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MECHANICS
Mechanics is a branch of the physical sciences that is concerned with the state of rest or motion of bodies subjected to the action of forces. Engineering mechanics is divided into two areas of study, namely, statics and dynamics. STATICS: is concerned with the equilibrium of a body that is either at rest or moves with constant velocity. DYNAMICS: deals with the accelerated motion of a body. The subject of dynamics will be presented in two parts: Kinematics: which treats only the geometric aspects of the motion, and Kinetics: which is the analysis of the forces causing the motion. The discussion is followed as:
- The dynamics of a particle
- The dynamics of rigid-bodies in two and then three dimensions.
DYNAMICS
Historically, the principles of dynamics developed when it was possible to make an accurate measurement of time. Galileo Galilei ( 1564 - 1642) was one of the first major contributors to this field. His work consisted of experiments using pendulums and falling bodies. Isaac Newton (1642-1727), who is noted for his formulation of the three Fundamental laws of motion and the law of universal gravitational attraction. Important techniques for application of these laws were developed by Euler, D' Alembert, Lagrange, and others.
STEPS FORPROBLEM SOLVING
1. Read the problem carefully and try to correlate the actual
physical situation with the theory you have studied.
2. Draw any necessary diagrams and tabulate the problem data.
3. Establish a coordinate system and apply the relevant principles,
generally in mathematical form.
4. Solve the necessary equations algebraically as far as practical;
then, use a consistent set of units and complete the solution
numerically.
Report the answer with no more significant figures than the
accuracy of the given data.
PROBLEM SOLVING
Study the answer using technical judgment and common sense to determine whether or not it seems reasonable.
Once the solution has been completed, review the problem. Try to think of other ways of obtaining the same solution.
- In applying this general procedure, do the work as neatly as possible.
- Being neat generally stimulates clear and orderly thinking, and vice versa.
RECTILINEAR KINEMATICS.
The kinematics of a particle is
characterized by specifying,
at any given instant, the particle's
position, velocity, and acceleration.
POSITION
The straight-line path of a particle is defined using a single coordinate axis s. The origin O on the path is a fixed point, and from this point the position coordinate s is used to specify the location of the particle at any given instant. The magnitude of s is the distance from O to the particle, usually measured in meters (m) or feet (ft), and the sense of direction is defined by the algebraic sign on s. Although the choice is arbitrary, in this case s is positive since the coordinate axis is positive to the right of the origin. Likewise, it is negative if the particle is located to the left of O. Position is a vector quantity since it has both magnitude and direction. Here, however, it is being represented by the algebraic scalar s since the direction always remains along the coordinate axis.
VELOCITY
If the particle moves through a displacement ∆s during the time
interval ∆t, the average velocity of the particle during this time
interval is
If we take smaller and smaller values of ∆t, the magnitude of ∆s
becomes smaller and smaller.
Consequently, the instantaneous velocity is a vector defined as
or
VELOCITY
Since ∆t or dt is always positive, the sign used to define the sense of
the velocity is the same as that of ∆ s or ds.
For example, if the particle is moving to the right, Fig. , the velocity is
positive; whereas if it is moving to the left, the velocity is negative.
(This is emphasized here by the arrow written at the left of Eq.1.)
The magnitude of the velocity is known as the speed, and it is
generally expressed in units of m/s or ft/s.
ACCELERATION
Provided the velocity of the particle is known at two points, the average acceleration of the particle during the time interval ∆t is defined as Here ∆V represents the difference in the velocity during the time interval ∆t, i.e., ∆ v = v' - v The instantaneous acceleration at time t is a vector that is found by taking smaller and smaller values of ∆ t and corresponding smaller and smaller values of ∆ v, so that
Substituting Eq. 1 into this
result, we can also write
or ……………….. (2)
ACCELERATION
Both the average and instantaneous acceleration can be either positive or negative. In particular, when the particle is slowing down, or its speed is decreasing, the particle is said to be decelerating. In this case, v' in Fig. is less than v, and so ∆v = v' - v will be negative. Consequently, a will also be negative, and therefore it will act to the left, in the opposite sense to v. Also, note that when the velocity is constant, the acceleration is zero since ∆v = v - v = O. Units commonly used to express the magnitude of acceleration are m/s 2 or ft/s 2 .
EQUATIONS OF MOTIONS (CONSTANT ACCELERATION, A =A C .)
When the acceleration is constant, each of the three
kinematic equations :
a c = dv/dt v = ds/dt a c ds = v dv
can be integrated to obtain formulas that relate a
c
, v, s ,
and t.
Integrate: ac = dv/dt assuming that initially v = v 0 when t = O. EQUATIONS OF MOTIONS (VELOCITY AS A FUNCTION OF TIME) v vo t 0 dv acdt
v v at ( 4 )
o
Constant acceleration
v v a t a 0
o
EQUATIONS OF MOTIONS (VELOCITY AS A FUNCTION OF POSITION) either: solve for t in Eq. 4 and substitute into Eq. 5. Or integrate v dv = a c ds , assuming that initially v = v o at s = s o v v s s c o o vdv a ds
v v 2 a (s s ) ( 6 )
c o 2 o 2
( ) 2 2 2 2 c o o a s s v v s c s v v o o a s v 2 2
PROBLEM
The car in Fig. moves in a straight line such that for a short time its velocity is defined by v = (3t 2
- 2t) ft/s, where t is in seconds. Determine its position and acceleration when t = 3 s. When t = 0, s = o.
