The length of a simple pendulum is 0.77 m and the mass of the particle (the \"bo

Question

The length of a simple pendulum is 0.77 m and the mass of the particle (the "bob") at the end of the cable is 0.25 kg. The pendulum is pulled away from its equilibrium position by an angle of 8.60

Explanation / Answer

the equation of motion for a pendulum is d^2(theta)/dt^2 + g/l sin(theta) = 0 if we make a small angle approximation then sin(theta) = theta for clarity I will now replace theta with 'X' to make the equation look a bit neater d^2X/dt^2 = -X* g/l this is the same as our regular equation for harmonic motion except k/m = g/l as such the angular frequency(answer a) w = sqrt(g/l) (l is the length of the pendulum) the second part is finding the equation for X (it's really theta but it doesn't matter for the math) we know that X is of the form X = Ae^iwt A is given --> A = 8.6 so X = 8.6e^(i(sqrt(g/l)t) taking the real part X = 8.6cos(sqrt(g/l)t) take the derivative we have dX/dt = 8.6*iw*e^(iwt) part B asks us to find the mechanical energy when velocity is max dX/dt =Re[ 8.6iw*e^(iwt)] = -8.6w*sin(sqrt(g/l)t) from the equation for X we can see that it's minimum occurs at t = pi/2 *sqrt(g/l) plug this into our second equation and vmax = abs(-8.6*sqrt(g/l)) HOWEVER this is in radians per second where one radian is one radius = 0.77m so multiply this in to get vmax in m/s then T = 1/2 mv^2 this solves the second 2 parts

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