Rectilinear Kinematics Erratic Motion, Lecture notes of Engineering Dynamics

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TEGT

Engineering Mechanics II -

Dynamics

UNIT 3A RECTILINEAR KINEMATICS: ERRATIC MOTION

RECTILINEAR KINEMATICS:

ERRATIC MOTION

Objectives: Determine position, velocity, and

acceleration of a particle using graphs.

Applications: s-t, v-t, a-t, v-s & a-s graphs.

Slope of a Curve as Differential dy/dx A curve described by y = f(x)

y Line tangent

P Slope of the curve at any point P equals the differential dy/dx

0 x

Area Under a Curve as Integral ʃ f(x)dx

A curve described by y = f(x)

y 5 4 3 2 1

0 1 2 3 4 5 6 x

The shaded area equals the integral of f(x) over the limit x=0 to x=4 or

ʃf(x)dx (limit x=0 to x=4 )

V-T GRAPH

Also, the distance moved (displacement) of the particle is the area under the v-t graph during time t. Plots of velocity vs. time can be used to find acceleration vs. time curves. Finding the slope of the line tangent to the velocity curve at any point gives the acceleration at that point (or a = dv/dt). Therefore, the acceleration vs. time (or a-t) graph can be constructed by finding the slope at various points along the v-t graph.

A-T GRAPH

Given the acceleration vs. time or a-t curve, the change in velocity (v) during a time period is the area under the a-t curve. So we can construct a v-t graph from an a-t graph if we know the initial velocity of the particle.

V-S GRAPH

Here we present the velocity vs. distance or v-s graph. By reading velocity v at a point on the curve and multiplying it by the slope of the curve (dv/ds) at that point, we can obtain the acceleration at that point. Recall the formula:

a = v (dv/ds)

Thus, we can obtain an a-s plot from the v-s curve.