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

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!