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.  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 Velocity v(t) Acceleration a(t) DerivativeIntegral

Today’s Examples 1. Ball dropped from rest 2. Ball thrown upward from ground level 3. Position & velocity from acceleration (graphical)

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  The change in velocity between times t 1 and t 2 is equal to the area under the acceleration curve between t 1 and t 2 :

Change in position = Area under velocity curve  The change in position between times t 1 and t 2 is equal to the area under the velocity curve between t 1 and t 2 :

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

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  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  We saw in Week 6 that, for a capacitor,  Therefore, for a capacitor,

Current and Voltage in an Inductor  We saw in Week 6 that, for an inductor,  Therefore, for an inductor,

Today’s Examples 1. Current, voltage & energy in a capacitor 2. Current & voltage in an inductor (graphical) 3. Current & voltage in a capacitor (graphical) 4. Current & voltage in a capacitor (graphical)

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  The change in position between times t 1 and t 2 is equal to the area under the velocity curve between t 1 and t 2 :

Applying Graphical Interpretation to Inductors  For an inductor, the change in current between times t 1 and t 2 is equal to 1/L times the area under the voltage curve between t 1 and t 2 :

Applying Graphical Interpretation to Capacitors  For a capacitor, the change in voltage between times t 1 and t 2 is equal to 1/C times the area under the current curve between t 1 and t 2 :