(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I was reading a book and it had this problem:

"The input u is an acceleration provided by the control system and applied in the horizontal direction to the lower end of the rod. The horizontal displacement of the lower end is y. The linearized form of Newton's law for small angles gives

[tex]mL\ddot{\theta} = mg\theta - mu[/tex] . . . (1)

... (then it proceeds to part (a) of the problem.)"

3. The attempt at a solution

My question is about equation (1). Suppose I sum up the forces at the mass m:

First, I see two axes. One is pointing from the ball to the hinge. The other is tangential to the circular movement. The forces pointing to the hinge will be cancelled by the normal force from the rod. Therefore I ignore that. The forces on the tangential direction (clockwise) are:

F = ma = mg sin(θ) + mu cos(θ) . .. . (2)

Here, I moved the acceleration u to the ball because this acceleration is equivalent to one applied at the center of mass of the ball. Now, since a is the tangential acceleration, I could translate it into angular acceleration a = Lα, this gives me:

mLα = mg sin(θ) + mu cos(θ) . . . (3)

This is almost the same as (1). Now when I use small angle approximation:

mLα = mgθ + mu . . . (4)

Why is my sign for mu flipped compared to (1)?

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# Inverted pendulum

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