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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. Recall that differentiation and integration are inverse operations.

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EGR 1101: Unit 11 Lecture # 1

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  1. EGR 1101: Unit 11 Lecture #1 Applications of Integrals in Dynamics: Position, Velocity, & Acceleration (Section 9.5 of Rattan/Klingbeil text)

  2. 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.

  3. Position, Velocity, & Acceleration Position x(t) Derivative Integral Velocity v(t) Derivative Integral Acceleration a(t)

  4. Today’s Examples • Ball dropped from rest • Ball thrown upward from ground level • Position & velocity from acceleration (graphical)

  5. 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.

  6. 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:

  7. 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:

  8. EGR 1101: Unit 11 Lecture #2 Applications of Integrals in Electric Circuits (Sections 9.6, 9.7 of Rattan/Klingbeil text)

  9. 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),

  10. 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):

  11. Current and Voltage in a Capacitor • We saw in Week 6 that, for a capacitor, • Therefore, for a capacitor,

  12. Current and Voltage in an Inductor • We saw in Week 6 that, for an inductor, • Therefore, for an inductor,

  13. 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)

  14. 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.

  15. 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:

  16. 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:

  17. 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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