Topic Summaries: Mechanics

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Introduction
- Mechanics is the study of motion, forces, and energy.
- Mechanics can be divided into subdisciplines.
  - Kinematics is the branch of mechanics that deals with motion without regard to forces or energy.
  - Dynamics is the branch of mechanics that deals with motion and forces together.
  - Statics is the branch of mechanics that deals with forces in the absence of changes in motion or energy.
  - The work-energy theorem is the is the central principle of mechanics that relates forces and energy.
  - The conservation of energy is the central principle of science that deals with energy in all of its forms.
  - The conservation of mechanical energy is the central principle of mechanics that deals with the forms of energy related to motion.
- There are several types of motion …
  - Translational motion results in a change of location.
  - Oscillatory motion is repetitive and fluctuates between two locations.
  - Rotational motion occurs when an object spins.
  - Chaotic motion is predictable in theory but unpredictable in practice, which makes it appear random.

Kinematics

Distance & Displacement

Distance is a scalar measure of the interval between two locations measured along the actual path connecting them.
- As a scalar it has magnitude only.
- Δs (italic) is the symbol for distance.
  (The origin of this symbol is from the Latin word for space: spatium.)
Displacement is a vector measure of the interval between two locations measured along the shortest path connecting them.
- As a vector it must be stated with both magnitude and direction.
- Δr (boldface) is the symbol for displacement.
  (The origin of this symbol is from the Latin word for spoke: radius.)

The relation of distance to displacement

The distance traveled to get from one location to another is always greater than or equal to the magnitude of the displacement between the two locations.

Δs ≥ |Δr|

Distance approaches the magnitude of displacement as distance approaches zero.


Δs → 0	⇒	Δs → \|Δr\|

The SI unit of distance and displacement is the meter [m].

Speed & Velocity

Speed is the rate of change of distance with time.
- As a scalar it has magnitude only.
- Average speed …
  - is measured over a non-zero time interval and
  - is represented by the symbol v_ave or v (overline)
- Instantaneous speed …
  - is the limit of average speed as the time interval approaches zero,
  - is the first derivative of distance with respect to time, and
  - is represented by the symbol v (italic)
Velocity is the rate of change of displacement with time.
- As a vector it must be stated with both magnitude and direction.
- Average velocity …
  - is measured over a non-zero time interval and
  - is represented by the symbol v_ave or v (overline)
- Instantaneous velocity …
  - is the limit of average velocity as the time interval approaches zero,
  - is the first derivative of displacement with respect to time, and
  - is represented by the symbol v (boldface)

The various forms of speed and velocity are defined by the following equations …

v =	Δs
	Δt

average
speed

v =	lim	Δs	=	ds
v =	Δt → 0	Δt	=	dt

instantaneous
speed

v =	Δr
	Δt

average
velocity

v =	lim	Δr	=	dr
v =	Δt → 0	Δt	=	dt

instantaneous
velocity

The relation of speed to velocity

An object's average speed approaches the magnitude of its average velocity as the time interval approaches zero.


Δt → 0	⇒	v → \|v\|

The instantaneous speed of an object is the magnitude of its instantaneous velocity.

v = |v|

The SI unit of speed and velocity is the meter per second [m/s].

Acceleration

Acceleration is the rate of change of velocity with time.
- As a vector it must be stated with both magnitude and direction.
Acceleration occurs anytime an object's …
- speed increases,
- speed decreases, or
- direction of motion changes.
Average acceleration …
- is measured over a non-zero time interval and
- is represented by the symbol a_ave or a (overline)
Instantaneous acceleration …
- is the limit of average acceleration as the time interval approaches zero,
- is the first derivative of velocity with respect to time,
- is the second derivative of displacement with respect to time, and
- is represented by the symbol a (boldface)

The various forms of acceleration are defined by the following equations …

a =	Δv
	Δt

average
acceleration

a =	lim	Δv	=	dv	=	d²r
a =	Δt→0	Δt	=	dt	=	dt²

instantaneous
acceleration

The SI unit of acceleration is the meter per second squared [m/s²].
The acceleration due to gravity …
- is a natural unit of acceleration,
- is represented by the symbol g (roman),
- is equal to 9.80665 m/s² by definition,
- is often rounded to 9.8 m/s² for convenience, and
- is sometimes called the "g force" even though it is not a measure of force.

Equations of Motion

The equations of motion are valid only when …
- acceleration is constant and
- motion is constrained to a straight line.

traditional name		equation		relationship
The One Dimensional Equations of Motion for Constant Acceleration
1st	equation	v =	v₀ + at	velocity	– time
2nd	equation	x =	x₀ + v₀t + ½ at²	displacement	– time
3rd	equation	v² =	v₀² + 2a(x − x₀)	velocity	– displacement
merton rule		v =	½ (v + v₀)	average velocity

Falling Bodies
- Free fall occurs whenever an object is acted upon by gravity alone.
  - An object in free fall experiences an acceleration due to gravity.
- The acceleration due to gravity …
  - is a natural phenomena,
  - is represented by the symbol g (italic),
  - varies with location,
  - is effectively 9.8 m/s² over the entire surface of the earth, and
  - is independent of mass.
- The acceleration due to gravity …
  - is a natural unit of acceleration,
  - is represented by the symbol g (roman),
  - is equal to 9.80665 m/s² by definition,
  - is often rounded to 9.8 m/s² for convenience, and
  - is sometimes called the "g force" even though it is not a measure of force.
Graphs of Motion
- On a displacement-time graph …
  - slope equals velocity.
  - the "y" intercept equals the initial displacement.
  - when two curves coincide, the two objects have the same displacement at that time.
  - straight lines imply constant velocity.
  - curved lines imply acceleration.
  - an object undergoing constant acceleration traces a portion of a parabola.
  - average velocity is the slope of the straight line connecting the endpoints of a curve.
  - instantaneous velocity is the slope of the line tangent to a curve at any point.
  - positive slope implies motion in the positive direction.
  - negative slope implies motion in the negative direction.
  - zero slope implies a state of rest.
  - The area under the curve is meaningless
- On a velocity-time graph …
  - slope equals acceleration.
  - the"y" intercept equals the initial velocity.
  - when two curves coincide, the two objects have the same velocity at that time.
  - straight lines imply uniform acceleration.
  - curved lines imply non-uniform acceleration.
  - an object undergoing constant acceleration traces a straight line.
  - average acceleration is the slope of the straight line connecting the endpoints of a curve.
  - instantaneousacceleration is the slope of the line tangent to a curve at any point.
  - positive slope implies an increase in velocity in the positive direction.
  - negative slope implies an increase in velocity in the negative direction.
  - zero slope implies motion with constant velocity.
  - the area under the curve equals the change in displacement.
- On an acceleration-time graph …
  - slope is meaningless.
  - the"y" intercept equals the initial acceleration.
  - when two curves coincide, the two objects have the same acceleration at that time.
  - an object undergoing constant acceleration traces a horizontal line.
  - zero slope implies motion with constant acceleration.
  - the area under the curve equals the change in velocity.
- The mathematical transformations between graphs of motion are shown below.
  
  [magnify]
Kinematics in Two and Three Dimensions
- Kinematics problems in two and three dimensions are essentially geometry problems. To solve them you should be able to …
  - represent a kinematic event with a geometric diagram;
  - use geometry to determine unknown magnitudes (lengths) and directions (angles); and
  - identify the magnitudes and directions needed to determine quantities like distance, displacement, speed, velocity, and acceleration from their definitions.
- Review the kinematic definitions presented earlier in this chapter.
  - First the scalar quantities …
    - Distance is a scalar measure of the interval between two locations measured along the actual path connecting them.
    - Speed is the rate of change of distance with time.
  - And then the vector quantities …
    - Displacement is a vector measure of the interval between two locations measured along the shortest path connecting them.
    - Velocity is the rate of change of displacement with time.
    - Acceleration is the rate of change of velocity with time.

Parametric Equations

N-dimensional motion can be completely described by n one-dimensional algebraic expressions along n mutually perpendicular directions (where n is any whole number greater than zero).
- Two-dimensional motion can be completely described by two, one-dimensional algebraic expressions along two perpendicular directions.
- Three-dimensional motion can be completely described by three, one-dimensional algebraic expressions along three mutually perpendicular directions.

r =

ˆi

ˆj

ˆk

r² =

x²

y²

z²

v =

v_x

ˆi

v_y

ˆj

v_z

ˆk

v² =

v_x²

v_y²

v_z²

a =

a_x

ˆi

a_y

ˆj

a_z

ˆk

a² =

a_x²

a_y²

a_z²

x =

x(t)

y =

x(t)

z =

z(t)

v_x =

Δx

v_y =

Δy

v_z =

Δz

Δt

a_x =

Δv_x

a_y =

Δv_y

a_z =

Δv_z

Δt

x =

x(t)

y =

x(t)

z =

z(t)

v_x =

lim

Δx

v_y =

lim

Δy

v_z =

lim

Δz

Δt → 0

Δt

Δt → 0

Δt

Δt → 0

Δt

a_x =

lim

Δv_x

a_y =

lim

Δv_y

a_z =

lim

Δv_z

Δt → 0

Δt

Δt → 0

Δt

Δt → 0

Δt

x =

x(t)

y =

x(t)

z =

z(t)

v_x =

v_y =

v_z =

a_x =

dv_x

d²x

a_y =

dv_y

d²y

a_z =

dv_z

d²z

dt²

Projectiles

A projectile is any object …
- with an initial non-zero, horizontal velocity
- whose acceleration is due to gravity alone.
The path of a projectile is called its trajectory.
A projectile is said to be …
- a simple projectile if the acceleration due to gravity may be assumed constant in both magnitude and direction throughout its trajectory.
- a satellite if it follows a closed path that never brings it in contact with a celestial body (like the earth).
- a general projectile no matter where its trajectory may take it.

The kinematic equations for a simple projectile are those of an object traveling with …

constant horizontal velocity and
constant vertical acceleration.


horizontal		vertical
a_x	= 0	a_y	= −g	acceleration
v_x	= v_0x	v_y	= v_0y − gt	velocity-time
x	= x₀ + v_0xt	y	= y₀ + v_0yt − ½ gt²	displacement-time
		v_y²	= v_0y² − 2g(y − y₀)	velocity-displacement

The horizontal distance traveled by a projectile is called its range.
A projectile launched on level ground with an initial speed v₀ at an angle θ above the horizontal …
- will have the same range as a projectile launched with an initial speed v₀ at 90° − θ. (Identical projectiles launched at complementary angles have the same range.)
- will have a maximum range when θ = 45°.

Dynamics I: Force

Forces

Newton's first law of motion also known as the law of inertia states that …
- An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity unless acted upon by a net external force.
- concept-map-1.pdf [pdf]
- special-principia.html
In general, inertia is resistance to change.
In mechanics, inertia is the resistance to change in velocity
or, if you prefer, the resitance to acceleration.
- The "normal" state of an object is to continue moving at a constant velocity.
  (A constant velocity of zero -- meaning, at rest for an extended period of time -- is one type of constant velocity.)
- Objects resist changes in their motion.
- Objects do not need to be pushed to keep going.
In general, a force is an interaction that causes a change.
In mechanics, a force is an interaction that causes a change in velocity
(or, if you prefer, an interaction that causes acceleration).
- The motion of an object won't change until a force is exerted from outside the object to cause a change.
- When more than one force is present, it is the net external force that matters.

name/symbol(s)		when/where	direction
Some Common Forces
weight	W, F_g	due to gravity	down
normal	N, F_n	surfaces in contact	normal to surface
friction	ƒ, F_ƒ	surfaces in contact	tangent to surface
tension	T, F_t	strings, ropes, cables, etc	along the axis
elasticity	F_s, F_e	springs, elastic bands, etc.	along the axis
buoyancy	B, F_b	immersed in a fluid	up
drag	R, D, F_d	moving through a fluid	opposite velocity of object
lift	L	moving through a fluid	perpendicular to flow
thrust	T	pushing a fluid	opposite velocity of fluid

Force & Mass
- Newton's second law of motion also known as the force law states that …
  - acceleration is directly proportional to net force when mass is constant, and
  - acceleration is inversely proportional to mass when net force is constant, and consequently
  - net force is directly proportional to mass when acceleration is constant.
  - concept-map-2 [pdf]
  - special-principia.html
- Newton's second law of motion is more compactly written as the equation.
  ∑ F = m a
- Mass
  - Mass is a measure of resistance to acceleration.
    (More generally, mass is a measure of resistance to all sorts of change.)
  - Mass is a scalar quantity associated with matter.
  - When a system is composed of several objects it is the total mass that matters.
  - The SI unit of mass is the kilogram [kg].
- Force
  - A force is an interaction that causes acceleration.
    (More generally, a force is an interaction that causes a change.)
  - Force is a vector quantity associated with an interaction.
  - When several forces act on a system it is the net, external force that matters.
  - The SI unit of force is the newton [N = kg·m/s²].
Action-Reaction
- Newton's third law of motion also known as the law of action and reaction states that …
  - for every action there is an equal and opposite reaction.
  - concept-map-3.pdf [pdf]
  - special-principia.html
- A force is an interaction between two objects.
- Forces always occur in pairs that …
  - are arbitrarily assigned the names action and reaction,
  - are of the same type,
  - have the same magnitude,
  - act on different objects,
  - act in opposite directions, and
  - may have different effects (since acceleration is inversely proportional to mass).

Weight

Weight is the force of gravity on an object
Derive formula from the free fall thought experiment.
Things that weigh a newton.

mass	weight
Mass vs. Weight
a measure of inertia (resistance to change in motion)	the force of gravity acting on an object due to its mass
invariant (does not change with location)	variable (depends upon location)
scalar (has magnitude only)	vector (directed downward)
kg (SI base unit)	N (SI derived unit)

Friction
- Definition
  - Friction is the force between surfaces in contact that resists their relative tangential motion.
  - "Relative tangential motion" is a fancy way to say "slipping".
  - Its direction is opposite the relative velocity (or intended velocity).
- Types
  - Dry Friction
    - The resistive force between clean dry solid surfaces.
    - The phenomena one normally associates with the word friction. Friction is normally synonymous with dry friction.
  - Viscous Friction
    - The resistive force between surfaces in relative motion through a fluid (liquids & gases).
  - Rolling Resistance
    - The resistive force experienced by rolling objects.
    - Since rolling does not does not necessarily involve slipping, rolling resistance is not really a form of friction.
- Factors affecting dry friction
  - Dry friction is directly proportional to the normal force between the two surfaces in contact.
  - Dry friction depends on the materials in contact. This factor is measured by the quantity known as the coefficient of friction which is …
    - the ratio of the friction force to the normal force.
    - unitless
    - always greater than 0
    - usually less than 1 for most everyday materials
  - Dry friction is subdivided into two types.
    - Static friction …
      - occurs when the two surfaces in contact are not in relative motion; that is, when one surface is stationary relative to the other surface,
      - varies in strength from zero (when no external force is trying to force slippage) to some maximum value (just before slippage occurs)
    - Kinetic friction …
      - occurs when two surfaces in contact are in relative motion; that is when one surface is slipping or sliding across another surface,
      - is always weaker than the maximum static friction.
- Factors that don't affect dry friction
  - Friction is largely independent of surface roughness (despite what you may have read in other textbooks).
    - Protrusions or rough spots may provide microscopic ledges where one surface can rest upon another and apply a normal force. This is not friction.
    - The friction associated with sandpaper is no greater than the friction associated with quartz. Friction and abrasion are different phenomena.
    - Ice, glass, and rubber can all be made smooth but ice has a low coefficient of friction, glass a medium coefficient, and rubber a high coefficient. The material is what determines the amount of friction, not is surface texture.
    - Sanding a slippery surface may increase its friction by removing the low friction surface material and exposing an underlying high friction material.
  - Friction is independent of speed once an object is moving.
    - Faster does not mean more friction.

Equilibrium

Statics is the branch of mechanics dealing with forces acting on an object at rest.
Static Equilibrium (Point Bodies)
- A point body is in static equilibrium when the net force on it is zero.
- A point body in static equilibrium will not accelerate.
  It will remain at rest or continue moving with a constant velocity.
2 Forces in Equilibrium
- Two forces are in equilibrium if they are
  - equal (have equal magnitudes) and
  - opposite (180° apart).
3 Forces in Equilibrium
- Resultant-Equilibrant
  - Select 2 forces and find their resultant.
  - The remaining force is called an equilibrant if it is equal and opposite the resultant.
- Triangle of Forces
  - Three forces are in equilibrium if they can be arranged to form a triangle.
N Forces in Equilibrium
- Polygon of Forces
  - Three or more forces are in equilibrium if they can be arranged to form a polygon.
- Components
  - Resolve all forces into components in some convenient coordinate system.
  - Combine the components along each axis.
    (Add the components parallel to the axis and
    subtract the components anti-parallel to the axis.)
  - If the resultant along each axis is zero the object is in equilibrium.


∑ F = 0 ⇒	⎧ ⎪ ⎨ ⎪ ⎩	∑ F_+x = ∑ F_−x
		∑ F_+y = ∑ F_−y
		∑ F_+z = ∑ F_−z
No Net Force in Any Direction

Force in Two and Three Dimensions
- Force is a vector quantity.
  - Forces have direction and don't you ever forget that.
  - Probably the best way to handle the vector nature of forces is to resolve them into components.
    - Go easy on yourself. Pick the most convenient coordinate system.

Centripetal Force

Circular motion in which the speed is constant is called uniform circular motion.
A centripetal acceleration …
- occurs whenever a moving object changes direction,
- does not change the speed of an object,
- acts at right angles to the velocity at any instant, and
- is directed toward the center of a circle.
A centripetal force …
- is the force that makes a moving object change direction,
- is not a particular force, but the name given to the net force responsible for circular motion,
- acts at right angles to the velocity at any instant, and
- is directed toward the center of a circle.
Directions in circular motion:
- Velocity is tangential (lies on a tangent to the path).
- Centripetal acceleration and centripetal force are radial (point toward the center of a circle).
- Centripetal acceleration and velocity are always perpendicular.
- Centripetal force and centripetal acceleration are always parallel.

Magnitudes in circular motion


F_c = ma_c = m	v²	= rω²	v = rω =	2πr	ω = 2πf =	2π	ƒ =	1
	r			T		T		T

A centrifugal force …
- is experienced by an object in a rotating reference frame,
- is a fictitious or apparent force,
- ceases to exist when an object stops moving in a circle, and
- feels as though it is directed away from the center of a circle.

Frames of Reference
- bullet

Energy

Work
- bullet
Equations for Work

using an average force W = F_∥Δs

using a variable force W = ∫ F · dr
Power
- bullet
Energy
- bullet
Kinetic Energy
- bullet

Potential Energy

Potential energy …
- is energy associated with position in a field (a force that exists at many locations)
- comes in four fundamental types, one for each of the fundamental forces, and several subtypes
  - gravitational
  - electromagnetic
    - electrostatic
    - chemical
    - elastic
  - strong
    - primary: between quarks, within nucleons (and other hadrons)
    - residual: between nucleons, within the nucleus
  - weak

Gravitational potential energy

can be computed through one of two formulas

the simplified formula

the more general formula

ΔU = mgΔh

U = −	Gm₁m₂
	r²

assumes that

is dealt with in a later section of this book

acceleration due to gravity is nearly constant
height change is small compared to the separation between centers

Forces and Potential Energy
- nonconservative forces
  - work done depends on path
  - W = ∮ F·dr > 0
  - cannot be associated with a potential energy function
- conservative forces
  - work done is independent of path
  - W = ∮ F·dr = 0
  - can be associated with a potential energy function

Forces and Potential Energy

work is the force-displacement integral
- this is the work energy theorem
a conservative force is the gradient of potential energy


one-dimensional		three-dimensional						compact notation
F(r) = −	dU	F(r) = −	∂U	ˆi −	∂U	ˆj −	∂U	F(r) = − ∇U
	dr		∂x		∂y		∂z

Potential energy curves (or surfaces, or their higher order equivalents) are useful problem solving tools

motion of a particle in a field
- constant total energy, horizontal line above curve
- kinetic energy is difference between line and curve
- bound and unbound states, binding energy

stability of equilibrium


stable equilibrium	unstable equilibrium	neutral equilibrium
[diagram]	[diagram]	[diagram]
local maximum	local minimum	constant potential energy

Conservation of Energy
- bullet
Simple Machines
- bullet

Dynamics II: Momentum

Impulse & Momentum
- bullet

Conservation of Momentum

bullet

	I	II
1st law	inertia m	momentum p = mv
2nd law	force law F = m a F = m dv / dt	impulse-momentum theorem J = Δp ∫ Fdt = m ∫ dv
3rd law	action-reaction F₁ = -F₂	conservation of momentum ∑ p = ∑p₀

Momentum & Energy
- bullet
Momentum in Two and Three Dimensions
- bullet

Rotational Motion

Rotational Kinematics
- bullet
Rotational Inertia
- bullet
Rotational Dynamics
- bullet

Rotational Equilibrium

Statics is the branch of mechanics dealing with forces acting on an object at rest.
Static Equilibrium (Extended Bodies)
- An extended body is in static equilibrium when the net force and net torque on it are zero.


∑ F = 0 ⇒	⎧ ⎪ ⎨ ⎪ ⎩	∑ F_+x = ∑ F_−x	∑ τ = 0 ⇒	⎧ ⎪ ⎨ ⎪ ⎩	∑ τ_+x = ∑ τ_−x
		∑ F_+y = ∑ F_−y			∑ τ_+y = ∑ τ_−y
		∑ F_+z = ∑ F_−z			∑ τ_+z = ∑ τ_−z
No Net Force in Any Direction			No Net Torque About Any Axis

The translational and rotational acceleration of an object in equilibrium equal zero.
- It will remain at rest or continue moving with a constant translational velocity.
- It will remain at rest or continue rotating with a constant angular velocity.

Center of Mass (Center of Gravity)
- is the effective location of the force of gravity acting on an extended body
- is the pivot about which an extended body will always be in equilibrium in a gravitational field
- is the point about which an extended body will always rotate when free of external torques
- may lie outside of an extended body

The center of mass is computed from the mass distribution.			The center of gravity is computed from the weight distribution.

r_cm =	∑ m_ir_i	= (x, y, z)	r_cg =	∑ W_ir_i	= (x, y, z)
	∑ m_i			∑ W_i
Discrete Collection of Objects

r_cm =

⌠⌠⌠
⌡⌡⌡

r dm =

⌠⌠⌠
⌡⌡⌡

r dV

r_cg =

⌠⌠⌠
⌡⌡⌡

r dW =

⌠⌠⌠
⌡⌡⌡

r dV

Continuous Distribution of Matter

Stability of Equilibrium
- The equilibrium of an object in a gravitational field is said to be …
  - stable if any small disturbance would raise its center of mass.
  - unstable if any small disturbance would lower its center of mass.
  - neutral if any small disturbance would neither raise nor lower its center of mass.

Angular Momentum
- bullet
Rotational Energy
- bullet

Rolling

Symbols used in this section

r	radius in the general sense (distance from the center or axis of rotation)
R	the outer radius of a round object (often just called the radius of the object)
v_cm	translational speed of the center of mass
ω	rotational or angular speed

Rolling is a combination of translational and rotational motion.
- When an object experiences pure translational motion, all of its points …
  - move with the same velocity as the center of mass; that is …
    - in the same direction and
    - with the same speed (v = v_cm)
  - move in a straight line in the absence of a net external force,
- When an object experiences pure rotational motion about its center of mass, all of its points …
  - move at right angles to the radius in a plane perpendicular to the axis of rotation, thus …
    - points on opposite sides of the axis of rotation move in opposite directions
  - move with a speed proportional to radius (v = rω), thus …
    - the center of mass does not move (since r = 0 there) and
    - points on the outer radius move with speed v = Rω
  - move in a circle centered on the axis of rotation
- When an object experiences rolling motion …
  - the point of the object in contact with the surface …
    - is instantaneously at rest
    - is the instantaneous axis of rotation
  - the center of mass of the object …
    - moves with speed v_cm = Rω
    - moves in a straight line in the absence of a net external force
  - the point fathest from the point of contact …
    - moves with twice the speed of the center of mass v = 2v_cm = 2Rω
Rolling and Slipping
- rolling without slipping
- slipping
  - and rolling forward
    - v_cm < Rω
      - accelerating on ice or mud
      - "burnout" or "burn rubber" while driving
      - "top spin" in billiards (a.k.a. "top" or "follow")
    - v_cm > Rω
      - decelerating on ice or mud
  - and rolling backward
    - v_cm > 0 and ω < 0
      - "back spin" in billiards (a.k.a. "bottom" or "draw")
- pure translation
- pure rotation

The path of a point on a rolling object is a cycloid (or a trochoid).

The cycloid generated by a point on an object rolling over the +x axis is described by the following parametric equations …

rolling	=	translation	+	rotation
x	=	v_cmt	+	r cos (θ − ωt)
y	=	R	+	r sin (θ − ωt)

where …

r, θ	cylindrical coordinates of the point
R	outer radius
v_cm	translational speed of the center of mass
ω	rotational or angular speed
t	time (the parameter of the parametric equation)

Types
- A basic cycloid …
  - is traced out by …
    - points on the surface of a generating circle that is …
    - rolling without slipping
    - over a straight line
  - has cusps (points with two tangents)
- A cycloid is curtate (or contracted) if …
  - it is traced out by …
    - points inside a generating circle (r < R) that is rolling without slipping or
    - points on the surface of the generating circle that is slipping while rolling with v_cm > Rω
  - does not have cusps or loops
- A cycloid is prolate (or extended) if …
  - it is traced out by …
    - points outside a generating circle (r > R) that is rolling without slipping or
    - points on the surface of the generating circle that is slipping while rolling with v_cm < Rω
  - it has loops
- A cycloid formed by rolling a generating circle on another circle is called …
  - an epicycloid if the generating circle rolls on the outside of the other circle
  - a hypocycloid if the generating circle rolls on the inside of the other circle

Rotation in Two and Three Dimensions
- bullet
Coriolis Force
- bullet

Periodic Motion
1. Elasticity
  - bullet
2. Elastic Potential Energy
  - bullet
3. Simple Harmonic Oscillator
  - bullet
4. Pendulum
  - bullet
5. Resonance
  - bullet
Planetary Motion
1. Geocentrism
  - the ancient solar system consists of …
    - a night sky filled with hundreds of fixed stars that …
      - remain in the same positions relative to one another
      - appear to belong to groups called constellations named after supernatural beings
      - rotate as a group across the sky about once every day
      - shift gradually in the sky when viewed at the same time from night to night
      - return to their original positions relative to the horizon after one year
    - two primary moving objects …
      - sun
        
        moves across the sky once every day
        moves across the background of the fixed stars once every year
      - moon
        
        moves across the sky about once a every day
        cycles through apparent changes (called phases) roughly once every month
    - five planets visible to the unaided eye
      - named in English after gods of the Ancient Rome
        
        mercury
        venus
        mars
        jupiter
        saturn
      - wander across the background sky of fixed stars
        
        generally moving in the same direction as the sun, moon, and stars
        but occasionally moving backward (retrograde)
    - a total of seven moving objects each of which is associated with a day of the week in English and other languages
2. Heliocentrism
  - bullet
3. Universal Gravitation
  - bullet
4. Orbital Mechanics I
  - bullet
5. Gravitational Potential Energy II
  - bullet
6. Orbital Mechanics II
  - bullet
7. Gravity of Extended Bodies
  - bullet

prev | up | next

Another quality webpage by

Glenn Elert

Equations for Work
using an average force	W = F_∥Δs
using a variable force	W = ∫ F · dr