Engineering Mechanics (Dynamics) LECTURE NOTES BY: ENGR. FRANCIS F. VILLAREAL
D L S U - D / C E A T / 1 ST S E M S Y 2 0 1 4 - 2 0 1 5
INTRODUCTION • Dynamics includes: -
Kinematics: study of the geometry of motion. Kinematics is used to relate displacement, velocity, acceleration, and time without reference to the cause of motion.
-
Kinetics: study of the relations existing between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by given forces or to determine the forces required to produce a given motion.
12.1 Introduction Mechanics
Rigid-body
Deformable-body
Static
Dynamics
Equilibrium body
Accelerated motion body
Kinematics (Geometric aspect of motion)
Kinetics (Analysis of force causing the motion)
fluid
KINEMATICS OF PARTICLES Rectilinear motion: position, velocity, and
acceleration of a particle as it moves along a straight line. Curvilinear motion: position, velocity, and acceleration of a particle as it moves along a curved line in two or three dimensions.
KINEMATICS OF PARTICLES
Road Map Kinematics of particles
Rectilinear motion
x-y coord.
Curvilinear motion
Relative motion
n-t coord.
r- coord.
RECTILINEAR MOTION
• Particle moving along a straight line is said to be in rectilinear motion.
Determination of the Motion of a Particle • Recall, motion of a particle is known if its position is known for all time t. • Typically, conditions of motion are specified by the type of acceleration experienced by the particle. Determination of velocity and position requires successive integrations. • Three types of motion may be defined for: - acceleration given as a function of time, a = f(t) - acceleration given as a function of position, a = f(x) - acceleration given as a function of velocity, a = f(v)
Graphical Solution of Rectilinear Motion
• Given the x-t curve, the v-t curve is equal to the x-t curve slope. • Given the v-t curve, the a-t curve is equal to the v-t curve slope.
Kinematic Equations Consider particle which occupies position P at time t
and P’ at t+Dt,
Average velocity Instantaneous velocity
Dx Dt
Dx v lim Dt 0 Dt
• Instantaneous velocity may be positive or negative. Magnitude of velocity is referred to as particle speed.
Kinematic Equations Consider a particle with velocity v at time t and v’ at
t+Dt,
Instantaneous acceleration
Dv a lim Dt 0 Dt
RECTILINEAR MOTION FORMULAS Average velocity:
V = dS/dt Average acceleration: a = d2S/dt2 = dV/dt Constant acceleration: V – V0 = at S = V0t + ½ (at2 ) V2 – V02 = 2aS This applies to a freely falling object:
a ds v dv 2 a 9 . 8 1 m / s 3 2 . 2 ft / s 2
Application Problem 1 Consider a particle moving a straight line and
assume that its position is defined by the equation where x is in meters and t in seconds.
x 6t
2
t
3
Show the graphical representation of the 3 motion
curves where x is a function of t, v as a function of t and a as a function of t.
Application Problem 2 The brake mechanism used to reduce recoil in certain types of guns as
shown in the given figure consists essentially of a piston attached to the barrel and moving in a fixed cylinder filled with oil. As the barrel recoils with an initial v0, the piston moves and oil is forced through orifices in the piston, causing the piston and the barrel to decelerate at a rate proportional to their velocity; that is a = -kv. Express a.) v in terms of t b.) x in terms of t c.) v in terms of x and d.) draw the corresponding motion curves.
Application Problem 3 Cars A and B approached each other on a straight
road from a point where the 2 cars are 450 meters apart. Car A has an initial velocity of 70 kph and is being decelerated at a rate of 0.40m/s2. Cars B has an initial velocity of 20 kph and is accelerating at a rate of 0.30m/s2. When will the cars meet and how far will Car A have traveled? Show accompanying figure.
FREELY FALLING BODIES In the absence of air resistance, it is found that all bodies at the same location above the earth fall vertically with the same acceleration. Furthermore, if the distance of the fall is small compared to the radius of the earth, the acceleration remains essentially constant throughout the fall. This idealized motion, in which air resistance is neglected and the acceleration is nearly constant, is known as free-fall.
Since the acceleration is constant in free-fall, the equations of kinematics can be used.
Galileo Galilei (1564-1642) Father of Kinematics Concluded that all objects fall
at same rate of acceleration. Demonstrated the scientific method in developing the kinematics of free fall motion. Tested his hypothesis through experimentation.
Sir Isaac Newton (1642-1727) Father of dynamics
(why) Published ‘Three laws of motion’ and universal law of gravitation in 1687. Inertia F=ma Action/reaction
Acceleration Due to Gravity
Galileo calculated that all freely falling objects
accelerate at a rate of
9.8
2 m/s
This value, as an acceleration, is known as g
Free Fall – An Object Dropped Initial velocity is zero Use the kinematic equations Generally use y instead of x since y is vertical vo= 0
Acceleration is ay = g = 9.80 m/s2
a=g
Section 2.7
Free Fall – Object Thrown Upward Initial velocity is upward,
so positive The instantaneous velocity at the maximum height is zero. ay = -g = -9.80 m/s2 everywhere in the motion
Section 2.7
v=0
vo≠ 0 a = -g
Application Problem 1
A ball is tossed with a velocity of 10m/s directed vertically upward from a window of a building located 20 meters above the ground. Determine the following: Velocity v of the ball at any time t with graphical motion diagram Elevation y of the ball at any time t with graphical motion diagram Highest elevation in meters reached by the ball and value of time in seconds Time in seconds when the ball hits the ground
V0= 10m/s
20 m
Application Problem 2 A stone is thrown vertically upward over the top of a well with a velocity of
21m/s and the splash is heard in 5.05 sec. If the velocity of sound is constant at 350m/s, determine the depth of the well to which the stone falls.
v0= 21m/s
depth
water
CURVILINEAR MOTION Particle moving along a curve other than a straight
line is in curvilinear motion Position vector of a particle at time t is defined by a vector between origin O of a fixed reference frame and the position occupied by particle.
Plane Curvilinear Motion
Speed and Velocity
Acceleration
Visualization of Motion
The Coordinate System
RECTANGULAR, x-y NORMAL –TANGENTIAL, n-t
POLAR, r-
Rectangular Coordinate System
Projectile Motion (x-y coordinate )
Application Problem 2 A rocket has expended all its fuel when it reaches point A, where it has
velocity u at angle with respect to the horizontal. It then begins unpowered flight and attains a maximum added height h at position B after traveling a horizontal s from A. Determine the expression for h and s, the time t of flight from A to B and the equation of the path. For the interval concerned, assume a flat earth with a constant acceleration g and neglect any atmospheric resistance.
Normal-Tangential Coordinate System
Acceleration (n-t coordinate )
Acceleration (n-t coordinate )
Direction of Acceleration (n-t coordinate)
Circular Motion (n-t coordinate)
Application Problem 1 When a skier reaches point A along the parabolic path, he has a speed of 6m/s
which is increasing at 2m/s2. Determine the direction of his velocity and direction and magnitude of his acceleration at this instant. Neglect the size of the skier.
Polar Coordinate System
Velocity and Acceleration (r-)
Geometric Interpretation (r-)
Circular Motion (r-)
Application Problem 1 The searchlight shown in the given figure casts a spot of light along the face of a
wall that is located 100m from the searchlight. Determine the magnitudes of the velocity and acceleration at which the spot travels across the wall at the instant = 450. The searchlight at a constant rate of 4 rad/sec.
