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Chapter 4

Chapter 4. Forces and Newton’s Laws of Motion F=ma ; gravity. 0) Background. Galileo inertia (horizontal motion) constant acceleration (vertical motion) Descartes & Huygens Conservation of momentum: mass x velocity = constant Kepler & Braha laws of planetary motion (kinematics only)

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Chapter 4

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  1. Chapter4 Forces and Newton’s Laws of Motion F=ma; gravity

  2. 0) Background • Galileo • inertia (horizontal motion) • constant acceleration (vertical motion) • Descartes & Huygens • Conservation of momentum: mass x velocity = constant • Kepler & Braha • laws of planetary motion (kinematics only) Question of the day: Explain planetary motion

  3. 1) Newton’s first law: the law of inertia A free object moves with constant velocity • Free object --> no forces acting on it • Constant velocity --> at rest or motion in a straight line with constant speed • Natural state is motion with constant velocity • Aristotle: rest is natural state • Galileo: circular motion (orbits) is natural state • Inertial reference frames • A reference frame in which the law of inertia holds • does not hold on a carousal, or an accelerating car • Requires ability to identify a free object: If no force acts on a body, a reference frame in which it has no acceleration is an inertial frame.

  4. 1) Newton’s first law: the law of inertia A free object moves with constant velocity e) Velocity is relative • All frames moving at constant velocity with respect to an inertial frame are also inertial frames • No local experiment can determine the state of uniform motion • Cannot define absolute rest: No preferred reference frame • (Principle of Relativity)

  5. 2) Newton’s second law: F=ma a) Mass • quantity of matter (determined with a balance) • quantity that resists acceleration (inertial mass) • Define 1 kg as mass of a standard cylinder • Addition of masses (scalar): m = m1 + m2 • in particular two identical masses have twice the mass, to satisfy quantity of matter definition (iii) Observe acceleration vs mass for a given force: mass acceleration 1 kg 1 m/s2 2 kg 1/2 m/s2 3 kg 1/3 m/s2 mass is inversely proportional to acceleration

  6. 2) Newton’s second law: F=ma b) Force • push or pull • disturbs “natural” state: causes acceleration • Define 1 N (newton) as force required to accelerate 1 kg by 1 m/s2 • Addition of forces (vector): Identical forces in opposite direction produce no acceleration Two identical forces at 60º produce the same acceleration as a third identical force at 0º (cos(60º)=1/2) Two identical parallel forces corresponds to twice the force.

  7. 2) Newton’s second law: F=ma electroweak (iii) Observe acceleration vs. force for a given object Force Acceleration 1 N 1 m/s2 2 N 2 m/s2 3 N 3 m/s2 Force is proportional to acceleration (iv) Types of force: - gravity - electromagnetic - weak nuclear -strong nuclear

  8. 2) Newton’s second law: F=ma c) Second Law Define proportionality constant =1. Then, For m = 1 kg, and a = 1 m/s2, F = 1 N by definition, and F = ma gives F = 1 kg m /s2, so 1 N = 1 kg m/s2

  9. 2) Newton’s second law: F=ma • F = ma can be used as the defining equation for force and inertial mass, but only because of the physical observation that force is proportional to acceleration (for a given mass), and mass is inversely proportional to acceleration (for a given force). • Inertia is the tendency of an object not to accelerate • Newton’s second law formally refers to the rate of change of momentum: For constant mass, • Special case:

  10. 2) Newton’s second law: F=ma FN F2 F2 F1 F1 m mg d) Free-body diagrams Replace object(s) by dot(s). Represent all forces from the dot. Solve F=ma for each object

  11. 2) Newton’s second law: F=ma 10 N 10 N 10 N 10 N 10 N scale scale m m m d) Free-body diagrams ? N

  12. 2) Newton’s second law: F=ma sum of all forces e) Components of force

  13. 2) Newton’s second law: F=ma y F1  x F2 e) Components of force Example: F1 = 15 N Find acceleration. m = 1300 kg º m F2 = 17 N

  14. 3) Newton’s third law FAB A B FBA For every action, there is an equal and opposite reaction Conservation of momentum: FAB = -FBA

  15. 4) Gravity5) Normal Force6) Friction

  16. 7) Tension and pulleys • Tension: force exerted by rope or cable • For an ideal (massless, inextensible) line, the same force is exerted at both ends (in opposite directions) • objects connected by a line (no slack) have the same acceleration • Pulley: changes direction of force • For an ideal pulley (massless, frictionless) the magnitude of the tension is the same on both sides • magnitude of acceleration of connected objects is the same

  17. 7) Tension and pulleys +a T1 T2 • T1 = T2 = T • a1 = a2 = a • For the example, a1y = -a2y • Simplify problem, by choosing sign for a sense of the motion m1 m2

  18. 7) Tension and pulleys T T T1 m1g m2g T2 +a m1 m2

  19. 7) Tension and pulleys T1 T2 e.g.m1 = 5 kg; m2 = 10 kg +a m1 m2

  20. 7) Tension and pulleys +a m1 m1g m2 m2g Acceleration can be determined by considering external forces (tension is an internal force holding objects together)

  21. Example T2 T1 T1 m2g m1g T2 If m1 = m2, and rope and pulley are ideal, what happens when the monkey climbs the rope? m1 Since T1 = T2, any change in T2 to cause the monkey to ascend, results in a change in T1, causing the bananas to ascend at the same rate. m2

  22. Example T2 T1 m2g m1g If m1 = m2, and rope and pulley are ideal, what happens when the monkey climbs the rope? Since T1 = T2, any change in T2 to cause the monkey to ascend, results in a change in T1, causing the bananas to ascend at the same rate.

  23. Example T2 T1 m2g m1g If m1 = m2, and rope and pulley are ideal, what happens when the monkey climbs the rope? Since T1 = T2, any change in T2 to cause the monkey to ascend, results in a change in T1, causing the bananas to ascend at the same rate.

  24. Example T2 T1 m2g m1g If m1 = m2, and rope and pulley are ideal, what happens when the monkey climbs the rope? Since T1 = T2, any change in T2 to cause the monkey to ascend, results in a change in T1, causing the bananas to ascend at the same rate.

  25. Example T2 T1 m2g m1g If m1 = m2, and rope and pulley are ideal, what happens when the monkey climbs the rope? Since T1 = T2, any change in T2 to cause the monkey to ascend, results in a change in T1, causing the bananas to ascend at the same rate.

  26. 8) Equilibrium applications • Equilibrium means zero acceleration • Balance forces in x and y directions

  27. 8) Equilibrium applications Free body diagram for pulley: T Free body diagram for weight: T mg Example: Find tension on leg (F) T=mg

  28. 9) Non-equilibrium applications • Non-equilibrium means non-zero acceleration • Determine acceleration from 2nd law: • Solve kinematic equations

  29. Example: Apparent weight FN W Apparent weight (measured by scale) is the normal force At rest or moving with constant velocity

  30. Example: Apparent weight FN W Apparent weight (measured by scale) is the normal force Accelerating up

  31. Example: Apparent weight FN W Apparent weight (measured by scale) is the normal force Accelerating down

  32. Example: Apparent weight FN = 0 W Apparent weight (measured by scale) is the normal force Free fall weightlessness

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