3.4 by Paul is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

# 3.4

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3.4 by Paul is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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when you take the derivative in the work energy principles, why does 1/2*k*x^2 only result in k*x as opposed to k* xdot

Nevermind, I answered my own question...is it because we are differentiating with respect to x and not time?

You're right, the derivative is with respect to and thus the derivative of is .

This problem is the same as 3.4 in the 11th edition

Just a addition point worth mentioning to possibly simplify the method of adding the spring coefficients.

For springs in series, use the same method of adding capacitors in series and you will arrive at the same equivalent stiffness coefficient. Example: for part a) k1+(k2^-1+k3^-1)^-1 = k1+ k2*k3/(k2+k3). Intuitively, this can be understood by comparing a springs behavior under a load as storing mechanical energy in the same respect as a capacitor under an electrical load stores electrical energy. Thus the same series addition can be applied.

For springs in parallel, you simply add their values together... like adding capacitors in parallel in an electric circuit. Hints why in the above example you add k1 to the inverse sum of the inverses.

You are absolutely correct. In fact each of the mechanical elements has an electrical (and fluidic) counterpart.

Spring -> Capacitor

Damper -> Resistor

Mass -> Inductor

The analysis of mechanical systems and electric circuits can both be reduced to the same fundamental idea. In fact there are methods of analysis which are completely agnostic of the type of system you are investigating, they focus entirely on how each element stores, transfers, or dissipates energy.