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Transcript
Chapter 4 Newton’s Laws of Motion Sir Isaac Newton Philosophiae Naturalis Principia Mathematica (1687) Opticks (1704) "Nature and nature's laws lay hid in night; God said 'Let Newton be' and all was light." Newton’s Three Laws Inertia: “Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by a force impressed on it.” Force, Mass, Acceleration (F=ma): “The change in motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.” “Action = Reaction”: “To every action there is always opposed an equal reaction; or, the mutual actions of two bodies are always equal, and directed to contrary parts.” Newton’s Law Summary 1. Velocity is zero or constant when net force is zero. 2. F=ma 3. Action = Reaction (in opposite direction) Net Force Net force (or total force) is the sum of all the forces applied to an object. FA FB FC For example, if there are three people, A, B and C pushing the crazy kid. The net force on him is: r r r Fnet FA FB FC First Law and Net Force The First Law deals with cases when there is no net force. r r r r Fnet FA FB FC 0 FA FB FC “Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by a force impressed on it.” Puck on ice Ice has very little friction (so no net force), so the inertia keeps the puck moving once it is set in motion. Mass or Inertia Inertia is the tendency of an object to remain at rest or in motion with constant speed along a straight line. Mass (m) is the quantitative measure of inertia. Mass is the property of an object that measures how hard it is to change its motion. Units: kg Mass vs. Weight Mass is an intrinsic property of an object. A rock has same mass whether it is on the moon or on Earth. Mass does not change Weight is the force exerted on an object by gravity: W=mg This is different depending upon the strength of the gravitational force. You weigh less on the Moon than on Earth. Example What is the weight of a man of mass 70kg on Earth? W mg (70kg)(9.8m / s ) 686N 2 Weight is measured in N. (Pound: 1lb = 4.448N) We sometimes use the symbols W or Fgravity to denote weight. Same mass, different weight Newton’s 1st Law There are many forces act on the plane, including weight (gravity), drag (air resistance), the thrust of the engine, and the lift of the wings. At some point the velocity of the plane is constant. At this time, the total (or net) force on the plane: lift 1. is pointing upward 2. is pointing downward 3. is pointing forward 4. is pointing backward correct 5. is zero drag thrust weight r r r r Fnet Fdrag Fweight Fthrust Flift ??? Newton’s 1st Law Newton's first law states that if no net force acts on an object, then the velocity of the object remains constant. Since the velocity is constant, the total force on the plane must be zero, according to Newton's first lift law. r r r r Fdrag Fweight Fthrust Flift 0 drag thrust weight Newton’s Second Law Unit: N (Newton) F = ma Deals with the effect of a non-zero net force. Non-zero net force causes acceleration. r r r r Fnet FA FB FC 0 FA FB FC Some Math r In vector notation: r F ma What does the vector symbol mean? F Fx iˆ Fy ĵ Fz k̂, r a ax iˆ ay ĵ az k̂ Fx iˆ Fy ĵ Fz k̂ max iˆ may ĵ maz k̂ Fx max Fy may F ma z z F = ma applies to each component independently. F means NET FORCE!!! The F in F = ma is the net force on the object. If you are careful, you may write instead: Fnet = ma Always remember to find the net force first! r r r r Fnet FA FB FC ma FA FB FC Simple Examples: Find a F = 200N m = 10kg F 200N F ma a 20m / s 2 m 10kg F = 200N m = 10kg F = 150N F 200N 150N a 5m / s 2 m 10kg Find the magnitude of a First write the forces as vectors : r F1 5cos20 oiˆ 5sin 20 o ˆj 4.70iˆ 1.71 ˆj r F 8cos60 oiˆ 8sin 60 o ˆj 4.00iˆ 6.93 ˆj 2 m 2kg Then find the net force vector : r r r Fnet F1 F2 8.70iˆ 5.22 ˆj Apply the second law : r r Fnet 8.70iˆ 5.22 ˆj ma r 1 a (8.70iˆ 5.22 ˆj ) (4.35iˆ 2.61 ˆj )m /s2 2 Now we can find the magnitude : r a 4.35 2 2.612 5.07m /s2 r r r You can also use v v0 at to figure out the velocity. When the net force is zero r r r r r r If Fnet 0, using F ma, we have a 0. r r r r On the other hand, if a 0, we have Fnet 0. Zero Net Force on a Lamp T : Tension W : Weight ( Fgravity mg) T W 0N T W When Fnet is non-zero When Fnet ≠ 0, since F=ma, we have a≠0. No net force, no acceleration. Net force leads to acceleration. If an object is accelerating, there must be a non-zero net force. Example The mass of m = 2kg is accelerating upward at 4m/s2. Find the tension. Fnet T mg but Fnet ma T mg ma T ma mg m(a g) (2kg)(4 9.8)m / s 2 27.6N mg Example The mass of m = 2kg is accelerating downward at 4m/s2. Find the tension. a 4m / s 2 mg T ma T m(g a) (2kg)(9.8 4)m / s 2 11.6N T-mg or mg-T ? In non-vector notation, we usually assume the variable a represents the magnitude of the acceleration, and is therefore positive whether it is up or down. In other words, unless stated otherwise, we will not use the up/positive, down/negative convention. With this new convention, whenever you have opposing forces, the forces pointing in the same direction as a comes first, minus the forces in the opposite direction as a. T-mg or mg-T ? T mg ma mg T ma Another way to remember, if a > 0, you want: (big number) - (small number) = ma or (same direction) - (opposite direction) = ma Two forces F1 = 200N a m = 10kg F2 = 350N It is obvious that a is going to point toward the left: 350N 200N ma 350N 200N a 15m / s 2 : Accelerate toward the left 10kg Newton’s Third Law For every force, there is an equal and opposite force every “action” has a “back-reaction” these are precisely equal and precisely opposite Newton’s Third Law You cannot push without being pushed back just as hard In tug-of-war, each side experiences the same force (opposite direction) When you push on a brick wall, it pushes back on you! Force Pairs Illustrated Force on box by person Force on floor by box Force on person by box Force on box by floor Force on person Force on floor by person by floor Not shown are the forces of gravity and the associated floor forces Don’t all forces then cancel? How does anything ever move (accelerate) if every force has an opposing pair? Action and reaction force act on different objects. Force on box by person Net Force on box Force on box by floor Exercise: Action/Reaction Suppose a tennis ball (m= 0.1 kg) moving at a velocity v = 40 m/sec collides head-on with a truck (M = 500 kg) which is moving with velocity V = 10 m/sec. During the collision, the tennis ball exerts a force on the truck which is smaller than the force which the truck exerts on the tennis ball. TRUE or FALSE ? The tennis ball will suffer a larger acceleration during the collision than will the truck. TRUE or FALSE ? Suppose the tennis ball bounces away from the truck after the collision. How fast is the truck moving after the collision? < 10 m/sec = 10 m/sec > 10 m/sec ? Exercise: Action/Reaction solution During the collision, the tennis ball exerts a force on the truck which is smaller than the force which the truck exerts on the tennis ball. TRUE or FALSE ? Equal and opposite forces! The tennis ball will suffer a larger acceleration during the collision than will the truck. TRUE or FALSE ? Acceleration = Force / mass Suppose the tennis ball bounces away from the truck after the collision. How fast is the truck moving after the collision? < 10 m/sec = 10 m/sec > 10 m/sec ? Force from the ball causes deceleration. Normal Force Force from a solid surface (e.g. wall, ground) providing support for an object. “Normal” means “perpendicular”, Normal Force is always perpendicular to the solid surface. Notations: FN, Fn, N or n Example of Normal Force Suppose the elevator is not accelerating. m=70kg, what does the scale read? A scale actually measure the normal force, NOT mg!!! The forces must cancel each other: FN mg 686N This is a special case, in general FN mg!!! Example of Normal Force 2 Suppose the elevator is accelerating up. m=70kg, a=5m/s2 FN mg ma FN m(g a) 1036N Note that FN mg!!! In general, the normal force is what a scale measures. Example of Normal Force 3 Suppose the elevator is accelerating down. m=70kg, a=5m/s2 mg FN ma FN m(g a) 336N Note that FN mg!!! Make sure you see the difference in these 3 cases! Be careful with the normal force Fn Fy mg Fn Fy mg Fn mg Fy m 1.5kg Fn mg F sin 9.7N Note that mg 14.7N Fn mg in general! !! What is Fn now? Fn mg Fy m 1.5kg Fn mg F sin 21.8N Fn mg Fy Note that mg 14.7N Fn mg in general! !!