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Chapter 4 Forces and Newton’s Laws of Motion F=ma; gravity

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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

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1) Newton’s first law: the law of inertia a)Free object --> no forces acting on it b)Constant velocity --> at rest or motion in a straight line with constant speed c)Natural state is motion with constant velocity -Aristotle: rest is natural state -Galileo: circular motion (orbits) is natural state d)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. A free object moves with constant velocity

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1) Newton’s first law: the law of inertia 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) A free object moves with constant velocity

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2) Newton’s second law: F=ma a)Mass -quantity of matter (determined with a balance) -quantity that resists acceleration (inertial mass) (i)Define 1 kg as mass of a standard cylinder (ii)Addition of masses (scalar): m = m 1 + m 2 -in particular two identical masses have twice the mass, to satisfy quantity of matter definition (iii) Observe acceleration vs mass for a given force: massacceleration 1 kg1 m/s 2 2 kg1/2 m/s 2 3 kg1/3 m/s 2 mass is inversely proportional to acceleration

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2) Newton’s second law: F=ma b) Force -push or pull -disturbs “natural” state: causes acceleration (i)Define 1 N (newton) as force required to accelerate 1 kg by 1 m/s 2 (ii)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.

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2) Newton’s second law: F=ma (iii) Observe acceleration vs. force for a given object ForceAcceleration 1 N1 m/s 2 2 N2 m/s 2 3 N3 m/s 2 Force is proportional to acceleration (iv) Types of force: - gravity - electromagnetic - weak nuclear -strong nuclear electroweak

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

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2) Newton’s second law: F=ma Special case: 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,

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2) Newton’s second law: F=ma d) Free-body diagrams Replace object(s) by dot(s). Represent all forces from the dot. Solve F=ma for each object F1F1 F2F2 FNFN mgmg F1F1 F2F2 m

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

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

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2) Newton’s second law: F=ma e) Components of force Example: m F 1 = 15 N F 2 = 17 N º m = 1300 kg Find acceleration. x y F1F1 F2F2

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3) Newton’s third law For every action, there is an equal and opposite reaction A B F AB F BA F AB = -F BA Conservation of momentum:

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4) Gravity 5) Normal Force 6) Friction

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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

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7) Tension and pulleys T 1 = T 2 = T a 1 = a 2 = a For the example, a 1y = -a 2y Simplify problem, by choosing sign for a sense of the motion m1m1 m2m2 T1T1 T2T2 +a

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7) Tension and pulleys m1m1 m2m2 T1T1 T2T2 +a T m1gm1g T m2gm2g

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7) Tension and pulleys m1m1 m2m2 T1T1 T2T2 +a e.g. m 1 = 5 kg; m 2 = 10 kg

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7) Tension and pulleys m1m1 m2m2 +a Acceleration can be determined by considering external forces (tension is an internal force holding objects together) m2gm2g m1gm1g

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Example m1m1 m2m2 If m 1 = m 2, and rope and pulley are ideal, what happens when the monkey climbs the rope? T1T1 T2T2 T1T1 m1gm1g T2T2 m2gm2g Since T 1 = T 2, any change in T 2 to cause the monkey to ascend, results in a change in T 1, causing the bananas to ascend at the same rate.

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Example If m 1 = m 2, and rope and pulley are ideal, what happens when the monkey climbs the rope? T1T1 m1gm1g T2T2 m2gm2g Since T 1 = T 2, any change in T 2 to cause the monkey to ascend, results in a change in T 1, causing the bananas to ascend at the same rate.

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Example If m 1 = m 2, and rope and pulley are ideal, what happens when the monkey climbs the rope? T1T1 m1gm1g T2T2 m2gm2g Since T 1 = T 2, any change in T 2 to cause the monkey to ascend, results in a change in T 1, causing the bananas to ascend at the same rate.

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Example If m 1 = m 2, and rope and pulley are ideal, what happens when the monkey climbs the rope? T1T1 m1gm1g T2T2 m2gm2g Since T 1 = T 2, any change in T 2 to cause the monkey to ascend, results in a change in T 1, causing the bananas to ascend at the same rate.

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Example If m 1 = m 2, and rope and pulley are ideal, what happens when the monkey climbs the rope? T1T1 m1gm1g T2T2 m2gm2g Since T 1 = T 2, any change in T 2 to cause the monkey to ascend, results in a change in T 1, causing the bananas to ascend at the same rate.

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

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

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

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

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

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

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Example: Apparent weight Free fall F N = 0 W Apparent weight (measured by scale) is the normal force weightlessness

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