Please use MATLAB code to solve the following:
3 This and some of the following problems concern models for the motion of a pendulum, which consists of a weight attached to a rigid arm of length L that is free to pivot in a complete circle. Neglecting friction and air resistance, the angle ) that the arm makes with the vertical direction satisfies the differential equation g 9"(t) + sin ( ( t)) = 0, (D.2) where g -32.2 ft/sec is the gravitational acceleration constant. We will assume the arm has length 32.2 ft and so replace D.2) by the simpler form sin D.3) (Alternatively, one can rescale time, replacing t by approximated by the linear g / L t, to convert (D.2) to (D.3) ) For motions with small displacements ( small), sin 9, and (D.3) can be equation D.4) This equation has general solution ) =A cos t-d) with amplitude A and phase shift Hence all the solutions to the linear approximation (D.4) have period 2T, independent of the amplitude. In this problem we consider solutions of equation (D.3 satisfying the initial conditions 0 A 0 0 If , these solutions are periodic. However, in contrast to the linear equation (D.4), their periods depend on the amplitude A. We do expect that, for small displacements A, the solutions to the pendulum (D.3) will have periods close to 2 (a) Investigate how the period depends on the amplitude A by plotting a numerical solution of equation (D3) using initial conditions 0-4, (0) = 0 on an appropriate interval for various A. Estimate the periods of the pendulum for the amplitudesA-0.1, 0.7, 1.5, and 3.0. Confirm these results by displaying the displacements at a sequence of times, and finding the time at which the pendulum returns to its original position. (b) The period is given by the formula where k = sin(A/2). This formula may be derived in your text; it can be found in Section 9.3 of Boyce & DiPrima, Problem 29. The integral is called an elliptic integral. It cannot be evaluated by an elementary formula, but it can be computed by int in terms of a MuPAD function ellipticK, or can be evaluated numerically using quadl. Calculate the period for the values of A we are considering. Do the values agree with those obtained in part (a)? (c) Redo the numerical calculations in part (a) with different tolerances, choosing the tolerances so that the values you get agree with those calculated in part (b). (d) How does the period depend on the amplitude of the initial displacement? For A small, is the period close to 2r? What is happening to the accuracy of the linear approximation as the initial displacement increases? 3 This and some of the following problems concern models for the motion of a pendulum, which consists of a weight attached to a rigid arm of length L that is free to pivot in a complete circle. Neglecting friction and air resistance, the angle ) that the arm makes with the vertical direction satisfies the differential equation g 9"(t) + sin ( ( t)) = 0, (D.2) where g -32.2 ft/sec is the gravitational acceleration constant. We will assume the arm has length 32.2 ft and so replace D.2) by the simpler form sin D.3) (Alternatively, one can rescale time, replacing t by approximated by the linear g / L t, to convert (D.2) to (D.3) ) For motions with small displacements ( small), sin 9, and (D.3) can be equation D.4) This equation has general solution ) =A cos t-d) with amplitude A and phase shift Hence all the solutions to the linear approximation (D.4) have period 2T, independent of the amplitude. In this problem we consider solutions of equation (D.3 satisfying the initial conditions 0 A 0 0 If , these solutions are periodic. However, in contrast to the linear equation (D.4), their periods depend on the amplitude A. We do expect that, for small displacements A, the solutions to the pendulum (D.3) will have periods close to 2 (a) Investigate how the period depends on the amplitude A by plotting a numerical solution of equation (D3) using initial conditions 0-4, (0) = 0 on an appropriate interval for various A. Estimate the periods of the pendulum for the amplitudesA-0.1, 0.7, 1.5, and 3.0. Confirm these results by displaying the displacements at a sequence of times, and finding the time at which the pendulum returns to its original position. (b) The period is given by the formula where k = sin(A/2). This formula may be derived in your text; it can be found in Section 9.3 of Boyce & DiPrima, Problem 29. The integral is called an elliptic integral. It cannot be evaluated by an elementary formula, but it can be computed by int in terms of a MuPAD function ellipticK, or can be evaluated numerically using quadl. Calculate the period for the values of A we are considering. Do the values agree with those obtained in part (a)? (c) Redo the numerical calculations in part (a) with different tolerances, choosing the tolerances so that the values you get agree with those calculated in part (b). (d) How does the period depend on the amplitude of the initial displacement? For A small, is the period close to 2r? What is happening to the accuracy of the linear approximation as the initial displacement increases
