Transcript Sect. 2.4, Part I

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.
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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!