Dynamics: Ch 11 Kinematics

initial

0

PROJECTILE MOTION: consider a general projectile set into motion at angle theta from horizontal plan & initial velocity, vo. in the absence of air drag, what rules apply to the case of travel over horizontal plane?

1. Parabolic trajectory. 2. impact velocity = initial velocity. 3. range is maximum when theta = 45 degrees. 4. time for the projectile to travel from launch to apex = to time traveled from apex to impact point. 5. time for the [projectile] to travel from [apex] of its flight path to [impact] = same time an [initially stationary object] would take to fall straight down from that height.

PROJECTILE MOTION: Equations

1. ax = 0. 2. ay = -g. 3. vx = vo cos (theta). 4. vy = -gt + vo sin (thetat). 5. x = vo cos (theta) t + xo. 6. y = -gt^2/2 + vo sin (theta) t + yo.

RECTANGULAR COORDINATES: how many coordinates define 3D and 2D spaces?

3, 2

CURVILINEAR MOTION: PLANE CIRCULAR MOTION. Polar Coordinates. Acceleration.

Ar = r 2 dots = r theta dot squared. A theta = r theta 2 dots + 2 r theta dot. A z = z 2 dots.

RELATIVE MOTION: AKA

Newtonian, Intertial frame of reference

PROJECTILE MOTION: NCEES Handbook note

Vosin(theta)t term = product of 3 terms. = (Vo)(t)(sin(theta)).

CURVILINEAR MOTION: PLANE CIRCULAR MOTION. Polar Coordinates. Velocity.

Vr = r dot. V theta r theta dot. Vz = z dot.

CONSTANT ACCELERATION:

a = constant in many cases. free falling body = constant g

acceleration

a = m/s^2

RECTILINEAR MOTION: a ds = v dv can be derived from

a dt = v dt = ds by eliminating dt

CONSTANT ACCELERATION: Velocity & Displacement w. NON-Constant ANGULAR acceleration

a(t) is calculated finding integral of v(t), s(t) for certain time intervals

PROJECTILE MOTION: Variables

acceleration in x & y. velocity in x & y. x = horizontal distance. y = vertical distance.

CONSTANT ACCELERATION: Variable Angular Acceleration

alpha (t) = angular acceleration. omega = angular velocity. theta= displacement. calculated by finding integral of omega (t) & theta (t) for certain time intervals.

angular acceleration

alpha = rad/s^2

CONSTANT ACCELERATION: Velocity & Displacement w. constant ANGULAR acceleration variables

alpha, omega, theta, time,

instantaneous acceleration

also the 2nd derivative of posititon

PROJECTILE MOTION: A projectile is placed into motion by

an initial impulse. KINEMATICS deals only with [dynamics] during the flight. the [force] acting on the projectile during the launch phase is covered in KINETICS.

CURVILINEAR MOTION: PLANE CIRCULAR MOTION. Angular Motion. behavior of rotating particle defined by?

angular position theta. angular velocity omega. angular acceleration alpha. = to s, v, a variables for linear systems.

RECTANGULAR COORDINATES: vector form

bold r or bold or with arrow over r. magnitude & direction

PROJECTILE MOTION: neglecting air drag, once projectile is in motion, it is acted upon only

by the downward gravitational acceleration. its own weight.

constant

c

CONSTANT ACCELERATION: if a is constant what happens to acceleration term?

can be taken out of integrals

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. force _

constrains the particle to curved path is directed TOWARD CENTER of ROTATION. particle experiences INWARD ACCELERATION PERPENDICULAR to tangential v & a.

RECTANGULAR COORDINATES: position of a particle is specified with a reference to a?

coordinate system

instantaneous v & a

derivatives of position and velocity respectively

CURVILINEAR MOTION: PLANE CIRCULAR MOTION. Polar Coordinates. position

described by a radius, r & angle, theta. has dots have variables.

CURVILINEAR MOTION: DEFINE

describes the motion of a particle along a path that is NOT A STRAIGNT LINE

parts or particles of a rigid body can have

different displacements, velocities, accelerations if the body has rotational as well as translational motion

DISTANCE & SPEED: net change in particle's position

displacement = linear displacement. r(t) = position function

DISTANCE & SPEED : which is greater? distance or displacement

distance always > or equal to displacement

DISTANCE & SPEED : distance

distance traveled is the accumulated length of the path traveled during all direction reversals. found by adding the path lengths covered during periods which the velocity sign doesn't change

rigid body

does not deform when loaded and can be a combo of two or more particles that remain at a fixed finite distance from each other

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. unit vectors

e n & e t

final

f

coefficient of friction

f =

frequency

f = Hz

RECTILINEAR MOTION: Cartesian Velocity & Acceleration

first 2 derivatives of the position vector

gravitational acceleration

g = 9.81 m/s^2

PROJECTILE MOTION: equations derived from

laws of uniform acceleration & conservation of energy.

RECTILINEAR MOTION: subscripts for Dynamics review section

location of accelerating point. c = centroid. x= x-direction. n= normal. 0= initial.

DISTANCE & SPEED : velocity

magnitude and direction = vector.

DISTANCE & SPEED : speed

magnitude of velocity = scalar.

normal

n

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. perpendicular

normal

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. magnitudes

omega = angular velocity. alpha = angular acceleration.

angular velocity

omega = rad/s

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. instantaneous radius of curvature

p

radius of curvature

p = m

RELATIVE MOTION: with rotating axis

particle moves in circular path.

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. Velocity & Acceleration

particle moving on curved path will have instantaneous linear velocity & linear acceleration. linear variables directed TANGENTIALLY to the path. = tangental velocity vt. and tangential acceleration at.

RECTILINEAR MOTION: DEFINE

particles move only in straight line. linear system

CURVILINEAR MOTION: examples

plane circular motion. projectile motion.

PROJECTILE MOTION: consider a general projectile set into motion at angle theta from horizontal plan & initial velocity, vo. Apex=

point where projectile is at max elevation.

RECTANGULAR COORDINATES: coordinate can represent an angular position in

polar system

RELATIVE MOTION: with translating axis

position = r A. velocity = v A. acceleration = aA. angular velocity = omega & angular acceleration = alpha are MAGNITUDES OF RELATIVE POSITION VECTOR, r A/B. cross products!! particle moves in linear path.

CURVILINEAR MOTION: TRANSVERSE & RADIAL COMPONENTS FOR PLANAR MOTION.

position in polar coordinate system can be expressed as vector of magnitude r. direction specified by UNIT VECTOR = e r.

CURVILINEAR MOTION: for particles traveling along curved paths, the _ may be specified in _

position, velocity, acceleration. rectangular coordinates. or may be MORE CONVENIENT to express the kinematics variables in terms of POLAR COORDINATES.

radial

r

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. vector forms

r , v, a

radius

r = m

position

r = m or s = m

RECTANGULAR COORDINATES: Cartesian unit vector form

r= xi + yj + zk

CURVILINEAR MOTION: TRANSVERSE & RADIAL COMPONENTS FOR PLANAR MOTION. 2 components

radial = parallel. = e r. transverse = perpendicular = e theta . to unit radial vector of particle.

RECTANGULAR COORDINATES: Cartesian coordinate system

rectangular coordinate form. (x,y,z)

RECTANGULAR COORDINATES: coordinate can represent a linear position in

rectangular coordinate system

kinematics deals only with

relationships among position, velocity, acceleration, and time.

particle

rotation of the body is absent or insignificant. do not have rotational kinetic energy. all parts have same instantaneous displacement velocity, and acceleration.

CURVILINEAR MOTION: PLANE CIRCULAR MOTION

rotational particle motion. angular motion. circular motion. motion of particle around a FIXED circular path. radial & tangential.

CONSTANT ACCELERATION: Velocity & Displacement w. constant ANGULAR acceleration units

rpm, rad/s^2, radians

displacement

s = m

distance

s = m

kinematics

study of body's motion independent of the forces on the body. study of geometry of motion without consideration of the causes of motion

dynamics

studying if moving objects, divided into kinematics & kinetics

RECTILINEAR MOTION: NCEES Handbook inconsistent uses of subscripts

subscript c is designated for constant acceleration, centroid, and mass center for equations 11.7 - 11.9. velocity, position, and v^2 equations.

tangential

t

time

t = s

LINEAR & ROTATIONAL VARIABLES: Linear variables will be directed

tangentially to the path. can be obtained by MULTIPLYING rotational variables by the path radius.

LINEAR & ROTATIONAL VARIABLES: relationships btw linear & angular variables. if path radius, r, is constant like in rotational motion,

the linear distance traveled, s, can be calculated. example = arc length.

transverse

theta

angular position

theta = rad

PROJECTILE MOTION: special case of motion...

under constant acceleration.

LINEAR & ROTATIONAL VARIABLES: relationships btw linear & angular variables

used to calculate tangential velocity, tangential acceleration, & normal acceleration from their angular variables.

velocity

v = m/s

CURVILINEAR MOTION: PLANE CIRCULAR MOTION. Rectilinear Forms. relationships of a, v, position in an nth coordinate system.

v= s dot. at = v dot = dv/ds. an = v squared / p. p = ...... / ......

RECTANGULAR COORDINATES: a particle's position, velocity, acceleration can be specified in 3 primary forms:

vector form, rectangular coordinate form, unit vector form

CURVILINEAR MOTION: NORMAL & TANGENTIAL COMPONENTS. resultant acceleraton

vector sum of t and n accelerations.

LINEAR & ROTATIONAL VARIABLES: relationships btw linear & angular variables. vt, at, an, s =?

vt = r omega. at = r alpha. an = - r omega squared. = toward center of circle. s = r theta.

RELATIVE MOTION: DEFINE

when motion of a particle's POSITION, VELOCITY, ACCELERATION. is described WRT something else in motion.

RECTILINEAR MOTION: when is a ds = v dv needed

where acceleration depends on position is a particle being accelerated or decelerated by a compression spring. spring force depends on spring extension. so does acceleration

horizontal

x

horizontal distance

x = m

vertical

y

elevation

y = m