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AP C UNIT 3
WORK & ENERGY
SCALAR PRODUCT
or DOT PRODUCT
Dot Product is defined
as the

magnitude of 1st ( a ) times
scalar
component of 2nd vector

( b ) along direction of 1st.
Essentially, the dot product
gives you information about how
much of each vector lies along
the direction of the other.
 
a  b  ab cos 
it’s a scalar result where Ф is
the angle between a and b.

a


b
THIS IS NOT THE SAME
AS ADDING 2 VECTORS
YIELDING A RESULTANT
The reason the dot product is
used in physics is because the
operation between certain vector
quantities produce meaningful
physical answers such as WORK.
Since the dot product involves the cosine then
IF Ф = 90o THEN PRODUCT = 0
IF Ф = 0o THEN PRODUCT = MAXIMUM
This is consistent in that when vectors are perpendicular neither lies along
the other, therefore an answer of zero results.
Directional properties of the Dot Product include:






i  i  j j  k  k  1

and





i  j  j k  i  k  0
Unit vector form of dot product:






 
a  b  (a x i  a y j  a z k ) (bx i  by j  bz k )
if you distribute, this would reduce to…
 
a  b  axbx  a y by  az bz
*Note that there is no direction associated with result but
answer can be negative depending on angle.
Calculate the dot product of the following
vectors and find the angle between them:
A = -3i + 5j
B = 6i +14j
More Calculus  - Derivative of a Product:
When taking the derivative of two functions multiplied
together, the derivative is: The 1st function times the derivative
of the 2nd plus the derivative of the 1st times the 2nd function.
d
dv
du
uv  u  v
dx
dx
dx

 u v  v u
'
'
Example…find dy/dx
y  (3x  1)( x  2)
2
u
v
y  (3x 2  1)( x  2)
dy
 (3 x 2  1)  1  ( x  2)  6 x
dx
dy
2
 9 x  12 x  1
dx
*Could have ‘foiled’ and then performed
power rule as well
Chain Rule:
The chain rule is used when there is a function within a function.
f‘ (x) = f‘( g(x) ) (g'(x))
f (x) = f ( g(x) )
Think of the functions f and g as ``layers'' of a problem.
Function f is the ``outer layer'' and function g is the ``inner
layer.'' Thus, the chain rule tells us to first differentiate the outer
layer, leaving the inner layer unchanged (the term f'(g(x))) ,
then differentiate the inner layer (the term g'(x) ).
f ( x)  ( x  5)
3
The inner layer, g(x) is
3
1
2

f  2( x  5)  3x
2
(x3
+ 5)
The outer layer f(x) is (x)2
Derivative of outer times derivative
of inner with respect to x
f   6 x  30 x
5
2
Example:
Find the
derivative of f’
function of a
function
1
f ( x)  2
x 3
Work Done by a Constant Force
Work is defined as an external force
(F) moving through a displacement
(Δr). How much force lies along the
movement of an object.
Positive & Negative Work
In all 4 cases, the force has the same magnitude
and the displacement of the object is to the right
with the same magnitude. Rank the situations from
most positive to most negative.
A crate of mass, M, is dragged along a level
rough surface a distance, x, by a force, F as
shown. The coefficient of friction is uk.
Find the net work done on the crate in
terms of given variables and constants.
Work done by a varying force
x2
W   F ( x)dx
x1
If F(x) = 4x2 then find
the work done on a
particle that moves from
x = 1m to x = 5m.
Example:
Suppose a mass moves with a trajectory defined by
the position vector
r (t )  (te
t

10
2 ˆ
ˆ
)i  (t  3t ) j
Find the work done by the force, F  10iˆ  4 ˆj
over the interval from t = 1 to t = 2.
Work-Energy Theorem
x2
W   F ( x)dx
x1
A force, F(x) = 2-4x, acts on a 7.0kg mass. What is
the final speed of the mass as it is moved from x=5m
to x=2? Assume mass starts from rest at t=0.
Hooke’s Law
Work done by Spring
Negative means that the force opposes
the displacement from equilibrium
If block is pulled to right, the
force by spring is NOT
constant via Hooke’s Law.
Therefore, the work done by
spring must use avg force or
be integrated as:
Fs
A plot of spring force vs
displacement reveals a slope
equal to spring constant, k
k1
k1
kk 2
k1
2
k2
Two springs are attached to a block in series and
parallel as shown above. Determine the effective spring
constant for each situation in terms of k1 and k2 .
What would spring
constant be if a mass was
attached to a massless
spring that stretched a
distance x?
What if same mass was
attached to 2 springs as
shown? How would stretch,
x, differ for each spring?
Power
Rate at which work is done or energy is
transferred
A 4kg particle moves along the x-axis. Its position
varies with time according to x = t + 2t3, where x is in
meters and t is in seconds. Find the the power
being delivered to the particle at any time t
OR
If a projectile thrown directly upward reaches a
maximum height h and spends a total time in the air T, the
average power of the gravitational force during the
trajectory is:
a) 2mgh / T
b) -2mgh / T
c) 0
d) mgh / T
e) -mgh / T
Potential Energy
As height above Earth increases…
Conservative &
Non-conservative Forces
The work a conservative force does on an object in
moving it from A to B is path independent - it depends
only on the end points of the motion.
Force of gravity and the spring force are conservative
forces. Conservative forces ‘store’ energy…available
for kinetic energy
The work done by non-conservative (or dissipative)
forces in going from A to B depends on the path
taken. Friction is non-conservative. Nonconservative forces don’t ‘store’ energy.
CONSERVATION OF NRG
Total energy in a closed system remains unchanged
Work done by NC forces or friction is positive in the
above formula. No need to put in minus sign. Work
done by friction occurs on left side as minus but
becomes + when taken to other side.
Energy worksheet
Potential Energy Function
Diagrams
Us (x) vs Restoring Force
Relationship
f
o
r
c
e
position
position
Potential Energy and Conservative
Restoring Forces (gravity, spring)
The instantaneous
restoring force is equal to
the negative derivative of
the potential energy
function.
In the case of a mass oscillating on a
horizontal frictionless surface we can
verify this relationship:
Example1
A certain spring is found not to obey Hooke’s Law; it
exerts a restoring force F(x) = -60x-18x2.
a) Calculate the potential energy function U(x) for
this spring. Let U = 0 when x = 0.
b) An object with mass 0.90kg on a frictionless,
horizontal surface is attached to this spring, pulled a
distance 1.00m to the right to stretch the spring and
released. What is the speed of the object when it is
0.50m to the right of x=0?
Potential energy
diagram states of
equilibrium:
Points of equilibrium
are where the force is
zero (slope = zero).
x3 and x5 are points of stable equilibrium or energy wells. If the system is
slightly displaced to either side the forces on either side will return the
object back to these positions.
x6 is a position of neutral equilibrium. Since there is no net force acting
on the object (slope of U(x) = 0) it must either possess only potential
energy and be at rest or, it also possesses kinetic energy and must be
moving at a constant velocity.
x4 is a position of unstable equilibrium. If the object is displaced ever so
slightly from this position, the internal forces on either side will act to
encourage further displacement instead of returning it back to x4.
Example2
A 5-kg mass moving along the xaxis passes through the origin with
an initial velocity of 3m/s. Its
potential energy as a function of its
position is given in the graph.
a) How much total energy does
the mass have as it passes
through the origin?
b) Between 2.5m and 5m, is the
mass gaining speed or losing speed?
c) How fast is it moving at 7.5m?
d) How much potential energy would have
to be present for the mass to stop moving?
Turning points
Positions where
potential energy
equals the total
mechanical energy,
Umax= E, are called
turning points
A particle moves along the x-axis according to
the following potential energy function:
2.4 Nm
U ( x)  (0.60 N ) x 
x
2
Find the positions of equilibrium for the particle.
Find F(x) when
 ax
U ( x)  2
2
b x
CENTER OF MASS
Center of Mass (COM)
COM = average position of mass of an object. A
point where all of the mass could be considered to
be located.
Similarly we can show that:
Example: A rectangular wire frame
lies in the xy plane. If the linear
mass density of the wires is λ, except
for the bottom segment which has
twice the density, where is the center
of mass of the wire frame?
a )( a / 2, b / 2)
a  2a  2b  b
b) 
,

2  3a  2b  2
a 2b
c) ,
2 3
a b  2a  2b 
d) , 
2 2  3a  2b 
a b  3a  2b 
e) , 
2 2  2a  2b 
(a, b)
Example
A 40kg woman and a 55kg man stand at the left and
right ends of a 3.0m long plank. The plank is floating
in water. Ignore frictional effects between the water
and the plank. The mass of the plank is 15.0kg. The
woman is intrigued by the amazing man and walks
over to him on the right side. How far does the plank
move?
• C.O.M. acts like mass of system…it can have vcom
and acom
• C.O.M. remains constant through problem
• Ie; explosion: com of fragments would follow
parabola
Linear Momentum
F  ma
Momentum is a vector
Units = kgm/s
Impulse
Impulse equals the change in momentum and is
also equal to the area under a force-time curve
given by:
Units = Ns
Impulse is a vector
Avg Force:
Same area that
is under Ft
curve.
Favg
Example1
Momentum of object moving along x-axis is
given by p(t) = 2t2 -5t + 3.
a) Find net force at t = 0.50s
b) Find net force when object first reverses
original direction.
c) Find net impulse between 0 and 0.80s.
Example2:
A batter strikes a ball (m =0.145kg) with force
given by F = (1.6x107 t – 6.0x109 t2 )i between 0
and 2.5ms. At t =0, v = -(40i + 5j).
a) Find impulse from bat on ball from t= 0 to t = 2.5ms.
b) Find the impulse of gravity on ball for same time.
c) Find avg force on ball by bat
d) Find momentum & velocity of ball at t = 2.5ms in
unit vector notation.

Conservation of p
Internal forces
External forces
Types of
Collisions:
ELASTIC: Both momentum & KE are both
conserved. Whatever KE is spent deforming
objects is recovered after objects separate.
INELASTIC: KE is not conserved (some is
used to produce sound, heat, etc…nonconservative processes)
TOTALLY INELASTIC: Same as above, but
objects remain stuck together after collision.
Example 1:
A block of mass M is resting on a horizontal,
frictionless table and is attached as shown to a
relaxed spring of spring constant k. A second block of
mass 2M and initial speed vo collides with and sticks
to the first block. Develop expressions for the
following quantities in terms of M, k, & vo.
A) Find v, the speed of the blocks immediately
after impact.
B) x, the maximum distance the spring is compressed
EXAMPLE 2:
A track consists of a frictionless arc XY, which is a quarter-circle
of radius R, and a rough horizontal section YZ. Block A of
mass M is released from rest at point X, slides down the curved
section of the track, and collides instantaneously and
inelastically with identical block B at point Y. The two blocks
move together to the right, sliding past point P, which is a
distance l from point Y. The coefficient of kinetic friction
between the blocks and the horizontal part of the track is  .
Express your answers in terms of M, l, , R, & g.
A) Determine the speed of the combined blocks
immediately after the collision.
B) Determine the total KE lost from point X to
point P.
Example 3:
2 identical masses are connected by a spring (k)
resting on a frictionless horizontal surface. The
right mass is initially in contact with the wall. A brief
blow to the left block leaves it with an initial velocity
vo to the right. After the spring is maximally
compressed, it moves to the left, away from the
wall continuing to oscillate.
a) What is the net momentum of the two masses after they
leave the wall?
b) What is the maximum compression of the spring after
the two masses have left the wall?
Example4
A railroad handcar is moving along straight, frictionless tracks
with negligible air resistance. In the following cases, the car
initially has a total mass (car + contents) of 200kg and is
traveling at 5.00m/s, east. Find the final velocity of the car in
each case:
a) A 25.0kg mass is thrown sideways out of the car with a
velocity of 2.00m/s relative to the car’s initial velocity.
b) A 25.0kg mass is thrown backward out of the car with a
velocity of 5.00m/s relative to the initial motion of the car.
c) A 25.0kg mass is thrown into the car with a velocity of 6.00m/s relative
to the ground and opposite in direction to the initial velocity of the car.
A massive frog drops vertically from a tree branch onto
a skateboard that moves horizontally below. When the
frog lands, the skateboard slows, consistent with the
conservation of momentum. The impulse that slows the
skateboard is
a) The friction force of the frog’s feet acting backward on
the skateboard x time during which the speed changes
b) equal and opposite to the impulse
that brings the frog up to speed
c) both of these
d) neither of these
2D Collisions:
Momentum must be conserved in both x and y
directions.
Example: A car with mass 1000kg is moving north at
15m/s. At an intersection, it collides with an SUV with
mass 2000kg, traveling east at 10m/s. Both vehicles
become entangled and move off together after the
collision. You can ignore any external forces.
a) Determine the velocity of the wreckage
immediately after the collision.
b) Determine the % of KE lost during the collision.
Example2
Spheres A (0.020kg), B (0.030kg), and C (0.050kg) are each
approaching the origin as they slide across a frictionless air
table. The initial velocities of A and B are given. All 3
spheres arrive at the same time at origin and stick together.
vB = 0.50m/s
60o
vA=1.50m/s
C
a)What must be the velocity of C if all three objects
end up moving at 0.50m/s east after the collision?
b) Find the change in KE of the system as a result
of the collision
Example3
An elastic collision occurs between 2 pucks on a frictionless
air hockey table. Puck A has initial speed of 4.00m/s east
(mass 0.500kg) and final speed of 2.00m/s in an unknown
direction. Puck B has a mass of 0.300kg and is initially at
rest. Find the final speed of B and the unknown angles.
A
α
β
B
Example: A mass m sits at top of smooth
semicircular cutout of larger mass M. M sits on a
frictionless surface.
m
R
M
a) When m is released from rest, describe the motion of
the 2 masses.
b) When the mass is at the bottom of the circle, what is its
speed with respect to the table?
c) If M were fixed to the table, would this increase or
decrease the maximum speed of m?
d) When m has reached the top left side of the cutout, how
far has M been displaced horizontally?
Example
A projectile is fired from the edge of a cliff 100m high with
initial speed of 60m/s at an angle of 45o.
A) Calculate the x-y location and velocity coordinates of
the particle at t = 5s.
Suppose the projectile experiences an internal explosion at
t=4s with an internal force purely in the y-direction, causing it
to break into a 2kg and 1kg fragment.
B) If the 2kg fragment is 77m above the height of the
cliff at t = 5s, what is the y position of the 1kg piece?
C) If the speed of the 2kg fragment is 46m/s and the fragment
is falling at t=5s, what is the y component of the velocity of the
1kg fragment?