equation of motion for a particle sect. 2.4 2 nd law (time independent mass): f = (dp/dt) =...
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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)!
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!)
• After the block begins to slide, the coefficient of kinetic friction is μk = 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 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
• 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!