the child on the swing PROBLEM 2 What is the optimum strategy for pushing a chil

Question

the child on the swing PROBLEM 2 What is the optimum strategy for pushing a child on a swing to maximize the amplitude? Can we develop a model that will allow us to answer this question? E.g. consider the simpler horizontal spring-mass system (k, m). Lets apply a constant force of magnitude E to the mass for a time 2T during each cycle where is a constant equal to and T is the period equal to 2n/Wn. The constant force will be applied to the mass starting at time aT - AT and lasting until aT + AT where a is a constant that defines when the impulse is applied each cycle 1000 To find the optimum forcing strategy, you may start with the fact that the work done by the constant force in each cycle is given by Fox(t) dt (a-A)T Use a sinusoidal response function x(t) Asin(ant) and ignore force-induced modifications to the sinusoidal response function during the duration of the force. Calculate the power delivered to the system by one impulse (W(Fo)/T) and, as in Eq. 2, cast it in the most physically transparent form you can. You might find it useful to use the sum to product trig. identity, sinu - sinv - 2 cos sin 2

Explanation / Answer

given

spring mass system, mass = m, springk constant= k
applied constnat force = Fo
time period = T = 2*pi/w
delta = 1/1000

hence
work done in each cycle = W
W = integrate(Fo*x'(t)dt) from t = (a - delta*T) to t = (a + delta*T)

consider x = Asin(wt)
then
x' = Awcos(wt)
hence
W = integrate(Fo*x'(t)dt) = integrate(Fo*Aw*cos(wt)dt)
W = Fo*A*sin(wt)
applying limits

W = Fo*A*[sin(wa + w*delta*T) - sin(wa - w*delta*T)]
W = Fo*A*[sin(wa + w*delta*2*pi/w) - sin(wa - w*delta*2*pi/w)]
W = Fo*A*[sin(wa + delta*2*pi) - sin(wa - delta*2*pi)]
using
sin(u) - sin(v) = 2cos((u + v)/2) *sin((u - v)/2)
W = Fo*A*2*cos(wa)*sin(2*pi*delta)

now, W/T = W*w/2*pi [ W is the work done by impulse, w is the natural frequency of the spring mass system]
hence
W/T = Fo*A*wcos(wa)*sin(2*pi*delta)/pi
power deliverd per time period = P = W/T = Fo*A*wcos(wa)*sin(2*pi*delta)/pi
now, as delta = 1/1000
hence 2*pi*delta < < 1
hence
sin(2*pi*delta) = 2*pi*delta
hence
P = Fo*A*w*cos(wa)*2*pi*delta/pi
P = 2Fo*A*w*cos(wa)*delta

so to maximise the power
wa -> 0
hence
a-> 0
hence the force has to be applied as the begining of every osscilation if we take x = Asin(wt)

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