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Equation of Motion for a Particle Sect. 2.4 • 2nd Law (time independent mass): F = (dp/dt) = [d(mv)/dt] = m(dv/dt) = ma = m(d2r/dt2) = m r (1) • A 2nd order differential equation for r(t). Can be integrated if F is known & if we have the initial conditions. • Initial conditions (t = 0): Need r(0) & v(0) = r(0). • Need F to be given. In general, F = F(r,v,t) • The rest of chapter (& much of course!) = applications of (1)! Problem Solving • Useful techniques: – Make A SKETCH of the problem, indicating forces, velocities, etc. – Write down what is given. – Write down what is wanted. – Write down useful equations. – Manipulate equations to find quantities wanted. Includes algebra, differentiation, & integration. Sometimes, need numerical (computer) solution. – Put in numerical values to get numerical answer only at the end! Example 2.1 • A block slides without friction down a fixed, inclined plane with θ = 30º. What is the acceleration? What is its velocity (starting from rest) after it has moved a distance xo down the plane? (Work on board!) Example 2.2 • Consider the block from Example 2.1. Now there is friction. The coefficient of static friction between the block & plane is μs = 0.4. At what angle, θ, will block start sliding (if it is initially at rest)? (Work on board!) Example 2.3 • After the block begins to slide, the coefficient of kinetic friction is μk = 0.3. Find the acceleration for θ = 30º. (Work on board!) Effects of Retarding Forces • Unlike Physics I, the Force F in the 2nd Law is not necessarily constant! In general F = F(r,v,t) • Arrows left off of all vectors, unless there might be confusion. • For now, consider the case where F = F(v) only. • Example: Mass falling in Earth’s gravitational field. – – – – Gravitational force: Fg = mg. Air resistance gives a retarding force Fr . A good (common) approximation is: Fr = Fr(v) Another (common) approximation is: Fr(v) is proportional to some power of the speed v. Fr(v) -mkvn v/v ( Power Law Approx.) n, k = some constants. • Approximation: (which we’ll use): Fr(v) -mkvnv/v • Experimentally (in air) usually n 1 , v ~ 24 m/s n 2 , ~ 24 m/s v vs where vs = sound speed in air ~ 330 m/s • A model of air resistance drag force W. Opposite to direction of velocity & v2: W = (½)cWρAv2 (“Prandtl Expression”) where A = cross sectional area of the object ρ = air density, cW = drag coefficient Free Body Diagram for a Projectile (Figure 2-3a) Measured Values for Drag Coeff. Cw (Figure 2-3b) Calculated Air Resistance, Using W = (½)cWρAv2 (Figure 2-3b) Note the scales! • Example: A particle falling in Earth’s gravitational field: – Gravity: Fg = mg (down, of course!) – Air resistance gives force: Fr = Fr(v) = - mkvn v/v • Newton’s 2nd Law to get Equation of Motion: (Let vertical direction be y & take down as positive!) F = ma = my = mg - mkvn – Of course, v = y • Given initial conditions, integrate to get v(t) & y(t). Examples soon!