2. A pendulum is a mechanical system in which a mass mis attached to a massless incxtensible string of lengthwhich is in turn connected to a frictionless pivot, asshown at right.The swing angle (t) of the pendulum satisfies thesecond-order nonlinear differential equationd'e22²+ sin 0-0.2sineThe transformationv --allows us to transform thisequation into the separable first-order differential equa-tion(1)U +sin 0-0.sinopivot1v(0)'mass 77(a) If the mass m is initally at rest at an angle of — T/6 degrees, solve the differ-ential equation (1) to find v = v(8).(b) Given that(2)deintegrate both sides of this separable differential equation to find the generalsolution tt(0) in the form of an integral (with respect to 0).(c) Hence, write down a definite integral for the period 7' of the pendulum (that is,the time for the mass m to complete one full swing and return to its startingposition. (Note: v <0 on this section because is decreasing.)(Hint: The period is four times the amount of time it takes for the mass to reachthe "vertically down" position from the starting position. You should integrate(2) from the starting point (- 0, 0 = π/6) to the "vertically down" position(t-T/4, 0-0). Note: v <0 on this section because is decreasing.)(d) If = 1.6 metres, use Simpson's rule with four strips to estimate T. (Note: Youshould assume, for simplicity, that g = 10.)

vdvdθ+glsinθ=0 above equation can be re written as vdv= -glsinθdθintegrating on both side∫vdv=…

2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length which is in turn connected to a frictionless pivot, as shown at right. The swing angle (t) of the pendulum satisfies the second-order nonlinear differential equation de + sin 0-0. The transformation v = allows us to transform this equation into the separable first-order differential equa- tion (1) 1- sin 0-0. pivot mass m

2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length which is in turn connected to a frictionless pivot, as shown at right. The swing angle (t) of the pendulum satisfies the second-order nonlinear differential equation de + sin 0-0. The transformation v = allows us to transform this equation into the separable first-order differential equa- tion (1) 1- sin 0-0. pivot mass m

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter4: Numerical Analysis Of Heat Conduction

Section: Chapter Questions

Problem 4.21P

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