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) Question of the day: Explain planetary motion

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

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

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

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.

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

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

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,

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

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

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

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

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:

4) Gravity 5) Normal Force 6) Friction

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

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

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

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

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

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.

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.

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.

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.

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.

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

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

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

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

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

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

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