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

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2. A pendulum is a mechanical system in which a mass m
is attached to a massless incxtensible 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
d'e
22²
+ sin 0-0.
2sine
The transformation
v --
allows us to transform this
equation into the separable first-order differential equa-
tion
(1)
U +
sin 0-0.
sino
pivot
1
v(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)
de
integrate both sides of this separable differential equation to find the general
solution 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 starting
position. (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 reach
the "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: 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 incxtensible 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 d'e 22² + sin 0-0. 2sine The transformation v -- allows us to transform this equation into the separable first-order differential equa- tion (1) U + sin 0-0. sino pivot 1 v(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) de integrate both sides of this separable differential equation to find the general solution 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 starting position. (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 reach the "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: You should assume, for simplicity, that g = 10.)
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