Energy in simple harmonic motion--car problem

[Brian_D]

Homework Statement

A 1400-kg car with poor shock absorbers is bouncing down the highway at 20 m/s, executing vertical harmonic motion at 0.67 Hz. If the amplitude of the oscillations is 18 cm, what is the total energy in the oscillations? What fraction of the car's kinetic energy is this? Neglect rotational energy of the wheels and the fact that not all the car's mass participates in the oscillations.

Relevant Equations

I am only working on the first part of this problem (total energy in the oscillations). The relevant equations are: $$total energy=.5*m*(Vmax)^2+.5kA^2$$ $$k=mw^2$$ $$Vmax=A*sqrt(k/m)$$

Using the above equations and the given information, I get $k=2.48*10^-4$ 2.47∗10−4max=7.56∗10−5$Vmax=7.58*10^-5$ and $total energy=8*10^-6 joules$. My answer is clearly the wrong order of magnitude. The book answer key says 400 joules. My calculation for total energy was: ##.5*1400*(7.58*10^-5)^2+.5*(2.48*10^-4)*.18^2 = 8*10^-6 joules. Where am I going wrong?

 

[haruspex]

The relevant equations are: $$total energy=.5*m*(Vmax)^2+.5kA^2$$ $$k=mw^2$$ $$Vmax=A*sqrt(k/m)$$

That is not right. At any instant, velocity v, displacement x, the total energy is $\frac 12mv^2+\frac 12kx^2$. At max velocity, x=0, and at displacement A, v=0. So it $=.5*m*(V_{max})^2=.5kA^2$.

How do you get that value for k?

 

[Brian_D]

Thank you, haruspex. Your post is not showing up on my computer (I get an ad for software instead). But I saw a version of your post in the email from PhysicsForums and in that you said that the answer is $.5*m*(V_{max})^2=.5kA^2$ When I calculate the first expression, I get $.5*1400*(7.58*10^-5)^2$, which equals $4.0*10^-6 joules$ and I get a similar value for the second expression. So why is the order of magnitude so far off base? The answer that the book gives is 400 joules.

 

[Brian_D]

haruspex, now your post is displaying correctly; I am checking my calculation for k.

 

[Brian_D]

Another mystery solved. I made a simple arithmetic error in calculating k (I used $10^-4$ instead of #10^4##. I checked this twice and made the same mistake both times! Thanks for your help, haruspex.