Understanding the Pendulum Dynamics Equation for a Cart

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The discussion focuses on understanding the pendulum dynamics equation for a cart, specifically how it relates to the forces acting on the center of mass of the rod. The equation involves terms for tangential acceleration (##\ddot\theta##) and radial (centripetal) acceleration (##\dot\theta^2##), which are derived from basic principles of force and acceleration. There is clarification that the length of the rod should be considered as 2l instead of l in the equation. Additionally, the centripetal force is noted to act radially, but the equation specifically addresses the force component in the x direction. Overall, the discussion aims to clarify the relationship between these forces and their representation in the equation.
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pendulum2.png


InvertedPendulum_SystemModeling_eq17002.png


Didn't get this equation which is written for the pendulum part of the cart.
Must be easy but couldn't get it.
 
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It is just "force = mass x acceleration" in the x direction, for the center of mass of the rod. (The length of rod isn't shown on your diagram, but if the equation is correct it is 2l, not l).

The ##\ddot\theta## term is from the tangential acceleration of the rod.

The ##\dot\theta^2## term is the from the radial (centripetal) accleration.
 
AlephZero said:
It is just "force = mass x acceleration" in the x direction, for the center of mass of the rod. (The length of rod isn't shown on your diagram, but if the equation is correct it is 2l, not l).

The ##\ddot\theta## term is from the tangential acceleration of the rod.

The ##\dot\theta^2## term is the from the radial (centripetal) accleration.

It's a little clearer now, but didn't understand the centripetal force component.

Isn't it

df86712e000fe347516b8f39b9490815.png
 
Yes.

You also know ##v = \omega r## and ##\omega = \dot\theta##

The centripetal force is radial (along the length of the rod), but the equation is for the component of the force in the X direction.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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