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2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length I which is in turn connected to a frictionless pivot, as shown at right. The swing angle 0(t) of the pendulum satisfies the second-order nonlinear differential equation pivot + sin 6 = 0. dt2 de allows us to transform this equation into the separable first-order differential equa- The transformation v = mass m tion dv (1) do + sin 0 = 0. (a) If the mass m is initally at rest at an angle of 00 = 7/3 degrees, solve the differ- ential equation (1) to find v = v(0). (b) Given that dt 1 (2) OP v(0)' integrate both sides of this separable differential equation to find the general solution t = t(0) in the form of an integral (with respect to 0). (c) Hence, write down a definite integral for the period T of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: v < 0 on this section because 0 is decreasing.) (Hint: The period is four times the amount of time it takes for the mass to reach the "vertically down" position from the starting position. You should integrate (2) from the starting point (t = 0, 0 = 1/3) to the "vertically down" position (t = T/4, 0 = 0). Note: v < 0 on this section because 0 is decreasing.) (d) If l = 1.2 metres, use Simpson's rule with four strips to estimate T. (Note: You should assume, for simplicity, that g = 10.)

2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length I which is in turn connected to a frictionless pivot, as shown at right. The swing angle 0(t) of the pendulum satisfies the second-order nonlinear differential equation pivot + sin 6 = 0. dt2 de allows us to transform this equation into the separable first-order differential equa- The transformation v = mass m tion dv (1) do + sin 0 = 0. (a) If the mass m is initally at rest at an angle of 00 = 7/3 degrees, solve the differ- ential equation (1) to find v = v(0). (b) Given that dt 1 (2) OP v(0)' integrate both sides of this separable differential equation to find the general solution t = t(0) in the form of an integral (with respect to 0). (c) Hence, write down a definite integral for the period T of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: v < 0 on this section because 0 is decreasing.) (Hint: The period is four times the amount of time it takes for the mass to reach the "vertically down" position from the starting position. You should integrate (2) from the starting point (t = 0, 0 = 1/3) to the "vertically down" position (t = T/4, 0 = 0). Note: v < 0 on this section because 0 is decreasing.) (d) If l = 1.2 metres, use Simpson's rule with four strips to estimate T. (Note: You should assume, for simplicity, that g = 10.)

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2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length I which is in turn connected to a frictionless pivot, as shown at right. The swing angle 0(t) of the pendulum satisfies the second-order nonlinear differential equation pivot + sin 0 = 0. dt? The transformation v = " allows us to transform this dt mass m equation into the separable first-order differential equa- tion dv sin 0 = 0. (1) OP (a) If the mass m is initally at rest at an angle of 60 ential equation (1) to find v = = T/3 degrees, solve the differ- = v(0). (b) Given that dt 1 (2) | do v(0)' integrate both sides of this separable differential equation to find the general solution t = t(0) in the form of an integral (with respect to 0). (c) Hence, write down a definite integral for the period T of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: v < 0 on this section because 0 is decreasing.) (Hint: The period is four times the amount of time it takes for the mass to reach the "vertically down" position from the starting position. You should integrate (2) from the starting point (t = 0, 0 = 1/3) to the "vertically down" position (t = T/4, 0 = 0). Note: v < 0 on this section because 0 is decreasing.) (d) If l = 1.2 metres, use Simpson's rule with four strips to estimate T. (Note: You should assume, for simplicity, that g = = 10.)
Transcribed Image Text:2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length I which is in turn connected to a frictionless pivot, as shown at right. The swing angle 0(t) of the pendulum satisfies the second-order nonlinear differential equation pivot + sin 0 = 0. dt? The transformation v = " allows us to transform this dt mass m equation into the separable first-order differential equa- tion dv sin 0 = 0. (1) OP (a) If the mass m is initally at rest at an angle of 60 ential equation (1) to find v = = T/3 degrees, solve the differ- = v(0). (b) Given that dt 1 (2) | do v(0)' integrate both sides of this separable differential equation to find the general solution t = t(0) in the form of an integral (with respect to 0). (c) Hence, write down a definite integral for the period T of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: v < 0 on this section because 0 is decreasing.) (Hint: The period is four times the amount of time it takes for the mass to reach the "vertically down" position from the starting position. You should integrate (2) from the starting point (t = 0, 0 = 1/3) to the "vertically down" position (t = T/4, 0 = 0). Note: v < 0 on this section because 0 is decreasing.) (d) If l = 1.2 metres, use Simpson's rule with four strips to estimate T. (Note: You should assume, for simplicity, that g = = 10.)
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