2. 3 D Kinematics
Position Vector = r = xî + yĵ + z k
Velocity vector = v = vxî + vyĵ + vzk
Speed is magnitude of velocity = pythag of velocity
Ex 1: Sally is chasing Calvin.
Calvin is moving along a path defined by rcalvin = (t-1) i+(11-t2) j
and Sally is moving along the path rsally = (5-5t) i + (5+5t) j
a) When does Sally catch Calvin?
b) What will be the angle between their two velocities when this happens?
Scalar Dot Product – a method of multiplying vectors that results in a scalar quantity
A ∙ B = AB cos θ
A ∙ B = AxBx + AyBy + …
3. Ex 2: A particle is moving in three dimensions with a velocity given by
v = 4t3 i – (cos t) j + 7 k
a) What is its acceleration as a function of time?
b) What is its speed at t = 2?
c) What is its displacement between t = 0 and t = 3?
Ex 1 An object is thrown off a cliff 50 m high with an initial velocity of 25 m/s and an
angle of elevation of 30 degrees. How far from the base of the cliff will it land?
Ex 2: Derive an equation for range as a function of angle and initial velocity for level
Ex 3: A ball is thrown up a hill with an initial speed of 20 m/s at an angle of elevation of
45 degrees. The hill is inclined at an angle of 30 degrees.
4. Uniform Circular Motion
Direction of velocity and acceleration
ac = v2/r
V object a relative to c = v object a relative to b + v object b relative to c
Ex. Boat crossing a river
5. Newton’s Laws
Newton’s First Law
Equilibrium – a = 0 and Net F = 0
Dynamic vs Static Equilibrium
Newton’s 2nd Law
Fnet = ma
Newton’s 3rd Law
Equal and opposite reaction forces
Mass vs weight
6. Common forces
1) Gravitational force (weight)
Magnitude = mg
Direction = straight down
2) Normal force
Direction = perpendicular to surface
Magnitude = weight (if there are no other forces acting on it!)
3) Friction Force
Direction = parallel to surface, opposite to incipient motion
Magnitude Fstatic friction <= µsFN
Fkinetic friction <= µk FN
Direction = in direction of rope (towards center of rope)
Magnitude = determined by SIN2
7. Problem Solving
Look for equilibrium in each direction (tilt axis if necessary)
Call direction of acceleration positive
Circular motion = there is always an acceleration toward the
ac = v2/r
8. 1) Suppose you apply the minimum force required to move an object
initially at rest on a horizontal surface. If you continue to apply this
force, once the object moves what will be the acceleration of the
object in terms of m, g, µs and µk?
2) A mass m is suspended by two ropes on either side of the mass.
The angles the ropes make with the horizontal are θ and ø. Calculate
the tension in each rope in terms of m, g, θ, and ø
3) An object is at rest on an inclined plane of angle θ.
a) Draw a free body diagram of the object.
b) Express the frictional force required to keep the object stationary
in terms of m, g, and θ.
c) Calculate the normal force in terms of….
d) What is the minimal coefficient of static friction required to keep
the object from slipping?
9. 4) The Atwood machine consists of two masses (m1>m2)attached to a
rope over a pulley.
a) What is the acceleration of the system of masses (assuming the
rope and pulley are massless)
b) What vertical force must the pin of the pulley exert on the wheel
of the pulley to hold it up?
5) A man of mass m is standing in an elevator that is accelerating
upward with an acceleration a. What is the reading on a scale the man
is standing on in terms of m and g?
6) A pendulum bob of mass m hanging in a truck is deflected through
an angle of θ as the truck accelerates. Find the tension and the
acceleration of the truck in terms of m, g, and θ.
10. 7) Two masses (m1 and m2) are connected by a rope hanging
from a pulley on an incline (θ and µ). What is the acceleration
of the masses?
8) A car is driven around a curve of radius R that is banked at
an angle of θ. If the coefficient of friction between the car’s
tires and the road is µ, what is the maximum speed the car can
go without flying off the road?
11. Motion with air resistance
Express variables in terms of velocity and time to obtain
a differential equation
Separate and integrate
An object of mass m is released from rest at t=0 and
experiences a force of air resistance given by F = -
cv, where c is a constant. Calculate the velocity as a
function of time. Sketch graphs of velocity and
12. Work, Energy and Power
--Displacement must occur for work to be done
--Unit is joules
W = ∫F ds where s is position/displacement
13. Total work can be found by finding the work done by each force
and adding OR finding the net force and compute the work done
If force is constant, W = F ∙ s
W = F s cos θ OR Fxsx + Fysy +…
Example: A mass m is dragged a distance s along a rough surface
by a force F applied at an angle of elevation θ. The coefficient of
friction is µ.
What is the work done by each force and the total work?
14. Work Energy Theorem
When work is done on an object, energy is changed.
W = ∆KE = ∆ ½ m v2
Example: A force F = (4x)i acts on an object of mass 2 kg as it
moves from x = 0 to x = 5m. If the object started at rest, find the
net work and the final speed of the object.
Conservation of Energy
Conservative forces (no friction): kinetic potential kinetic
Nonconservative forces: kinetic/potential heat, sound, etc
15. Gravitational PE
Ug = mgh
Spring PE (derived from Hooke’s Law (F = -kx))
Us = ½ kx2
F = - dU/ds
Energy Wells : Stable and unstable equilibrium
Rate at which work is done
P = W/t
Of course, any equation can be turned into a calculus
P = dW/dt
= F v cos θ
Vector in the direction of velocity
p = mv
Fnet = ma = m (dv/dt) = dp/dt
Force is slope of p vs t graph
When the net force on an object or system is zero, the
net momentum is constant (conserved).
18. Impulse = ∫F dt (= F∆t if the force is constant)
Impulse – area under F vs t graph
Example 1: During a collision with a wall lasting from t =
0 to t =2s, the force acting on a 2 kg object is given by
the equation F = 4t(2s-t) i.
a) Calculate the impulse of the force on the object.
b) What is the average force on the object?
c) If the object starts from rest, what is its final
19. Elastic collision – kinetic energy is conserved. Energy
stored in deforming the object is completely recovered.
Spring between, very special situations
Inelastic collision – kinetic energy is not conserved
(converted to heat, sound, deformation)
Completely inelastic collision – objects stick together
after the collision, KE not conserved.
Solving collision problems
1) Decide if collision is elastic or inelastic.
2) Momentum is a vector.
3) Use separate cons of p equations for x and y
4) Change in KE in collision is final KE – initial KE
20. Example 1
A bullet of mass 0.005 kg moving at a speed of 100 m/s
imbeds in a 1kg block of wood resting on a frictionless
surface and attached to a horizontal spring of k = 50 N/m.
a) What is the velocity of the block the instant after the
bullet strikes it?
v = 0.50 m/s
b) What is the maximum compression of the spring?
x = 0.07 m
Example 2 A mass of 3 kg moving to the right at a speed of 5
m/s collides elastically with a 10 kg mass moving at 1 m/s to
the left. What is the velocity of the 10 kg mass after the
v = 1.769 m/s
21. Center of mass
The center of mass is the weighted average of the
location of masses in a system.
rCM = ∑mr/ ∑ m
rCM = ∫x dm/ ∫dm This is extremely rare. You know
what that means…
A 1 m long linear segment of wire has a linear mass
density of ρ= a x2 where a is a constant. Find the
location of the center of mass.
xCM = ¾ m
23. Angular acceleration ( α )
α = d ω /dt = d2θ /dt2
ω = ∫ α dt
The Big Three and the Forgotten Fourth (mutated
ω = ωo + αt
θ = θ o + ωo t + ½ αt2
ω2 = ωo2+ 2α θ
θ = ((ω + ωo)/2) t
Relationship between angular and linear quantities
24. Moment of Inertia (rotational inertia)
I = ∫ r2dm
Kinetic Energy of a rotating object
KE = ½ I ω2
-the ability of a force to cause an object to accelerate angularly
Units – Newton meters
τ = r x F = rF sin θ
Where r is the vector from the axis of rotation to the point where
the force is applied.
25. Vector cross product
A method of multiplying vectors that results in a vector
A x B = A B sinθ
Torque can also be found if the moment arm (perpendicular
distance from the axis to the line of action of the force) is known
τ = (moment arm) F
Newton’s Second Law analog
Work analog Power analog
26. Rotational Motion 2
Moment of Inertia
I = ∫ r2dm
Review the integral derivation of the MOI equation for
I = ∑mr2 for a mass moving around a point
Parallel Axis Theorem
Iparallel axis = I CM + MD2
Given that the I CM for a rod about its center is 1/12
ml2, what is the MOI for a rod around its end?
27. Angular Momentum
L = r x p = rp sin θ
Conservation of Angular Momentum
Rolling without Slipping
Torque Equilibrium problems