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Newton’s Second Law: Motion Along a Line Readings: Chapter 6 (2nd edition) 1 1) Object – as a particle 2) Identify all the forces 3) Find the net force (vector sum of all individual forces) 4) Introduce convenient co-ordinate system 5) Find the acceleration of the object (second Newton’s law) 6) With the known acceleration find kinematics of the object 2 The First Class of Problems: Equilibrium 1. Static Equilibrium: no motion (velocity = 0, then acceleration = 0 ) 2. Dynamical Equilibrium: no acceleration (velocity = constant) - In both cases acceleration = 0 - Second Newton’s Law – Net Force = 0 Static Equilibrium: Convenient coordinate system!! Fnet T1 T2 T3 0 T1 T3 cos 0 T2 T3 sin 0 3 Special The First Class Class of of Problems: Problems: Equilibrium Equilibrium 1. Static Equilibrium: no motion (velocity = 0, then acceleration = 0 ) 2. Dynamical Equilibrium: no acceleration (velocity = constant) Dynamic Equilibrium: Convenient coordinate system!! Fnet n T w f k 0 T f k w sin 0 n w cos 0 fk Kinetic friction 4 The Second Acceleration SpecialClass Classof ofProblems: Problems:Find Equilibrium Important: - Introduce convenient co-ordinate system!! - Understand what the direction of acceleration is!! Fnet a m a fk Kinetic friction Fnet n T w f k ma T f k w sin ma n w cos 0 5 The Second Acceleration SpecialClass Classof ofProblems: Problems:Find Equilibrium Fnet a m a Fnet n T w f k ma fk T f k w sin ma n w cos 0 T 20 N is given m 1kg w mg 10 N 300 is given n w cos 10cos 300 8.7 N is given Then: T f k w sin 20 f k 5 a m 1 ? 6 Friction s f s f s ,max s n Static friction: - coefficient of static friction (it is usually given in the problem) n Kinetic friction: f k k n k Rolling friction: f r r n r Usually: - normal force - coefficient of kinetic friction (it is usually given in the problem) - coefficient of rolling friction (it is usually given in the problem) s k r 7 very small Friction Static friction: f s f s ,max s n Find the maximum tension, s n - coefficient of static friction - normal force Tmax y Equilibrium: x Fnet n w T f s 0 n T n w 0 fs fs T 0 weight, friction tension normal nw w f s ,max s n s w f s T f s ,max Condition of equilibrium: T s w Tmax s w 8 Friction Static friction: f s f s ,max s n Find the maximum tension, Equilibrium: Fnet n w T f s 0 n w 0 fs T 0 s n - coefficient of static friction - normal force y Tmax x f s ,max s n s w n T f s T f s ,max fs w fs T Condition of equilibrium: f s ,max s w T s w Tmax s w Tmax T 9 Friction Kinetic friction: Find acceleration, f k k n k n - coefficient of kinetic friction - normal force (T Tmax ) y Fnet n w T f k ma x n w 0 f k T ma a n T fs w weight, friction tension normal n w then f k k n k w T f k T k w T a k g m m m 10 Friction Static friction: f s f s ,max s n Find the maximum angle, s n - coefficient of static friction - normal force max Equilibrium: Fnet n w f s 0 n w cos 0 f s w sin 0 n w cos f s ,max s n s w cos f s w sin f s ,max Condition of equilibrium: w sin s w sin tan s tan max s 11 Friction Static friction: f s f s ,max s n Find the maximum angle, max s n - coefficient of static friction - normal force f s ,max s n s w cos f s w sin f s ,max f s w sin f s ,max s w cos max Condition of equilibrium: tan s 12 Friction Kinetic friction: Find acceleration k n f k k n - coefficient of kinetic friction - normal force ( max ) Fnet n w f k ma a n w cos 0 f k w sin ma n w cos then f k k n k w cos f k w sin k w cos w sin a g(sin k cos ) m m 13 Weight and Apparent Weight Weight – gravitational force - pulls the objects down w mg m - mass of the object (the same on all planets) g 9.8 m s2 - free-fall acceleration (different on different planets) How can we measure weight? 1. We can measure mass by comparing with the known mass munknown mknown 2. We can measure the weight by comparing with the known force w Fspring 14 Weight and Apparent Weight Apparent Weight – reading of the scale (or the normal force) In equilibrium: Fnet 0 then Fspring w Motion with acceleration: Fnet ma Fspring w ma a Fnet ma Fspring w ma then then Fspring m ( g a ) Fspring m ( g a ) The man feels heavier than normal while accelerating upward a The man feels lighter than normal 15 while accelerating upward