holding surfaces together • Frictional forces are different depending on whether surfaces are static or moving Kinetic Friction • When a body slides over a surface there is a force exerted by the surface on the body in the opposite direction to the motion of the body • This force is called kinetic friction • The magnitude of the force depends on the nature of the two touching surfaces • For a given pair of surfaces, the magnitude of the kinetic frictional force is proportional to the normal force exerted by the surface on the body. Kinetic Friction The kinetic frictional force can be written as F fr m k FN Where mk is a constant called the coefficient of kinetic friction The value of mk depends on the surfaces involved + + FN = mg Fp m= Ffr = mFN 20.0 kg Fg = -mg Static Friction A frictional force can arise even if the body remains stationary – A block on the floor - no frictional force FN = -mg m Fg = mg Static Friction – If someone pushes the desk (but it does not move) then a static frictional force is exerted by the floor on the desk to balance the force of the person on the desk FN = -mg Fp m Ffr = mFN Fg = mg Static Friction – If the person pushes with a greater force and still does not manage to move the block, the static frictional force still balances it FN = -mg Fp m Ffr = mFN Fg = mg Static Friction – If the person pushes hard enough, the block will move. • The maximum force of static friction has been exceeded FN = -mg Fp m Ffr = mFN Fg = mg Static Friction • The maximum force of static friction is given by F fr (max) m s FN • Static friction can take any value from zero to msFN – In other words F fr m s FN Forces on an incline • Often when solving problems involving Newton’s laws we will need to deal with resolving acceleration due to gravity on an inclined surface Forces on an incline y What normal force does the surface exert? mgsinq mgcosq W = mg q x Forces on an incline Forces on an incline F F x mg sin q y N mg cos q Forces on an incline F F x mg sin q ma y N mg cos q 0 Equilibrium Forces on an incline • If the car is just stationary on the incline what is the (max) coefficient of static friction? Fx mg sin q m s N ma 0 F y N mg cos q 0 mg sin q m s N m s mg cos q sin q ms tan q cos q ‘Equilibrium’ problems (SF = 0) ‘Equilibrium’ problems (SF = 0) 0 0 F T cos 53 T cos 37 0 x 2 1 0 0 F T sin 37 T sin 53 T3 0 y 1 2 Connected Object problems • One problem often posed is how to work out acceleration of a system of masses connected via strings and/or pulleys – for example - Two blocks are fastened to the ceiling of an elevator. Connected masses • Two 10 kg blocks are strung from an elevator roof, which is accelerating up at 2 m/s2. • Find T1 and T2 T1 m1=10kg T2 m2=10kg a 2 m/s2 Connected masses • What is the acceleration of the system below, if T is 1000 N? • What is T*? T* m2=10kg m1=1000kg a T Connected masses • θ is 250, what are T* and a now? a Connected masses • Add in friction: μ = 0.4 • What are T* and a now? T* a Motion over pulleys a • We know that the magnitude and direction of the acceleration should be the same for the whole system T a 10kg θ = 300 μ = 0.4 55. The figure below gives as a function of time t, the force component Fx that acts on a 3kg ice block, which can move only along the x axis. At t=0, the block is moving in the positive direction of the axis, with a speed of 3m/s. What are (a) its speed and (b) direction of travel at t=11s? A block of mass 3.00 kg is pushed up against a wall by a force P that makes a 50.0 angle with the horizontal as shown in Figure. The coefficient of static friction between the block and the wall is 0.250. Determine the possible values for the magnitude of P that allow the block to remain stationary. Two bodies, m1= 1kg and m2=2kg are connected over a massless pulley. The coefficient of kinetic friction between m2 and the incline is 0.1. The angle θ of the incline is 20º. Calculate: a) Acceleration of the blocks. (b) Tension of the cord. N f T m2 T m1 20º m2g m1g The three blocks in the figure below are connected by massless cords and pulleys. Data: m1=5kg, m2=3kg, m3=2kg. Assume that the incline plane is frictionless. (i) Show all the forces that act on each block. N T2 (ii) Calculate the acceleration of m1, m2, m3. T m2 Fg2x 2 m3 (iii) Calculate the tensions on the cords. (iv) Calculate the normal force acting on m2 Fg2y 30º T1 T1 m3g m2g m1 m1g 1B. (a) What should be the magnitude of F in the figure below if the body of mass m=10kg is to slide up along a frictionless incline plane with constant acceleration a=1.98 m/s2? (b) What is the y magnitude of the Normal force? N x 20º F 30º Fg Three forces, given by , F1 2.00ˆi 2.00 ˆj N , F2 5.00 ˆi 3.00 ˆj N and , F3 45.0ˆi N act on an object to give it an acceleration of magnitude 3.75 m/s2. (a) What is the direction of the acceleration? (b) What is the mass of the object? (c) If the object is initially at rest, what is its speed after 10.0 s? (d) What are the velocity components of the object after 10.0 s? A bag of cement of weight 325 N hangs from three wires as suggested in Figure. Two of the wires make angles q1 = 60.0 and q2 = 25.0 with the horizontal. If the system is in equilibrium, find the tensions T1, T2, and T3 in the wires. A block of mass 3.00 kg is pushed up against a wall by a force P that makes a 50.0 angle with the horizontal as shown in Figure. The coefficient of static friction between the block and the wall is 0.250. Determine the possible values for the magnitude of P that allow the block to remain stationary. Review problem. A block of mass m = 2.00 kg is released from rest at h = 0.500 m above the surface of a table, at the top of a 30.0 incline as shown in Figure. The frictionless incline is fixed on a table of height H = 2.00 m. (a) Determine the acceleration of the block as it slides down the incline. (b) What is the velocity of the block as it leaves the incline? (c) How far from the table will the block hit the floor? (d) How much time has elapsed between when the block is released and when it hits the floor? (e) Does the mass of the block affect any of the above calculations? A block is given an initial velocity of 5.00 m/s up a frictionless 20.0° incline. How far up the incline does the block slide before coming to rest? A woman at an airport is towing her 20.0-kg suitcase at constant speed by pulling on a strap at an angle q above the horizontal. She pulls on the strap with a 35.0-N force, and the friction force on the suitcase is 20.0 N. Draw a free-body diagram of the suitcase. (a) What angle does the strap make with the horizontal? (b) What normal force does the ground exert on the suitcase? Three objects are connected on the table as shown in Figure. The table is rough and has a coefficient of kinetic friction of 0.350. The objects have masses 4.00 kg, 1.00 kg and 2.00 kg, as shown, and the pulleys are frictionless. Draw free-body diagrams of each of the objects. (a) Determine the acceleration of each object and their directions. (b) Determine the tensions in the two cords. Gravitational Mass vs. Inertial Mass • In Newton’s Laws, the mass is the inertial mass and measures the resistance to a change in the object’s motion • In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth • Experiments show that gravitational mass and inertial mass have the same value Applications of Newton’s Law • Assumptions – Objects can be modeled as particles – Masses of strings or ropes are negligible • When a rope attached to an object is pulling it, the magnitude of that force, T, is the tension in the rope – Interested only in the external forces acting on the object • can neglect reaction forces – Initially dealing with frictionless surfaces Objects in Equilibrium • If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium • Mathematically, the net force acting on the object is zero F 0 F 0 and F x y 0 Equilibrium, Example 1a • A lamp is suspended from a chain of negligible mass • The forces acting on the lamp are – the force of gravity (Fg) – the tension in the chain (T) • Equilibrium gives F y 0 T Fg 0 T Fg Equilibrium, Example 1b • The forces acting on the chain are T’ and T” • T” is the force exerted by the ceiling • T’ is the force exerted by the lamp • T’ is the reaction force to T • Only T is in the free body diagram of the lamp, since T’ and T” do not act on the lamp Equilibrium, Example 2a • Example 5.4 • Conceptualize the traffic light • Categorize as an equilibrium problem – No movement, so acceleration is zero Equilibrium, Example 2b • Analyze – Need two free-body diagrams – Apply equilibrium equation to the light and find T3 – Apply equilibrium equations to the knot and find T1 and T2 Objects Experiencing a Net Force • If an object that can be modeled as a particle experiences an acceleration, there must be a nonzero net force acting on it. • Draw a free-body diagram • Apply Newton’s Second Law in component form Newton’s Second Law, Example 1a • Forces acting on the crate: – A tension, the magnitude of force T – The gravitational force, Fg – The normal force, n, exerted by the floor Newton’s Second Law, Example 1b • Apply Newton’s Second Law in component form: F T ma Fy n Fg 0 n Fg x x • Solve for the unknown(s) • If T is constant, then a is constant and the kinematic equations can be used to more fully describe the motion of the crate Note About the Normal Force • The normal force is not always equal to the gravitational force of the object • For example, in this case F y n Fg F 0 and n Fg F • n may also be less than Fg Multiple Objects, Example 3 • Draw the free-body diagram for each object – One cord, so tension is the same for both objects – Connected, so acceleration is the same for both objects • Apply Newton’s Laws • Solve for the unknown(s) 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 resistance is called the force of friction Forces of Friction, cont. • Friction is proportional to the normal force – ƒs µs n and ƒk= µk n – These equations relate the magnitudes of the forces, they are not vector equations • 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 Forces of Friction, final • The direction of the frictional force is opposite the direction of motion and parallel to the surfaces in contact • The coefficients of friction are nearly independent of the area of contact Static Friction • Static friction acts to keep the object from moving • If F increases, so does ƒs • If F decreases, so does ƒs • ƒs µs n where the equality holds when the surfaces are on the verge of slipping – Called impending motion Kinetic Friction • The force of kinetic friction acts when the object is in motion • Although µk can vary with speed, we shall neglect any such variations • ƒk = µk n Some Coefficients of Friction Friction in Newton’s Laws Problems • Friction is a force, so it simply is included in the SF in Newton’s Laws • The rules of friction allow you to determine the direction and magnitude of the force of friction Friction Example, 1 • The block is sliding down the plane, so friction acts up the plane • This setup can be used to experimentally determine the coefficient of friction • µ = tan q – For µs, use the angle where the block just slips – For µk, use the angle where the block slides down at a constant speed Friction, Example 2 • Draw the free-body diagram, including the force of kinetic friction – Opposes the motion – Is parallel to the surfaces in contact • Continue with the solution as with any Newton’s Law problem Friction, Example 3 • Friction acts only on the object in contact with another surface • Draw the free-body diagrams • Apply Newton’s Laws as in any other multiple object system problem