3. Consider a pendulum of length L, released from rest at an angle a. It can be shown that such a pendulum has period T given by T = 4, *xp: 1 – sin? (4) sin? x (a) Notice that the integrand has the form For simplicity, write A = sin? (9). Show that +Σ 1.3-5... (2n – 1) 2nn! 2n %3D V1- A sin? x n=1 (b) Calculate T/2 */2 */2 sin?: x dr, s sin“ r dr and sin® r x dr. (c) We can now determine the period T as a power series in A (which encodes the initial angle a) by integrating the power series from (a). Show that 1 1+-A+ 25 x² + 256 T = 27 64 (d) This is a great result, but the initial angle a is hidden from view. Let's turn this into a power series in a: i. We have the identity sin? t = (1 – cos(2t)). Use this to show that the Maclaurin series for sin? t is given for any t by (-1)n+12n-1 sin?t: (2n)! n=1 Write out the terms of this series up to degree 6. ii. Show that to degree 6, A is given in terms of a by 2 )"- 45 iii. Now show that to degree 6, the period T, in terms of a, is given by 11 173 T = 2n 1+ 16a +... 3072 737280 (e) You will often find the following statement: For small a, the period of a pendulum is given by T = 27/. Explain this based on your calculation.

3. Consider a pendulum of length L, released from rest at an angle a. It can be shown that such a pendulum has period T given by T = 4, xp: 1 – sin? (4) sin? x (a) Notice that the integrand has the form For simplicity, write A = sin? (9). Show that +Σ 1.3-5... (2n – 1) 2nn! 2n %3D V1- A sin? x n=1 (b) Calculate T/2 /2 */2 sin?: x dr, s sin“ r dr and sin® r x dr. (c) We can now determine the period T as a power series in A (which encodes the initial angle a) by integrating the power series from (a). Show that 1 1+-A+ 25 x² + 256 T = 27 64 (d) This is a great result, but the initial angle a is hidden from view. Let's turn this into a power series in a: i. We have the identity sin? t = (1 – cos(2t)). Use this to show that the Maclaurin series for sin? t is given for any t by (-1)n+12n-1 sin?t: (2n)! n=1 Write out the terms of this series up to degree 6. ii. Show that to degree 6, A is given in terms of a by 2 )"- 45 iii. Now show that to degree 6, the period T, in terms of a, is given by 11 173 T = 2n 1+ 16a +... 3072 737280 (e) You will often find the following statement: For small a, the period of a pendulum is given by T = 27/. Explain this based on your calculation.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

answer qs 9
M2. Subject:- Advance Maths
please send correct answer of Q 21
I need Part d and e. Thanks!
4.b- Please I want answer for this question by typing it. Thanks
Activity 6.2 Driving around For this activity, you need a paper, a ruler, and a writing implement. Consider three cars originally in the same location in a city-we set this location to be the origin. The city streets are designed such that all travel occur along straight lines and streets span either directly eastward or directly northward. The cars then drove to different directions and took the following paths: Car A: 450 m East Car B: 50 m North →300 m East → 100 m South → 50 m West →→ 150 m North Car C: 300 m West → 150 m South → 50 m West → 50 m South Draw the paths taken by all cars on a clean sheet of paper (with scale 50 m: 1 cm). Calculate the total distance travelled by each car. Then using a ruler, draw a straight line from the starting point to where the paths of each car ended. Measure the length of that line. Rank the cars in descending order, first by total distance travelled, then by the straight-line distance from the origin. Do the two rankings agree with each other?…
Physics. Please help!
Please provide typed answer. I NEED TYPED ANSWERS. Thank you.
I need help with my physics hw, thank youuu
Please answer letter H, I, and J only.
Please do b to e. Don't worry about A
Answer Qn 3