Republic of the Philippines
CAMARINES SUR POLYTECHNIC COLLEGES
Nabua, Camarines Sur
MODULE 1 – INTRODUCTION TO DYNAMICS
OBJECTIVE:
1.
2.
3.
4.
5.
Define the position, velocity, and acceleration of a particle
Apply formulae in determining the change in motion of particles
Define rectilinear motion
Explain the concept and type of rectilinear motion
Apply the equations of rectilinear motion in calculating the change in motion of a particle
INTRODUCTION
The DYNAMICS OF RIGID BODIES is part of mechanics that deals with the analysis of bodies in motion. It
includes:
1. Kinematics – the study of the geometry of motion. Kinematics is used to relate displacement, velocity,
acceleration, and time, without reference to the cause of the motion.
2. Kinetics – the study of the relation 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.
POSITION, VELOCITY AND ACCELERATION
Rectilinear Motion – a particle moving along a straight line.
At any given time t, to define the position P of the particle, we
choose a fixed origin O on the straight line and measure the distance x
from O to P, with the appropriate sign, completely defines the position of
the particle; it is called the position coordinate of the particle considered.
For example, the position coordinate corresponding to P in Fig. 11.1a is x
= +5 m; the coordinate corresponding to P’ in Fig. 11.1b is x’= -2 m.
Consider the position P occupied by the particle at time t and the
corresponding coordinate x (Fig. 11.2). Consider also the position P’
occupied by the particle at a later time t+∆t; the position coordinate of P’
can be obtained by adding to the coordinate x of P the small
displacement ∆x, which will be positive or negative according to whether
P’ is to the right or to the left of P. Distance is the total distance travelled.
The average velocity of the particle over the time interval ∆t is
defined as the quotient of the displacement ∆x and the time interval ∆t:
The instantaneous velocity v of the particle at the instant t is
obtained from the average velocity by choosing shorter and shorter time
intervals ∆t and displacements ∆x:
The velocity v is represented by an algebraic number which can be
positive or negative. A positive value of v indicates that x increases, i.e.,
that the particle moves in the positive direction (Fig. 11.3a); a negative
value of v indicates that x decreases, i.e., that the particle moves in the
negative direction (Fig. 11.3b). The magnitude of v is known as the speed
of the particle.
Consider the velocity v of the particle at time t and also its velocity
v+∆v at a later time t+∆t (Fig. 11.4).
The average acceleration of the particle over the time interval ∆t
is defined as the quotient of ∆v and ∆t:
Engineering Mechanics | Page 1 of 4
Republic of the Philippines
CAMARINES SUR POLYTECHNIC COLLEGES
Nabua, Camarines Sur
The instantaneous acceleration a of the
particle at the instant t is obtained from the average
acceleration by choosing smaller and smaller values
for ∆t and ∆v. The limit of the quotient, which is by
definition the derivative of v with respect to t,
measures the rate of change of the velocity. We write
or, substituting for v from (11.1),
Another expression for the acceleration
Example 1. Consider a particle moving in a straight line, and assume that
its position is defined by the equation
where t in seconds
x in meters.
The velocity v at any time t is obtained by differentiating x with
respect to t:
The acceleration a is obtained by differentiating again with respect
to t:
The position coordinate, the velocity, and the acceleration have
been plotted against t in Fig. 11.6. The curves obtained are known as
motion curves.
DETERMINATION OF THE MOTION OF A PARTICLE
Three common classes of motion:
1. a = f(t). Acceleration Is A Given Function of t. Solving for dv and substituting f(t) for a, we write
Integrating both members, we obtain the equation
Specify the initial conditions of the motion: v0 – velocity and x0 – position coordinate at t =0. Replacing
the indefinite integrals by definite integrals with lower limits corresponding to the initial conditions t = 0
and v = v0 and upper limits corresponding to t = t and v = v, we write
which yields v in terms of t
Solving for dx
and the expression just obtained substituted for v. Both members are then integrated, the left-hand
member with respect to x from x = x0 to x = x, and the right-hand member with respect to t from t = 0
to t = t. The position coordinate x is thus obtained in terms of t; the motion is completely determined.
Engineering Mechanics | Page 2 of 4
Republic of the Philippines
CAMARINES SUR POLYTECHNIC COLLEGES
Nabua, Camarines Sur
The case when a = 0 corresponds to uniform motion and the case when a = constant correspond to a
uniformly accelerated motion.
2. a = f(x). Acceleration Is A Given Function of x. Rearranging and substituting f(x) for a, we write
Since each member contains only one variable, we can integrate the equation. Denoting again by v0 and
x0, respectively, the initial values of the velocity and of the position coordinate, we obtain
which yields v in terms of x. We now solve for dt
and substitute for v the expression just obtained. Both members can then be integrated to obtain the
desired relation between x and t. However, in most cases this last integration cannot be performed
analytically and one must resort to a numerical method of integration.
3. a = f(v). Acceleration Is A Given Function of v. We substitute f(v) to obtain the following relations:
Integration of the first equation will yield a relation between v and t; integration of the second equation
will yield a relation between v and x. Either of these relations can be used to obtain the relation between
x and t which characterizes the motion of the particle.
Example 2
Solution:
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Republic of the Philippines
CAMARINES SUR POLYTECHNIC COLLEGES
Nabua, Camarines Sur
Sources:
1. Ferdinand P. Beer, Vector Mechanics for Engineers (Dynamics), 10th Edition, 2013
2. https://www.youtube.com/watch?v=ksmsp9OzAsI&list=PL8T_f_7Y4charkweVhlyYAxQuLJdjWj
Ui&index=15
3. https://www.youtube.com/watch?v=0HpAov6SVx0&list=PLmBPD1aEuq8JslHuVoupWealZzQ
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