Problem 2 a mechanical metronome is a device that is used to...

Problem 2: A mechanical metronome is a device that is used to mark time in music by producing a tone at regular and adjustable intervals of time. It consists of a double-weighted pendulum with masses m1 and m2 that rotates about the fixed-point O, as shown in Fig. 2, where the length $l_1$ is adjustable. It is assumed that the connecting rod is of negligible mass, and we consider small displacement angles $\theta$. i) Find the equation of motion of the metronome double-weighted pendulum model with respect to the rotating angle $\theta(t)$ in terms of the masses $m_1$ and $m_2$ and the lengths $l_1$ and $l_2$. Linearize the equation of motion about the vertical position $\bar{\theta} = 0$.

Transcript

00:01 Now here in this question, basically we look at a component pendulum, right? so let me try to draw this compound pendulum.

00:08 Properly is kind of like this.

00:10 Let me say it's kind of thin rod, right? so the same raw, let me try to put it like this.

00:18 So this kind of thin rod.

00:19 But the point is that this thing is not necessarily uniform, right? and so i try to draw it and not really that uniform, but it's thin rod, okay? like this.

00:28 And the center of mass of it, let me say somewhere here.

00:32 Okay, that's the center of mass, c .m.

00:36 And now this rod is pivoted at this point.

00:41 So this is a pivot point, okay? and it's freely to swing in the paper plane.

00:47 The paper plane, according to the question, and actually is the x y plane.

00:51 Let me try to enjoy it.

00:53 So i will put according to the question, this is x direction.

00:57 And this is the y direction.

01:00 And of course, the z directions actually come out of the plan.

01:04 I'm not going to draw it anymore.

01:07 So there's a distance here.

01:10 And you asked to write down the equation of motion for the angular momentum, right, of this rod.

01:18 While the angular momentum, when the rod is swinging in the x -y plane, the angular momentum, of course, is perpendicular.

01:28 To this x -y plane, that's on z direction.

01:31 So the angular momentum z dot, right, that would be actually given by, you know, the total torque acted on the lot, right? and that torque is just the torque due to the, due to, of course, the weight, right? so imagine this rod swans to, this place with angle theta to the vertical direction.

02:03 And then you can easily see that the torque, the torque due to this weight will be given by just a distance here.

02:13 No, sorry, not another distance there.

02:15 The distance here, right, multiplied by the weight, right? the weight is the m is the mass of the route and g is the acceleration required on earth.

02:27 And this is the distance.

02:28 And that, of course, is simply given by the a times sine cita.

02:33 So what we were found for talk is mg, the force multiplied by the distance.

02:39 That is a times sine cata, right? and, of course, the a .o .z is just given by the momentum of inertia, the moment of inertia of this route, right? the moment of inertia of this route multiply the angle speed, which is a time derivative of theta.

03:02 I will just write as theta dot, right? now you combine these two equations, you'll find that i times theta double dot, which is the angular acceleration, the second order time derivative of theta, must be equal to mga times sine theta.

03:20 Now i'm going to make the approximation that the theta actually small, so that same sata is approximately ceta.

03:26 So i just write that cita, right? and this gives us the equation.

03:33 We need to find out the frequency of the pendulum.

03:36 And what we need to do, obviously, is just to re -assume for some ceta oscillates kind of like cosine -om omega -t.

03:44 Then you plot this into this equation...

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