One-Dimensional Kinematics
MAIN POINTS
Definitions of Kinematic Quantities
x(t )
Displacement
Velocity
v(t ) ≡
dx(t )
dt
Velocity is the time rate of change of
displacement.
Acceleration
a (t ) ≡
dv(t )
dt
Acceleration is the time rate of change of
velocity.
Obtaining Displacement and Velocity from Acceleration
tf
Displacement
³
x(t f ) − x(t i ) = v(t ) dt
Displacement is the integral of the
velocity over time.
ti
tf
Velocity
v(t f ) − v(t i ) = ³ a(t ) dt
Velocity is the integral of the
acceleration over time.
ti
Special Case: Motion with Constant Acceleration
The displacement is
obtained by integrating
the velocity over time.
The velocity is obtained by
integratng the constant
acceleration over time.
Displacement
Velocity
Acceleration
v = v o + at
a = constant
x = xo + vo t +
1 2
at
2
Unit 2
MAIN POINTS
Kinematic Quantites as Vector Quantities
Vectors have magnitude and direction.
Displacement
&
x
Velocity
&
& dx
v≡
dt
Acceleration
&
& dv
a≡
dt
Vector Addition
&
&
The vector sum &of two vectors A and B is
another vector C .
C x = Ax + B x
C y = Ay + B y
Special Case: Projectile Motion
Projectile motion is the superposition of two independent motions:
1) Horizontal: constant velocity
2) Vertical: constant acceleration
Projectile motion can be understood simply as free fall viewed from a moving reference
frame.
Relative and Circular Motion
MAIN POINTS
Relative Motion
The velocity of an object in frame A can be found
from its velocity in frame B by adding (as vectors)
the relative velocity of the two frames.
&
&
&
vobject , A = vobject , B + vB , A
Uniform Circular Motion
An object moving with constant speed in a circular path has an acceleration whose
direction is always toward the center of the circle (centripetal acceleration) and whose
magnitude is proportional to the square of the speed and inversely proportional to the
radius of the circle.
Centripetal Acceleration
v2
ac =
= ω2R
R
Unit 4
MAIN POINTS
Force, Mass and Newton’s Second Law
Mass is the property of an object that determines how hard it is to change its velocity.
Force is the thing that is responsible for an object’s change in velocity.
Newton’s second law: A force that acts on an object causes that
object to accelerate in the same direction that the force acts, and
the magnitude of this acceleration is proportional to the
magnitude of the force
&
& FNet
a=
m
This law provides the link between mass and force.
Inertial Reference Frames and Newton’s First Law
Newton’s first law: An object subject to no external forces is at rest or moves with
constant velocity if viewed from an inertial reference frame. This law serves to define
“inertial reference frames.”
Newton’s Third Law
Newton’s third law: For every action there is
an equal and opposite reaction.
All forces come in pairs but act on different
objects.
&
&
FAB = − FBA
Unit 5
MAIN POINTS
Support Forces
Support forces, for example the normal force and the tension force, are what they have to
be to do what they have to do. Magnitudes are determined by Newton’s second law.
Spring Force
The spring force is a restoring force.
&
& &
Fspring = −k (x − x o )
Universal Gravitation
Any two objects with mass exert attractive forces on each
other with a magnitude that is proportional to the product
of the masses divided by the square of the distance
between them and whose direction lies along a line
connecting them.
Acceleration due to gravity (near the Earth’s surface):
g ≡G
M Earth
R
2
Earth
= 9.8 sm
2
Fgravity = G
m1 m 2
r2
Unit 6
MAIN POINTS
Friction Forces
The frictional force refers to the parallel component of the contact force between two
surfaces.
Kinetic Friction
Kinetic friction exists between surfaces in
relative motion.
fk = μk N
The direction of kinetic friction always
opposes the relative motion.
The magnitude of kinetic friction is
proportional to the normal force.
Static Friction
Static friction exists between surfaces not
in relative motion.
The direction of static friction always
opposes the relative motion that would
exist in the absence of friction.
The magnitude of static friction must be
determined from Newton’s second law. Its
maximum value is proportional to the
normal force.
The static frictional force is what it has to
be to do what it does!
fs ≤ μs N
Unit 7
MAIN POINTS
Work and Kinetic Energy Definitions
K≡
The kinetic energy of an object is defined to be ½ the
product of the mass and the square of its velocity.
The work done by a force as an object is moved between
two points along some path is defined to be the dot
product of the force with the displacement along the path
between those two points.
1
mv 2
2
&
r2
& &
W1→2 ≡ ³ F ⋅ dl
&
r1
The Work-Kinetic Energy Theorem
Integrating Newton’s second law, we obtain the workkinetic energy theorem: the work done by the net force on
an object as it moves between two points is equal to the
change in its kinetic energy.
W Net = ΔK
Conservative Forces
Conservative forces are defined to be those forces in which the work done by them does
NOT depend on the path, only on the endpoints.
Work Done by Two Conservative Forces
W = −mg Δy
Gravitational Force
Spring Force
§1 1·
W1→ 2 = GM E m¨¨ − ¸¸
© r2 r1 ¹
W1→ 2 = −
1
k ( x 22 − x12 )
2
(near Earth)
(general expression)
Conservative Forces and Potential Energy
MAIN POINTS
Potential Energy
The change in potential energy that is associated
with a specific conservative force as an object
moves between two locations is defined as
negative the work done by that force between
those two locations.
A potential energy function can be defined for the
object and the particular force by choosing a
specific location as the zero of the function.
ΔU = U B − U A ≡ −WA→B
U = −Wr →r + U o
o
Conservation of Mechanical Energy
The mechanical energy of an object is defined to
be the sum of its kinetic and potential energies.
E Mechanical ≡ K + U
The work-kinetic energy theorem can be reformulated as a conservation law. Whenever
the work done by non-conservative forces is zero, the mechanical energy of that object is
conserved.
W Non −Conservative Forces = 0 Ÿ ΔE Mechanical = 0
Unit 9
MAIN POINTS
Macroscopic Work Done by Friction
The work done by a kinetic friction force acting on a deformable object cannot be
calculated without an understanding of the nature of the interactions at the surfaces.
However, the work-kinetic energy theorem can
still be applied if we consider all forces acting at
the center of mass of the object. The integral of
the friction force through the full displacement of
the
object
is
called
the
macroscopic work done by
Wmacroscopic
friction. This macroscopic work
does determine the change in the
mechanical energy of the object.
Force from Potential Energy
For conservative forces, the force
acting on any object at any point is
equal to minus the spatial derivative of
the potential energy function of the
object at that point.
Equilibrium positions for a
particle
acted
on
by
conservative forces are those
locations where the slope of
the potential energy function
is zero.
& &
Wmacroscopic friction ≡ ³ f k ⋅ dx
friction
= ΔE
(1-D)
F ( x) = −
dU ( x)
dx
If friction is the only nonconservative force acting
(3-D)
&
&
F = −∇U
Center of Mass
MAIN POINTS
Definition of Center of Mass
The center of mass
of a system of
objects is defined
to be the massweighted average
of its components.
(Discrete Particles)
&
RCM =
1
M Total
&
¦ mi ri
N
(Continuous Mass Distribution)
&
RCM =
i =1
1
M Total
&
r
³ dm
Equation of Motion for the Center of Mass
Applying Newton’s second law to a system of
particles, we obtain the equation of motion
for the center of mass.
&
&
FNet , External = M Total ACM
The Center-of-Mass Equation
Integrating the equations of motion for
the center of mass, we obtain the
&
&
§1
2 ·
center-of-mass equation that relates the
Δ¨ MVCM ¸ = FNet , External ⋅ dlCM
©2
¹
change in the kinetic energy of the
center of mass (calculated as if the
system were a particle having the total
mass of the system and moving with the velocity of the center of mass) to the macroscopic
work done by the toal external force (calculated as if all forces were acting at the center
of mass).
³
Conservation of Momentum
MAIN POINTS
Conservation of Momentum
If the sum of the external forces acting on any system of particles is zero, then the total
momentum of the system, defined as the vector sum of the momenta of the individual
particles, is conserved.
&
&
When FNet , External = 0 Ÿ PTotal ≡
&
¦p
i
= Constant
i
Forces in a Collision
Internal forces determine the amount of energy
lost in a collision.
If only internal forces act during a collision, the
total momentum of the system will be
conserved.
Center-of-Mass Reference Frame
The center-of-mass reference frame is defined
to be that frame in which the total momentum of
all particles in the system is zero. In other
words, it is the frame in which the center of
mass of the system is at rest.
Conservation of momentum calculations are
often simplified in the center-of-mass frame.
Center-of-Mass Reference
&
&
PTotal = M Total VCM = 0
Elastic Collisions
MAIN POINTS
Elastic Collisions
If the only forces acting
during a collision are
conservative forces, then
the kinetic energy of the
system, defined to be the
sum of the kinetic energies
of the colliding objects, is
conserved. Such collisions
are called elastic collisions.
¦K = ¦K
i
f
Center-of-Mass Frame
Elastic collisions are most simply described in
the center-of-mass reference frame of the
colliding objects.
The collision may cause objects to be
deflected through some angle in the frame, but
their speeds will always remain the same.
Center-of-Mass Reference Frame
&
&
PTotal = M Total VCM = 0
Unit 13
MAIN POINTS
Elastic Collisions: Relative Speeds
The rate at which two
objects approach each
other before an elastic
collision is equal to the
rate at which they
separate afterward.
Relative Speeds Before and After an Elastic Collision
&
&
&
&
&
&
&
&
v 2, f − v1, f = v 2,i − v1,i = v 2*,i − v1*,i = v 2*, f − v1*, f
Impulse
The impulse is defined as the
integral of the force over the time
of the collision.
Integrating Newton’s second law
over time, we find that the
impulse is equal to the change in
momentum during the collision.
&
&
F
³ Net dt = Δp
t2
Impulse
t1
Kinetic Energy of a System of Particles
The kinetic energy of a
system
of
particles,
defined as the sum of the
kinetic energies of the
particles in the system, is
equal to the kinetic energy
of the particles relative to
the center of mass, a term
that is the same in all
reference frames, plus the
energy of the center of
mass, a term that does
depend on the reference
frame of the observer.
K system ,lab ≡ ¦
i
2
1
1
1
2
m i v i2 = ¦ m i v i* + M Total v CM
2
2
2
i
Unit 14
MAIN POINTS
Rotational Kinematics
Rotational motion is described in terms of the
(1) angular displacement , (2) the angular
velocity , and (3) the angular acceleration .
The displacement, velocity, and
acceleration of any point that is rotating
is proportional to the corresponding
angular parameter.
dθ
dt
ω≡
α≡
dω
dt
s = Rθ
v = Rω
a = Rα
Moment of Inertia & Kinetic Energy
The moment of inertia of a system of
particles about an axis is defined to be
the sum of the product of the mass and
the square of the distance from the axis
for all parts of the system.
Discrete Distributions
I≡
¦m r
Continuous Distributions
³
2
I = r 2 dm
i i
Rotational Kinetic Energy
The kinetic energy of a system of particles is equal to
one-half the product of the square of the angular
velocity and the moment of inertia about the axis of
rotation.
Moment of Inertia of
Cylinders and Spheres
The moment of inertia for any
cylindrical or spherical object is
proportional to the product of
the square of the radius of the
object and the total mass of the
object.
K system =
1 2
Iω
2
Parallel Axis Theorem and Torque
MAIN POINTS
Parallel Axis Theorem
The moment of inertia about a chosen axis is
equal to the moment of inertia about a parallel
axis passing through the center of mass plus
the moment of inertia of the center of mass,
treated as a point particle, about the chosen
axis.
I Total = I CM + MD 2
Torque
The concept of torque plays the role for
rotational motion that force does for
translational motion.
&
&
&
τ ≡r×F
τ = rF sin θ
The direction of the torque vector is
determined by a right-hand
rule: curl
&
&
fingers from r into F and then thumb
points in direction of torque.
Dynamics Equation for Rotational Motion
The equation that determines the dynamics of rotational
motion is derived from Newton’s second law.
The net torque about an axis is equal to the product of
the moment of inertia about that axis and the angular
acceleration.
&
&
τ = Iα
Unit 16
MAIN POINTS
Center of Mass Equation for Rotational Motion
Integrating the rotational dynamics equation, we
determined that the change in the rotational kinetic
energy is equal to the integral of the torque over the
angular displacement.
θf
³τ
Net
θi
§1
·
dθ = Δ¨ I CM ω 2 ¸
©2
¹
Translation + Rotation: Rolling Without Slipping
The total kinetic energy of an object is equal to
the sum of the translational kinetic energy of the
center of mass and the rotational motion about
the center of mass.
Applying
the
center-of-mass
equation and its rotational analog, we
determined that the change in the
total kinetic energy of a solid ball
rolling without slipping down the
ramp is equal to the work done by gravity.
K Total =
1
1
2
MvCM
+ I CM ω 2
2
2
§1
§1
2 ·
2·
Δ¨ MvCM
¸ + Δ¨ I CM ω ¸ = W gravity
¹
©2
¹
©2
Applying Newton’s second law and the rotational dynamics
equation, we determined the acceleration of a solid ball down
the ramp.
aCM =
5
g sin θ
7
Rotational Statics: Part I
MAIN POINTS
Torque Due to Weight of Object
The torque produced by the weight of a
solid object can be calculated by simply
assuming the entire mass of the object is
located at its center of mass.
&
&
&
τ gravity = RCM × Mg
Lever Arm Calculation of Torque
The magnitude of any torque can be
calculated as the product of the force
and its lever arm, the perpendicular
distance of a line through this force and
the rotation axis.
τ = r⊥ F ≡ (r sin θ ) F
Conditions for Static Equilibrium
The condition for static equilibrium is
that the acceleration of the center of
mass and the angular acceleration about
any rotation axis must be zero.
Therefore, for a system to be in static
equilibrium, the sum of the forces
acting on the system must be zero and
the sum of all torques produced by these
forces must also be zero.
&
&
¦ F = ma = 0
&
&
¦ τ = Iα = 0 (about any axis)
i
i
CM
Rotational Statics: Part II
MAIN POINTS
Gravitational Potential Energy
The gravitational potential energy
of an extended object is equal to
the product of the weight of the
object and the vertical displacement
of its center of mass from the
height chosen to be the zero of
potential energy.
U gravity = MgYCM
Condition for Static Equilibrium
The position for static equilibrium of a suspended
object can be determined in two equivalent ways:
1) The torque about the suspension axis is zero
( τ Net = 0 ).
2) The gravitational potential energy is
minimized ( dU gravity / dy = 0 ).
Condition for Stability
The condition for the stability of an extended object placed on a surface is that its center
of mass must be located over its footprint on the surface.
Unit 19
MAIN POINTS
Angular Momentum Definition
&
The angular momentum L of a
point particle about some axis is
defined to be the cross product of
&
r , the vector from the axis to the
&
particle, with p , the momentum
vector of the particle.
&
&
LTotal = ¦ Li
& & &
L≡r×p
The angular momentum of a
system of particles about a fixed
axis is equal to the vector sum of
the individual angular momenta.
The angular momentum for a
system of particles rotating about a
common axis with a fixed angular
velocity is equal to the product of
the moment of inertia of the system
about the axis and the angular
velocity vector.
&
&
L = I systemω
Rotational Equation of Motion
We used Newton’s second law to obtain the
rotational equation of motion, namely, that the
sum of the external torques acting on a system is
equal to the time rate of change of the angular
momentum of the system.
&
τ Net , External
&
dLTotal
=
dt
Consequently, if the sum of the external torques on a system is zero, the angular
momentum of the system is conserved.
Angular Momentum Vector and Precession
MAIN POINTS
Conservation of Momentum Examples
The angular momentum of a system is
conserved when there are no external
torques acting on it.
Example 1: If a student sitting on a stool
flips a spinning top over, he will begin to
rotate in the direction of the initial
orientation of the top.
Example 2: Both the angular velocity
and the kinetic energy of a particle
executing circular motion with a
slowly decreasing radius will increase
by a factor equal to the ratio of its
initial to its final moment of inertia
about the center.
ωf =
Ii
ωi
If
Kf =
Ii
Ki
If
Precession Rate of a Gyroscope
The weight of a spinning disk
provides a torque that changes the
angular momentum of the system.
In particular, the torque is
perpendicular
to
the
angular
momentum vector arising from the
spinning of the disk, causing the disk
to precess in the horizontal plane at a
rate that is given by the ratio of the
torque to the angular momentum due
to the spin.
Precession Rate
Ω=
τ weight
L
where
Ω≡
dφ
dt
Simple Harmonic Motion
MAIN POINTS
Simple Harmonic Motion
A mass, resting on a horizontal
frictionless surface connected to a
spring, will oscillate about its
equilibrium position with a frequency
that is determined by the spring
constant and the mass. This system is
an example of simple harmonic
motion.
The amplitude and the phase of these
oscillations must be determined from
the initial conditions, a specification
of how the oscillation got started.
x(t ) = A cos(ωt + φ )
v(t ) = −ωA sin(ωt + φ )
a(t ) = −ω 2 A cos(ωt + φ )
Angular Velocity
(or Angular Frequency)
ω=
k
m
Uniform Circular Motion
Simple harmonic motion specified by an angular frequency and an amplitude A can also
be represented as the projection of an object moving in a circular path of radius A with
constant angular velocity along an axis that passes through the center of the circle.
Simple and Physical Pendula
MAIN POINTS
Torsion Pendulum
A disk, suspended by a wire through its symmetry
axis, is an example of a torsion pendulum. The
restoring torque provided by the wire is
proportional to the angular displacement of the disk
about its axis. The disk executes simple harmonic
motion with a frequency proportional to the square
root of the torsion constant divided by the moment
of inertia about its symmetry axis.
Newton’s 2nd Law for Rotations
d 2θ
= −ω 2θ
dt 2
where
ω=
κ
I
Physical Pendulum
An object that executes oscillations, that are
determined by the restoring torque provided by the
weight of the object itself, is called a physical
pendulum.
In the limit that the angular displacement from
vertical is small, the motion can be described as
simple harmonic motion with a frequency
proportional to the square root of the weight of the
object to its moment of inertia about the pivot
times the distance between the pivot and the center
of mass of the object.
Newton’s 2nd Law for Rotations
d 2θ
= −ω 2θ
2
dt
where
ω=
MgRCM
I
Unit 23
MAIN POINTS
Transverse Harmonic Waves
y ( x, t ) = A cos( kx − ω t )
The displacement of an element of
the medium through which a
transverse harmonic wave propagates
is described as a sinusoidal function
of space and time.
2π
Period of Oscillation
P=
Wave Number
k=
Speed of Wave
vwave =
The speed of such a wave is given by
the product of its wavelength and its
frequency.
ω
2π
λ
λ
P
=
ω
k
The Wave Equation
Applying Newton’s second law to a string under
tension yields an equation that holds for any wave
on a string.
The velocity of the wave is identified as the square
root of the tension in the string divided by its mass
density.
Energy in Waves
Energy propagates with the wave and its maximum
value is proportional to the square of the product of
its amplitude and its frequency.
1 d2y
d2y
=
dx 2 v 2 dt 2
v wave =
T
μ
Max Kinetic Energy of a
String Element
K max ∝ ω 2 A 2
= fλ
Unit 24
MAIN POINTS
Wave Equation Solutions
Any function whose argument is
proportional to ( x ± vt ) is a
solution to the wave equation
with velocity equal to v.
y ( x, t ) = f ( x − vt )
y ( x, t ) = f ( x + vt )
Waves traveling in the +x direction
Waves traveling in the −x direction
The velocity of mechanical waves is obtained by applying Newton’s second law to
produce an appropriate equation of motion.
The velocity of the wave is determined totally by the properties of the medium.
Superposition and Standing Waves
y ( x, t ) = 2 A cos(kx) cos(ωt )
If more than one wave is excited in a
medium, the resulting motion is given
by the displacements of the individual
waves at each point at each time.
If two waves of equal amplitude and
frequency travel in the same medium,
but with opposite velocities, the
resultant motion will be a standing
wave.
Standing Waves in a String
An initial wave propagated on a string that has both ends fixed will be reflected from the
downstream fixed end, producing an additional wave which, when added to the initial
wave, gives a standing wave.
Unit 25
MAIN POINTS
Pressure and Depth Relationship
Pressure
The pressure in a fluid is defined to be equal
to the average force per unit area exerted by
the molecules of the fluid during their
collisions with the container walls.
Setting the net force on any volume of the
fluid, we obtain the result that the difference
in pressure between two points in the fluid is
proportional to the vertical separation of the
two points.
P=
Faverage
Area
P2 − P1 = ρgh
Archimedes’ Principle
The buoyant force exerted by the fluid on
any object placed in it is determined from
the relationship between the pressure and
depth.
Buoyant Force
FB = ρliquidVdisplaced g
Archimedes’ principle is derived from this
determination of the buoyant force. Namely,
the buoyant force exerted on an object is
equal to the weight of the fluid it displaces.
Floating
Archimedes’ principle can be used to
determine that the submerged fraction of a
floating object is equal to the ratio of the
mass density of the object to the mass
density of the fluid.
Submerged Fraction
of a Floating Object
Vdisplaced
Vobject
=
ρobject
ρliquid
Fluid Dynamics
MAIN POINTS
Continuity Equation
We used the fact that an ideal fluid is incompressible (i.e., its density does not change) to
derive the continuity equation: The product of the speed of the flow and the crosssectional area of the pipe through which an ideal fluid flows is a constant!
A1v1 = A2 v 2
Bernoulli’s Equation
We used the work-kinetic energy theorem to determine that the sum of the work done by
the pressure of the fluid and kinetic energy density is a constant at any point in the flow.
By considering ideal fluids that
flow from one height to
another, we used the workkinetic energy theorem to
derive Bernoulli’s equation:
that the change in pressure of
an ideal fluid between two
points is equal to the sum of
the changes in the kinetic
energy density and the
gravitational potential energy
density of the fluid.
P2 − P1 =
1
ρ (v12 − v 22 ) + ρg ( h1 − h2 )
2