A curve in R will be on a helix with central axis of the z-axis if...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
A curve in R³ will be on a helix with central axis of the z-axis if it can be given the a parametrization in the following form: c(t) = (r cos(t), r sin(t), kt) where r> 0 and k0 are constants (this parametrization is usually not by arc length.) The term r measures the radius of the helix (i.e. the radius of the cylinder x² + y² = r² which contains the curve) and the term k determines the "slope" of the helix. If k = 0 we will have a degenerate case where we actually have a circle, otherwise the sign of k will determine the direction in which the helix "twists." You can see what is meant by "slope" if you graph the angle t versus the height z = kt. Prove that both the curvature and torsion of a helix will be constant and that further- more we have: T K = k T A curve in R³ will be on a helix with central axis of the z-axis if it can be given the a parametrization in the following form: c(t) = (r cos(t), r sin(t), kt) where r> 0 and k0 are constants (this parametrization is usually not by arc length.) The term r measures the radius of the helix (i.e. the radius of the cylinder x² + y² = r² which contains the curve) and the term k determines the "slope" of the helix. If k = 0 we will have a degenerate case where we actually have a circle, otherwise the sign of k will determine the direction in which the helix "twists." You can see what is meant by "slope" if you graph the angle t versus the height z = kt. Prove that both the curvature and torsion of a helix will be constant and that further- more we have: T K = k T
Expert Answer:
Related Book For
Fundamentals of Physics
ISBN: 978-0471758013
8th Extended edition
Authors: Jearl Walker, Halliday Resnick
Posted Date:
Students also viewed these accounting questions
-
Graph the curve x = 2 cos cos 2 y = 2 sin sin 2. If this curve is rotated about the -axis, find the area of the resulting surface. (Use your graph to help find the correct parameter interval.)
-
A right circular cylinder of height h and radius r is inscribed in a right circular cone with a height of 10 ft and a base with radius 6 ft. (a) Express the height h of the cylinder as a function of...
-
Prove that Cos t = cos c using an argument similar to the one used in the proof that Sin t = sin c. im lim
-
Calculate the shape factor f for the wide flange beam shown in the figure if h = 12.2 in., b = 8.08 in., tf = 0.64 in., and tw = 0.37 in. Iw
-
A company is in the process of negotiating with a bank for an additional loan. Why will the bank be interested in the companys debt service coverage ratio?
-
How many different Boolean functions F(, , ) are there such that F(x, y, z) = F( x, y, z) for all values of the Boolean variables x, y, and z?
-
When should control risk be set at a maximum?
-
Pargo Company budgeted selling expenses of $30,000 in January, $35,000 in February, and $40,000 in March. Actual selling expenses were $31,000 in January, $34,500 in February, and $47,000 in March....
-
Item 2 : Entity B reported net income of $ 1 3 8 , 0 0 0 for the current year. Depreciation recorded on buildings and equipment amounted to $ 3 4 , 0 0 0 for the year. Balances of needed current...
-
How high h must the light bulb in Figure 5 be if the point (1, 25, 0) is on the edge of the illuminated region? Light hu b
-
Amy offered to sell a television to Holden for $400. Holden said "okay, as long as you deliver it." Amy said "no way, my car is in the shop." Holden says "fine then, we have a deal. I'll pick it up an
-
On July 1, 2023, Crane Corp., which uses IFRS, signs a 4-year, non-cancellable lease agreement to lease a equipment from Blossom Ltd. The following information concerns the lease agreement. 1. The...
-
Discussion One: Review the material presented in the course about how to conduct a job analysis, the various methods for collecting data, and the pros and cons of the various methods. Then, develop a...
-
Through a body of knowledge interpreted by the stakeholders Through a body of knowledge interpreted by government officials Through a body of knowledge interpreted by competent practitioners and...
-
Adam's AGI for the year 2020 is $100,000. Adam is single and age 58. Adam incurred the following medical expenses: Medical Insurance $ 7,200 Hospital & Doctors 14,500 Dentists 5,400 Rx Drugs 2,200...
-
Your generous grandmother has just announced that she has opened a savings account for you with a deposit of $14,000. Moreover, she intends to make 9 more similar gifts,at the end of this year, next...
-
Find the maximum and minimum values of the function g(?) = 2? - 8 sin(?) on the interval [0, ?]
-
If (x) 0 on the interval [a, b], the definite integral gives the exact area under the curve between x = a and x = b.
-
An x-ray beam of wavelength A undergoes first-order reflection from a crystal when its angle of incidence to a crystal face ts 23o, and an x-ray beam of wavelength 97 pm undergoes third-order...
-
A violin string 15.0 cm long and fixed at both ends oscillates in its n = 1 mode. The speed of waves on the string is 250 m/s, and the speed of sound in air in 348 m/s. What are the (a) Frequency and...
-
Figure shows four-capacitor arrangement that is connected to a larger circuit at points A and B. The capacitances are C1 = l0 p and C2 = C3 = C4 = 20F.The charge on capacitor 1 is 30C. What is the...
-
Who are the members of an audit committee? What things might an auditor discuss with them during the initial meeting? What things might be discussed during the course of an audit?
-
What client documents should be reviewed to obtain knowledge as suggested by ISA 310?
-
(Financial planning) You are 30 years old today and are considering studying for an MBA. You have just received your annual salary of $50,000 and expect it to grow by 3% per year. MBAs typically earn...
Study smarter with the SolutionInn App