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# EGR 1101: Unit 11 Lecture # 1 - PowerPoint PPT Presentation

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

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

(Section 9.5 of Rattan/Klingbeil text)

• 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 x(t)

Derivative

Integral

Velocity v(t)

Derivative

Integral

Acceleration a(t)

• Ball dropped from rest

• Ball thrown upward from ground level

• Position & velocity from acceleration (graphical)

• 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.

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

• 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

Applications of Integrals in Electric Circuits

(Sections 9.6, 9.7 of Rattan/Klingbeil text)

• 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),

• 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):

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

• Therefore, for a capacitor,

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

• Therefore, for an inductor,

• Current, voltage & energy in a capacitor

• Current & voltage in an inductor (graphical)

• Current & voltage in a capacitor (graphical)

• Current & voltage in a capacitor (graphical)

• 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.

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

• 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:

• 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: