Egr 1101 unit 11 lecture 1
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EGR 1101: Unit 11 Lecture # 1. Applications of Integrals in Dynamics: Position, Velocity, & Acceleration (Section 9.5 of Rattan/Klingbeil text). Differentiation and Integration. Recall that differentiation and integration are inverse operations.

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EGR 1101: Unit 11 Lecture # 1

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Egr 1101 unit 11 lecture 1

EGR 1101: Unit 11 Lecture #1

Applications of Integrals in Dynamics: Position, Velocity, & Acceleration

(Section 9.5 of Rattan/Klingbeil text)


Differentiation and integration

Differentiation and Integration

  • Recall 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.


Position velocity acceleration

Position, Velocity, & Acceleration

Position x(t)

Derivative

Integral

Velocity v(t)

Derivative

Integral

Acceleration a(t)


Today s examples

Today’s Examples

  • Ball dropped from rest

  • Ball thrown upward from ground level

  • Position & velocity from acceleration (graphical)


Graphical derivatives integrals

Graphical derivatives & integrals

  • Recall 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.


Change in velocity area under acceleration curve

Change in velocity = Area under acceleration curve

  • The change in velocity between times t1 and t2 is equal to the area under the acceleration curve between t1 and t2:


Change in position area under velocity curve

Change in position = Area under velocity curve

  • The change in position between times t1 and t2 is equal to the area under the velocity curve between t1 and t2:


Egr 1101 unit 11 lecture 2

EGR 1101: Unit 11 Lecture #2

Applications of Integrals in Electric Circuits

(Sections 9.6, 9.7 of Rattan/Klingbeil text)


Review

Review

  • Any 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),


Energy and power

Energy and Power

  • We 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):


Current and voltage in a capacitor

Current and Voltage in a Capacitor

  • We saw in Week 6 that, for a capacitor,

  • Therefore, for a capacitor,


Current and voltage in an inductor

Current and Voltage in an Inductor

  • We saw in Week 6 that, for an inductor,

  • Therefore, for an inductor,


Today s examples1

Today’s Examples

  • Current, voltage & energy in a capacitor

  • Current & voltage in an inductor (graphical)

  • Current & voltage in a capacitor (graphical)

  • Current & voltage in a capacitor (graphical)


Review graphical derivatives integrals

Review: Graphical Derivatives & Integrals

  • Recall 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.


Review change in position area under velocity curve

Review: Change in position = Area under velocity curve

  • The change in position between times t1 and t2 is equal to the area under the velocity curve between t1 and t2:


Applying graphical interpretation to inductors

Applying Graphical Interpretation to Inductors

  • For an inductor, the change in current between times t1 and t2 is equal to 1/L times the area under the voltage curve between t1 and t2:


Applying graphical interpretation to capacitors

Applying Graphical Interpretation to Capacitors

  • For a capacitor, the change in voltage between times t1 and t2 is equal to 1/C times the area under the current curve between t1 and t2:


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