Unit 11 Lecture 1 Applications of Integrals in Dynamics Position Velocity amp Acceleration Section 95 of RattanKlingbeil text Differentiation and Integration Recall that differentiation and integration are inverse operations ID: 526515
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Slide1
EGR 1101: Unit 11 Lecture #1
Applications of Integrals in Dynamics: Position, Velocity, &
Acceleration
(Section 9.5 of Rattan/Klingbeil text)Slide2
Differentiation and IntegrationRecall that differentiation and integration are inverse operations.Therefore, any relationship between two quantities that can be expressed in terms of derivatives can also be expressed in terms of integrals.Slide3
Position, Velocity, & Acceleration
Position
x(t)
Derivative
Velocity
v(t)
Acceleration
a(t)
Derivative
Integral
IntegralSlide4
Today’s ExamplesBall dropped from restBall thrown upward from ground level
Position & velocity from acceleration (graphical)Slide5
Graphical derivatives & integralsRecall that:Differentiating a parabola gives a slant line.
Differentiating a slant line gives a horizontal line (constant).
Differentiating a horizontal line (constant) gives zero.
Therefore:
Integrating zero gives a horizontal line (constant).
Integrating a horizontal line (constant) gives a slant line.
Integrating a slant line gives a parabola.Slide6
Change in velocity = Area under acceleration curveThe change in velocity between times t1 and
t
2
is equal to the area under the acceleration curve between
t
1 and t2
:Slide7
Change in position = Area under velocity curveThe change in position between times t1 and
t
2
is equal to the area under the velocity curve between
t
1 and t2
:Slide8
EGR 1101: Unit 11 Lecture #2
Applications of Integrals in Electric
Circuits
(Sections 9.6, 9.7 of Rattan/Klingbeil text)Slide9
ReviewAny relationship between quantities that can be expressed using derivatives can also be expressed using integrals.
Example: For position
x(t)
, velocity
v(t)
, and acceleration a(t), Slide10
Energy and PowerWe saw in Week 6 that power is the derivative with respect to time of energy:
Therefore
energy is the integral with respect to time of power (plus the initial energy):Slide11
Current and Voltage in a CapacitorWe saw in Week 6 that, for a capacitor,
Therefore
, for a capacitor,Slide12
Current and Voltage in an InductorWe saw in Week 6 that, for an inductor,
Therefore
, for an inductor,Slide13
Today’s ExamplesCurrent, voltage & energy in a capacitorCurrent & voltage in an inductor (graphical)
Current & voltage in a capacitor (graphical)
Current & voltage in a capacitor (graphical)Slide14
Review: Graphical Derivatives & IntegralsRecall that:Differentiating a parabola gives a slant line.
Differentiating a slant line gives a horizontal line (constant).
Differentiating a horizontal line (constant) gives zero.
Therefore:
Integrating zero gives a horizontal line (constant).
Integrating a horizontal line (constant) gives a slant line.
Integrating a slant line gives a parabola.Slide15
Review: Change in position = Area under velocity curveThe change in position between times t1 and
t
2
is equal to the area under the velocity curve between
t
1 and t2
:Slide16
Applying Graphical Interpretation to InductorsFor an inductor, the change in current between times t1 and
t
2
is equal to 1/
L
times the area under the voltage curve between t1 and
t2:Slide17
Applying Graphical Interpretation to CapacitorsFor a capacitor, the change in voltage between times t1 and
t
2
is equal to 1/
C
times the area under the current curve between t1 and
t2: