MATLAB PENDULUM LAB Theory Assume a pendulum consists of a mass n is suspended f
ID: 3598580 • Letter: M
Question
MATLAB PENDULUM LAB Theory Assume a pendulum consists of a mass n is suspended from a ceiling by a rigid massless rod of constant length L rotating about a pivot point (where e is the angular displacement in radians measured from vertical). Assume there is no damping or other external forces except for gravity (where g is acceleration due to gravity). mg Recall that the lengths of an are of a circle cut by an angle 8 (in radians) is given by sore (here it would be s= LO). After differentiating twice with respect to time we get the tangential acceleration d's dºe de a= =1. So by Newton's second law, the tangential force is Frma - mL . But we also d see that the tangential force is F=-img sina (the negative sign is because the direction of the force is de opposite of 8, ie, when 8>0 the force acts to decrease 8 and vis-versa). So, -mg sinQ = ml which after dividing by m and rearranging yields de g (1) +sina = 0. The equation (1) is not linear and cannot be solved using elementary functions. However for small 6-values, sina and we can approximate the differential equation (1) by the linear differential equation de g + 8 - 0. (2)Explanation / Answer
The equations given can be solved on matlab.
What is the amplitude of motion?
The Amplitude is the height from the center line to the peak. write the eq (2) of the question as -
d^2(theta)/d^2(t) = - (g/L) theta
= > theta = theta_max * Cos(wt)
Note here that for a given pendulum, g/L is a constant. The following matlab code solves this -
syms y(x)
Dy = diff(y);
ode = diff(y,x,2) == -(g/L)*x
ySol(x)=dsolve(ode);
ySol=simplify(ySol); %ysol has the first order diffrential of the motion it's root at 0 will give the amplitude.
fzero(ySol,0); %will the maximum amplitude.
Note - this is to find the amplitude with L, g and other values given i.e. to find the numerically maximum amplitude of a particular pendulum. the formula for the amplitude is given above.
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What is the period of motion?
The period of the motion for a pendulum is how long it takes to swing back-and-forth, measured in seconds.
Period of motion = 2*pie(L/g)^(1/2)
3) No, period doesn't depent on mass, as seen from the eq above.
4) no, it doesn;t.
5) if g increased by 4, The time period decreases by 2. Check eq above for verification.
6) if L increased by 4, the time period increases by 2. Check eq above for verification.
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