Chapter 11: Q. 55 (page 890)

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos h t, \sin h t, t>$

Short Answer

Expert verified

The torsion of the given vector is $τ (t) = \frac{1}{2} .$

Step by step solution

Step 1. Given Information.

The given vector function is $r (t) = <\cos h t, \sin h t, t> .$

Step 2. Compute the torsion.

To compute the torsion of the vector function, we will use the torsion formula $τ (t) = \frac{(r' (t) \times r'' (t)) \cdot r''' (t)}{{||r' (t) \times r'' (t)||}^{2}} .$

So,

$r (t) = <\cos h t, \sin h t, t> r' (t) = <- \sin h t, \cos h t, 1> r'' (t) = <- \cos h t, - \sin h t, 0> r''' (t) = <\sin h t, - \cos h t, 0> Now, r' (t) \times r'' (t) = |\begin{matrix} i & j & k \\ - \sin h t & \cos h t & 1 \\ - \cos h t & - \sin h t & 0 \end{matrix}| = i (\sin h t) - j (\cos h t) + k (1) = <\sin h t, - \cos h t, 1> Let' s compute the dot product ((r' (t) \times r'' (t)) \cdot r''' (t)), = <s i n h t, - c o s h t, 1> \cdot <\sin h t, - \cos h t, 0> = \sin^{2} h t + \cos^{2} h t = 1 . . . . . . . . (a) Now, ||r' (t) \times r'' (t)|| = \sqrt{{(\sin h t)}^{2} + {(- \cos h t)}^{2} + {(1)}^{2}} = \sqrt{1 + 1} = \sqrt{2} {||r' (t) \times r'' (t)||}^{2} = {(\sqrt{2})}^{2} {||r' (t) \times r'' (t)||}^{2} = 2 . . . . . . . . . . . (b)$

Step 3. Calculate.

Put the values of (a) and (b) in the torsion formula,

$τ (t) = \frac{(r' (t) \times r'' (t)) \cdot r''' (t)}{{||r' (t) \times r'' (t)||}^{2}} τ (t) = \frac{1}{2}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos h t, \sin h t, t>$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Compute the torsion.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.r(t)=cosht,sinht,t

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Compute the torsion.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos h t, \sin h t, t>$