Equation of Motion for a Particle Sect nd Law (time independent mass): F = (dp/dt) = [d(mv)/dt] = m(dv/dt) = ma = m(d 2 r/dt 2 ) = m r (1) A 2 nd 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 x o 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 μ s = 0.4. At what angle, θ, will block start sliding (if it is initially at rest)? (Work on board!)

After the block begins to slide, the coefficient of kinetic friction is μ k = 0.3. Find the acceleration for θ = 30º. (Work on board!) Example 2.3

Effects of Retarding Forces Unlike Physics I, the Force F in the 2 nd 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: F g = mg. –Air resistance gives a retarding force F r. –A good (common) approximation is: F r = F r (v) –Another (common) approximation is: F r (v) is proportional to some power of the speed v. F r (v)  -mkv n v/v (  Power Law Approx.) n, k = some constants.

Approximation: (which we’ll use): F r (v)  -mkv n v/v Experimentally (in air) usually n  1, v  ~ 24 m/s n  2, ~ 24 m/s  v  v s where v s = sound speed in air ~ 330 m/s A model of air resistance  drag force W. Opposite to direction of velocity &  v 2 : W = (½)c W ρAv 2 (“Prandtl Expression”) where A = cross sectional area of the object ρ = air density, c W = drag coefficient

Free Body Diagram for a Projectile (Figure 2-3a)

Measured Values for Drag Coeff. C w (Figure 2-3b)

Calculated Air Resistance, Using W = (½)c W ρAv 2 (Figure 2-3b)  Note the scales! 

Example: A particle falling in Earth’s gravitational field: –Gravity: F g = mg (down, of course!) –Air resistance gives force: F r = F r (v) = - mkv n v/v Newton’s 2 nd Law to get Equation of Motion: (Let vertical direction be y & take down as positive!) F = ma = my = mg - mkv n –Of course, v = y Given initial conditions, integrate to get v(t) & y(t). Examples soon!