A point mass m slides without friction from O = (0,0) to P = (a, b) on a curve C under the action of constant gravity (see Figure) with vanishing initial velocity. The time it takes for m to reach P is given by P 1 -ds, Jo T = where ds = V(dr)² + (dy)² and v is the speed. The goal is P=(a,b) y to find the curve C that minimises T. (a) Write T as T = Sº F[y(x), y'(x)]dx and determine the function F. (b) Making use of F – y = const (see Problem 1), derive the relation y' y ƏF dy' V# - 1, where d is a constant. (c) Use the parametric representation y(@) = d sin² = $(1 – cos ø) and determine r(6). %3D The extremal curve is a cycloid.

A point mass m slides without friction from O = (0,0) to P = (a, b) on a curve C under the action of constant gravity (see Figure) with vanishing initial velocity. The time it takes for m to reach P is given by P 1 -ds, Jo T = where ds = V(dr)² + (dy)² and v is the speed. The goal is P=(a,b) y to find the curve C that minimises T. (a) Write T as T = Sº F[y(x), y'(x)]dx and determine the function F. (b) Making use of F – y = const (see Problem 1), derive the relation y' y ƏF dy' V# - 1, where d is a constant. (c) Use the parametric representation y(@) = d sin² = $(1 – cos ø) and determine r(6). %3D The extremal curve is a cycloid.

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

A particle starts from the origin at t = 0 with a velocity of (16 - 12) m/s and moves in the xy plane with a constant acceleration of = (3.0 - 6.0) m/s2. What is the speed of the particle at t = 2s?
I do not know how to get started with my graph. Does my x axis need to be between 7.84 or 15.68 and my y axis between 0.779 or 1.558. If so how should I start my graph and how much should I go up by with both x and y axis's. Please help! I am confused.
Two position vectors are given by Á = (4.00m)î – (2.00m)j – (3.00m)k and B= (4.00m)i - (5.00m)k. What is the scalar product of (3Á) · B? O (16.0m²)î + (15.0m²)k O 31 m2 (48.0m2)i + (45.0m2)k O 93 m2 O None of the above.
A skater with mass 66 kg is skating on a horizontal surface at a constant speed 4.2 m/s. There is a ramp ahead, and the skater has just enough speed to make it to the top of the ramp (meaning the speed at the top of the ramp is 0 m/s). What is the height of the ramp in the unit of meters? Use g = 10 m/s2 for the acceleration due to gravity.
Find the time Tc it takes for a frictionless bead to slide along the cycloid x = a(t - sin t), y = a(1 - cos t) from t = 0 to t = π
A record is played by spinning the record so that a circular groove in the vinyl slides under the stylus. Suppose that a record turns at a rate of 33 1/3 rev. min, the groove being played is at a radius of 10.0 cm and the bumps in the grove are uniformly separated by 1.75 mm. At what rate (hits per second) do the bumps hit the stylus? Enter units as hertz (Hz). You will need v=rω ω is in rad/s and 1rev = 2 pi rad.
The force F has a magnitude of 770 N. Express F as a vector in terms of the unit vectors i and j. Identify the x and y scalar components of F. Assume F-770 N, 8-23° Answers: F- i i+ i j) N Fx- i N Fy- i N
In the figure, a force P acts on a block weighing 45.0 N. The block is initially at rest on a plane inclined at angle = 18.0° to the horizontal. The positive direction of the x axis is up the plane. The coefficients of friction between block and plane are μ = 0.540 and Uk = 0.340. In unit-vector notation, what is the frictional force on the block from the plane when Pis (a) (-5.30 N)î, (b) (-8.10 N)î, and (c) (-15.1 N)? (a) Number i (b) Number i (c) Number i i+ i+ i i i+ i j Units j Units j Units
The y-coordinate of a particle varies at a constant speed of 4.2 m/s. At t=0, the y-coordinate was found to be 2.7 m. Find an analytic expression for the function y(t).
Problem 3: Incline with Friction |A mass m = 2 kg is launched with an initial speed vo = 1 m/s down an incline with coefficient of static friction H, = 3/4 and coefficient of kinetic friction A = 2/3. It travels a distance L = 4 m down the incline before reaching the bottom of the incline with speed vy. The angle 0 such that the values of cos(8) and sin(0) are the ones provided below and you may use g = 10 m/s².