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Equation of Motion for a Particle Sect. 2.4

Equation of Motion for a Particle Sect. 2.4. 2 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.

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Equation of Motion for a Particle Sect. 2.4

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  1. Equation of Motion for a ParticleSect. 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)!

  2. 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!

  3. 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 xodown the plane? (Work on board!)

  4. 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!)

  5. 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!)

  6. 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) approximationis: Fr = Fr(v) • Another (common) approximationis: Fr(v) is proportional to some power of the speed v. Fr(v)  -mkvn v/v ( Power Law Approx.) n, k = some constants.

  7. 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 forceW. 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

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

  9. Measured Values for Drag Coeff. Cw (Figure 2-3b)

  10. Calculated Air Resistance, Using W = (½)cWρAv2 (Figure 2-3b)  Note the scales! 

  11. 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!

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