💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Numerade Educator

Like

Report

Problem 35 Hard Difficulty

(a) Write the formulas similar to Equations 4 for the center of mass $ (\bar{x}, \bar{y}, \bar{z}) $ of a thin wire in the shape of a space curve $ C $ if the wire has density function $ \rho(x, y, z) $.

(b) Find the center of mass of a wire in the shape of the helix $ x = 2 \sin t $, $ y = 2 \cos t $, $ z = 3t $, $ 0 \leqslant t \leqslant 2\pi $, if the density is a constant $ k $.

Answer

a) $$
\overline{x}=\frac{1}{m} \int_{C} x \rho(x, y) d s
$$
and
$$
\overline{y}=\frac{1}{m} \int_{C} y \rho(x, y) d s
$$
and
$$
\overline{z}=\frac{1}{m} \int_{C} z \rho(x, y) d s
$$
b) The center of mass of the helix is $(0,0,3 \pi)$

Discussion

You must be signed in to discuss.

Video Transcript

so the formal off center ofthe masses three dimension to damage is very similar We first have to compute the mass Ah which will be this? And the center ofthe mass? We'Ll just be one over him times What? Check through the s Why road? Yes zero t d s And here we can compute We're given this information We have computer tedious Off his squared of the ecstasy He square prostitute Why did he square past easy Did he square So square Rudo four plus nine square Rudo thirteen Oh sorry, I forgot Did he hear? So everything should be with the TT So has computed on the row is constant k so mass should be integral Tea for one zero to two Pi constant k He s his square of certainty So it should be two squared off thirteen Pi k Then we just have to integrate other stuff. Ah, ex road. Yes, Uh tea from zero to two Pi access to scientific e We're always kay the system squirrel thirteen Petey So this one got two squared of thirteen pi k No, no, no to square those thirteen k into her from zero to two Pi Sai Inti Petey However, we know this is zero. He was the entire derivative. Ah, negative. Cosa has the same value as your two pi. So this is zero by the similar argument Ah, this one We'LL get a similar thing with science course I'm being swamped. So this is also zero z times. Rozi is treaty so three thirteen time's thiss which should be Siri Square, the thirteen k integral from zero to two Pi ti ti We're t square over to prowl into pies off four pi square over too. So this should be to Pi square. So yes, we get sarees Anyway, we just keep it like this because we have to divide it by m l z. And so the moment Sinema was Ah So the centre ofthe mass should be zero zero and ah for Z should be thiss divided by two square root of three pie stew score of thirteen bike to score the thirteen k and the pie cake and so so were left with three pie. No