In class, we derived the displacement for a mass on a spring without friction as
$$x=x(t) = x_o\cos(\omega_ot) + \frac{v_o}{\omega_o}\sin(\omega_ot)$$
We derived this equation from conservation of energy
$$E = \frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + \frac{1}{2}kx^2$$
Now, if we were to introduce friction in the picture, Newton's equation becomes
$$-kx – \beta\frac{dx}{dt} = m\frac{d^2x}{dt^2}$$
instead of just
$$-kx = m\frac{d^2x}{dt^2}$$
My task is to introduce the energy function above and show that if $x=x(t)$ satisfies the above Newton equation with friction, then $\frac{dE}{dt} < 0$
From my understanding, $\frac{dE}{dt} < 0$ because friction causes energy to be lost overtime in the spring mass system. I am just having trouble proving that the energy function satisfies the above Newton equation w/ friction.
I attempted to take the first and second derivative of $x=x(t)$ and substituting into the above Newton equation with friction, but I could not really draw any conclusions from that.
Best Answer
Hint: The right side of your Newton equation should be $m\frac {d^2x}{dt^2}$ Having made that correction, take $\frac {dE}{dt}$ by taking the time derivative of the right hand side. You will get some terms times $\frac {dx}{dt}$ Those terms are in Newton's equation.