finding torsion of spiral

[david]

a spiral stair has radius 3 feet and rises 4 feet per revolution (360 degrees). find the length of the handrail that follows the stair if the stair climbs 10 feet. find the torsion of the handrail.

i used this formula to find the length of handrail, not sure if this is the right formula nor not , help please.

L= sqrt(height^2 + (2*pi*radius)^2)

 

[galactus]

Yes, it would appear that is the correct formula.

Except, you have to multiply that result by the number of revs in the 10 feet.

Since the staircase rises 4 feet in one rev, it turns 2.5 revs in 10 feet.

So, multiply your result by 2.5.

\(\displaystyle \displaystyle \frac{5}{2}\sqrt{16+(6\pi)^{2}}\)


We can also use a result from calculus.

If we represent the helix by \(\displaystyle x=a\cdot cos(t), \;\ y=a\cdot sin(t), \;\ z=ct=\frac{2}{\pi}\), then

the length of the helix is \(\displaystyle \displaystyle L=\int_{0}^{t_{0}}\sqrt{a^{2}sin^{2}t+a^{2}cos^{2}t+c^{2}}dt\)

\(\displaystyle \displaystyle =\int_{0}^{t_{0}}\sqrt{a^{2}+c^{2}}=t_{0}\sqrt{a^{2}+c^{2}}\)

Since the staircase turns \(\displaystyle 5\pi \) in the 10 feet, we have:

\(\displaystyle 5\pi \sqrt{(\frac{2}{\pi})^{2}+9}\)

 

[david]

i got it now, thank you.

so the torsion for this would be c/(a^2+c^2) right.

 

[galactus]

I am not sure about that one. I know nothing of torsion. SK can probably answer that one.

 

[pappus]

a spiral stair has radius 3 feet and rises 4 feet per revolution (360 degrees). find the length of the handrail that follows the stair if the stair climbs 10 feet. find the torsion of the handrail.

i used this formula to find the length of handrail, not sure if this is the right formula nor not , help please.

L= sqrt(height^2 + (2*pi*radius)^2)

1. Only one remark: A spiral staircase is not a spiral but a helix.

2. Have a look here: http://en.wikipedia.org/wiki/Helix