Chapter Objectives. To introduce the concepts of position, displacement, velocity, and acceleration. To study particle motion along a straight line and represent this motion graphically. To investigate particle motion along a curved path using different coordinate systems. To present an analysis of dependent motion of two particles. To examine the principles of relative motion of two particles using translating axes..
5. Introduction Mechanics – the state of rest of motion of bodies subjected to the action of forces
Static – equilibrium of a body that is either at rest or moves with constant velocity
Dynamics – deals with accelerated motion of a body
1) Kinematics – treats with geometric aspects of the motion
2) Kinetics – analysis of the forces causing the motion
6. Rectilinear Kinematics: Continuous Motion Rectilinear Kinematics – specifying at any instant, the particle’s position, velocity, and acceleration
1) Single coordinate axis, s
2) Origin, O
3) Position vector r – specific location of particle P at any instant
7. 4) Algebraic Scalar s in metres
Note : - Magnitude of s = Dist from O to P
- The sense (arrowhead dir of r) is defined by algebraic sign on s
=> +ve = right of origin, -ve = left of origin
8. Displacement – change in its position, vector quantity
9. If particle moves from P to P’
is +ve if particle’s position is right of its initial position
is -ve if particle’s position is left of its initial position
Instantaneous velocity is defined as
11. Representing as an algebraic scalar,
Velocity is +ve = particle moving to the right
Velocity is –ve = Particle moving to the left
Magnitude of velocity is the speed (m/s)
12. Average speed is defined as total distance traveled by a particle, sT, divided by the elapsed time .
The particle travels along
the path of length sT in time
13. Acceleration – velocity of particle is known at points P and P’ during time interval ?t, average acceleration is
?v represents difference in the velocity during the time interval ?t, ie
14. Instantaneous acceleration at time t is found by taking smaller and smaller values of ?t and corresponding smaller and smaller values of ?v,
15. Particle is slowing down, its speed is decreasing => decelerating => will be negative.
Consequently, a will also be negative, therefore it will act to the left, in the opposite sense to v
If velocity is constant,
acceleration is zero
16. Velocity as a Function of Time
Integrate ac = dv/dt, assuming that initially v = v0 when t = 0.
17. Position as a Function of Time
Integrate v = ds/dt = v0 + act, assuming that initially s = s0 when t = 0
18. Velocity as a Function of Position
Integrate v dv = ac ds, assuming that initially v = v0 at s = s0
19. PROCEDURE FOR ANALYSIS
Establish a position coordinate s along the path and specify its fixed origin and positive direction.
The particle’s position, velocity, and acceleration, can be represented as s, v and a respectively and their sense is then determined from their algebraic signs.
20. The positive sense for each scalar can be indicated by an arrow shown alongside each kinematics eqn as it is applied
21. 2) Kinematic Equation
If a relationship is known between any two of the four variables a, v, s and t, then a third variable can be obtained by using one of the three the kinematic equations
When integration is performed, it is important that position and velocity be known at a given instant in order to evaluate either the constant of integration if an indefinite integral is used, or the limits of integration if a definite integral is used
22. Remember that the three kinematics equations can only be applied to situation where the acceleration of the particle is constant.
27. A small projectile is forced downward into a
fluid medium with an initial velocity of 60m/s.
Due to the resistance of the fluid the
projectile experiences a deceleration equal to a =
(-0.4v3)m/s2, where v is in m/s2.
Determine the projectile’s
velocity and position 4s
after it is fired.
Coordinate System. Since the motion is
downward, the position coordinate is downwards
positive, with the origin located at O.
Velocity. Here a = f(v), velocity is a function of
time using a = dv/dt, since this equation relates v,
a and t.