3) Four masses m1, m2, m3, and m4 are each attached to a...

3) Four masses $m_1$, $m_2$, $m_3$, and $m_4$ are each attached to a uniform pole of length $L$ having mass $m_p$ and center of mass at $x_p$ such that $x_p > x_r$. If the masses are placed at distances $x_1$, $x_2$, $x_3$, and $x_4$ from the left end of the pole, at what distance from that end will the pole balance? Use torque methods and express your answer in terms of $m_p$, $m_1$, $m_2$, $m_3$, $m_4$, $x_p$, $x_1$, $x_2$, $x_3$, and $x_4$. Record your answer in the Derivation 3 table.

Transcript

00:01 So we have two masses with the weight of one newton and the other one with the weight of three newtons.

00:07 How far along should we put the fulcrum in order to balance it? and if we put the fulcrum at exactly the middle, what is the net torque about the axis of rotation? so in the problem, we have two masses upon a meter stick.

00:29 And they're both at the ends of the meter stick.

00:34 One of them, the first one, is going to be a weight or a force to the gravity of one nuden.

00:41 The second one has a force to the gravity of three nons.

00:47 So how far parts to the fulcrum or the middle be? well, let's say first we put it in the half waypoint of the meter stick.

01:00 So our r1 and our 2 is equal to each other.

01:05 R1 is equal to r2.

01:08 For simplicity stake, let's just call both of these r.

01:12 Remember, for a torque equilibrium, we have mgl1 is equal to mgl2.

01:23 Mass of 1 times gravity times the length to the center is equal to of massitude times the length two from the center.

01:32 So we have our mgs for both of them.

01:35 So we have 1 times l1 is equal to 3 times l2.

01:41 Remember, we did say that these links are the same.

01:45 So 1r is equal to 3r, which is false.

01:52 So how will we go about balancing these? well, what if we put the fulcrum maybe half r closer to the heavier weight? therefore, here we would no longer have r and r, we would have 1 .5 r because it is adding half an r and then 0 .5r, subtracting half an r...

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