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Review from Last Lecture

Review from Last Lecture. Kinematics in 2D Use same equations but with vectors Projectile Motion X and Y motion independent x = v i cos q t y = v i sin q t - ½gt 2 h = v i 2 sin 2 q / 2 g R = v i 2 sin 2 q / g Galilean Relativity

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Review from Last Lecture

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  1. Review from Last Lecture • Kinematics in 2D • Use same equations but with vectors • Projectile Motion • X and Y motion independent x = vi cosqt y = vi sinqt - ½gt2 h = vi2 sin2q / 2g R = vi2 sin2q / g • Galilean Relativity • Acceleration is the same in frames moving with constant velocity WRT one another • Uniform Circular Motion a = v2/r

  2. Forces • In kinematics we described motion • Position, displacement, velocity, acceleration • Now dynamics: What causes (changes in) motion? • Simple answer: Force causes motion (acceleration) • Types of forces • Contact forces • Springs, ropes, pushing • Field forces • Action-at-a-distance: gravity, magnetism, electrostaitcs

  3. Forces • How to measure? • Use a spring • The more it stretches the larger the force • Is it a good measure? • If we add forces, does the spring stretch linearly? • Yes it does: can use springs to measure forces • Scalar or Vector? • Forces add like vectors

  4. Newton’s 1st Law Statement 1: In the absence of any force, an object at rest remains at rest, an object in motion remains in motion with constant velocity. Statement 2: For an isolated object, one can identify a reference frame in which the object has zero acceleration. (Inertial frames exist.) • But what does this really mean? • Any object has a reference frame in which it is at rest • For an object isolated from its surroundings, this is called an inertial frame • Frames moving with constant relative velocity are also inertial (with identical measured accelerations)

  5. Newton’s 2nd Law In an inertial frame, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass • As an equationa = F/m or F = ma • Note the phrase “net force” • The net force is the vector some of all the forces acting on an object:F = SF = F1 + F2 + F3 + …

  6. Units for force • Units for force • a is measured in [L/T2] (SI: m/s2) • m is measured in [M] (SI: kg) • Since F=ma, force is measured in [ML/T2] (SI: kg m/s2) • Define new SI unit: Newton 1 N ≡1 kg m/s2 • In US, a pound is a weight (ie. a force) • 1lb ≡1 slug ft/s2 ≈ 4N • In general, don’t confuse weight and mass • Mass is an intrinsic property of an object and does not change • Weight is the force due to gravity on an object and depends on where the object is

  7. Inertial and Gravitational Mass • Inertial mass appears in Newton’s 2nd Law F = mia • Gravitational mass appears in defining force due to gravity Fg = mgag = -mgg • Inertial mass quantifies resistance to acceleration • Gravitational mass only specifies strength of one force (of possibly many) • Requires no change of motion • Yet mi = mg (!)

  8. Newton’s 3rd Law If two objects interact, the force F12 exerted by the first object on the second is equal in magnitude and opposite in direction to the force F21 exerted by the second on the first. • As an equation:F12 = -F21 • The two forces must be of the same type (e.g. gravity or contact) and act on different objects

  9. Newton’s Laws in a Nutshell • Inertial frames exist • F = ma • F12 = -F21

  10. Main Entry: 1nor·malPronunciation: 'nor-m&lFunction: adjectiveEtymology: Latin normalis, from normaDate: circa 1696 1: PERPENDICULAR;especially: perpendicular to a tangent at a point of tangency The Normal Force • Whenever one body exerts a contact force on another, there is a “normal force” as a reaction (3rd Law) force • A normal force is always perpendicular (normal) to the face of the object • The normal force prevents one object from passing through another and can thus have any magnitude (until something breaks)

  11. Problem Solving Solving problems using Newton’s Laws Always start by drawing a free body diagram! • Draw the object • Draw all the forces acting on the object • Choose a coordinate system • Find the components of all the forces in that coordinate system • Solve SF = ma for each coordinate SFx = max SFy = may

  12. Problem Solving Examples • The monitor is not moving, thus there is no net force • Pick horizontal (x) and vertical (y) axes • No horizontal forces • Vertically we have: • Gravity pulling down Fg = -mgg • Normal n keeping the monitor from falling 0 = n +Fg = n – mgg n = mgg

  13. Problem Solving Examples • Inclined plane • The Problem: • A car of mass m is on an icy driveway inclined at angle q • Find the acceleration of the car, assuming a frictionless driveway

  14. Problem Solving Examples • Draw the object

  15. Problem Solving Examples • Draw the object • Draw all the forces acting on the object

  16. Problem Solving Examples • Draw the object • Draw all the forces acting on the object • Choose a coordinate system

  17. Problem Solving Examples • Draw the object • Draw all the forces acting on the object • Choose a coordinate system • Find the components of all the forces in that coordinate system

  18. Problem Solving Examples • Draw the object • Draw all the forces acting on the object • Choose a coordinate system • Find the components of all the forces in that coordinate system • Solve SF = ma for each coordinate SFx = max = mg sinq SFy = may = n - mg cosq = 0

  19. Friction • Friction occurs when an objects is on a surface, or two objects are touching • Static friction: Equals applied force (along surface) up to some value, resisting motion fs£msn • Kinetic (sliding) friction: Proportional to normal force, but parallel to surface, resisting motion fk = mkn • Note that m independent of contact area

  20. Friction

  21. Problem Solving Examples A block of mass m1 on a rough, horizontal surface ks connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown. A force of magnitude F at angle q is applied to the block. The coefficient of friction between the block and the surface is mk. Determine the magnitude of acceleration of the two objects.

  22. Problem Solving Examples • Draw the objects • Draw all the forces acting on the objects n T fk T m1g m2g

  23. Problem Solving Examples • Draw the object • Draw all the forces acting on the object • Choose a coordinate system • Find the components of all the forces in that coordinate system n F sinq T F cosq fk T m1g m2g

  24. Problem Solving Examples • Draw the object • Draw all the forces acting on the object • Choose a coordinate system • Find the components of all the forces in that coordinate system • Solve SF = ma for each coordinate n F sinq T F cosq fk T m1g m2g

  25. Problem Solving Examples • Solve SF = ma for each coordinate • SFx = Fcosq – T – fk = Fcosq – T – mkn = m1a SFy = n + Fsinq - m1g= 0 Þn = m1g - Fsinq • SFx = 0 SFy = T – m2g = m2a ÞT = m2(g +a) m1a = Fcosq – m2(g+a) – mk(m1g-Fsinq ) a(m1+m2) = Fcosq–m2g–mk(m1g-Fsinq) n F sinq T F cosq fk T m1g m2g

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