# On the rotation of an object accelerated by gravity with a moving base [closed]

How could you describe the rotation/falling of an irregular object whose base/rotational point is being moved with constant acceleration to the left? How much will the angle between itself and the base change over time? Try avoiding using the moment of inertia of it on equations, as I donâ€™t have that information. If I know the pattern between the distances of N cylinders, could I get the moment of inertia? I tried to equalize two torque formulas (distance *F * sin(x) and I*a) but I could not get the moment of inertia as a function of position.

• You can always estimate the mass moment of inertia of an object from the shape and material. In fact, it is required to know this value for any dynamics type of problem that involves rotations. – ja72 Nov 17 '19 at 2:51
• Please show us some effort. Any sketches, free body diagrams, and attempts to the solution would be good. Otherwise, the question might be closed as do my homework type. – ja72 Nov 17 '19 at 2:53
• Sorry about that... I added what I have assessed so far. – 2 False Nov 17 '19 at 11:19
• Thanks for clarifying a bit more, but this raises more questions. It appears you have multiple bodies interacting here and not a single body as portrayed in the title. The concept of MMOI is valid for a single body, so unless you are willing to assume that all bodies move together as a single body, your problem is far more complex now. Anyway, yes you can synthesize MMOI of a complex arrangement of bodies using the parallel axis theorem. – ja72 Nov 17 '19 at 15:24
• Is the contact free-to-slip, with-friction or is it stuck together but allowed to rotate? – ja72 Nov 17 '19 at 19:07

This is a very complex problem to solve especially from someone with a cursory knowledge of dynamics professing from the get-go that no mention of mass moment of inertia is to be made when the very concept of MMOI is crucial in solving such problem.

So I am just going to describe the problem here in hopes of getting the ball rolling, without actually solving this problem. In dealing with dynamics, one must follow very specific steps.

1. Initial Conditions - describe all the important arrangements, dimensions, and assumptions when the system hasn't gone through any movement yet. Typically all allowed degrees of freedom have a zero value at initial conditions.

In this case, the bundle of pipes is assumed to be glued together, and act as a single rigid body. The combined center of mass of the body is offset by $$b$$ outside of the base end and at a height $$h$$ from the bottom. These values remain constant throughout time. If initially, the center of mass is on inside of the end of the base, then $$b$$ would be a negative value. Put an immovable coordinate system where the corner of the base is initially. Note also the contact point A and the center of mass C.

2. Kinematics - describe all degrees of freedom of the system and show them with positive displacements, and assume their values as well as the time derivatives of their values completely describe the positions and velocities of all the parts in the system.

In our case, the base displaces to the right by a prescribed amount $$x = \tfrac{1}{2} a t^2$$, the pipes all tilt together from horizontal by an angle $$\theta$$, and slide away from the contact point A by $$s$$. So the system has three degrees of freedom $$x$$, $$\theta$$ and $$s$$ that vary with time and have zero value initially.

3. Track centers of mass - The center of mass of the bundle C has location, velocity and acceleration described in terms of the degrees of freedom only

• Positions \begin{aligned} x_C & = x - (b+s) \cos\theta - h \sin \theta \\ y_C &= -(b+s) \sin \theta + h \cos \theta \\ \end{aligned} \tag{1}
• Velocities \begin{aligned} \dot{x}_C &= \dot{x} + \dot{\theta} (b+s) \sin \theta - (\dot{s}+h \dot{\theta}) \cos \theta \\ \dot{y}_C & = -\dot{\theta}(b+s) \cos\theta - (\dot{s} + h \dot{\theta}) \sin \theta \end{aligned} \tag{2}
• Accelerations \begin{aligned} \ddot{x}_C & = \ddot{x} + ( \dot{\theta}^2 (b+s) - h \ddot{\theta}-\ddot{s}) \cos \theta + (\ddot{\theta}(b+s) +\dot{\theta} (h \dot{\theta}+2 \dot{s})) \sin \theta \\ \ddot{y}_C & = ( \dot{\theta}^2 (b+s) -h \ddot{\theta}-\ddot{s}) \sin \theta - ( \ddot{\theta} ( b+s) + \dot{\theta} (h \dot{\theta} + 2 \dot{s})) \cos\theta) \end{aligned} \tag{3}

I am sure we have reached a level of complexity at this point far beyond of what you expected, and this is just the beginning.

4. Free Body Diagram - describe all forces acting on each body

In this case, there is the combined weight of $$W$$, the contact normal force $$N$$ and any friction present $$F$$.

5. Contact Conditions - describe how contacts behave. In this case, the contact at A is allowed to slip, which means $$x$$ and $$s$$ are independent of each other, but the frictional force is dependent on the normal force $$F = \mu N \tag{4}$$

If the contact wasn't allowed to slip, then $$F$$ would be independent and $$s$$ would be always zero. At least until contact was lost when $$N<0$$. More about this situation later.

6. Balance of Forces and Torques - describe the equations of motion for each body. Net forces and torques about the center of mass describe the motion of the center of mass as well as the rotation. Use the free body diagram and trigonometry to build these equations taking care of leading signs as positive for counter-clockwise sense and negative otherwise.

\begin{aligned} m \ddot{x}_C & = F \cos \theta - N \sin \theta \\ m \ddot{y}_C & = N \cos \theta + F \sin \theta - W \\ I_C \ddot{\theta} & = (b+s) N + h F \end{aligned} \tag{5}

Here $$m$$ is the combined mass of the pipes and $$I_C$$ the combined mass moment of inertia expressed at the center of mass C. How $$I$$ can be computed for a composite system of bodies is subject to a different question in [Physics.SE] and out of context for this answer.

7. Solution - count all equations and unknowns to see if the system is solvable. The goal is to get the acceleration of the degrees of freedom at each time frame and use a simulation solver to integrate over time.

Here there are 3 equations of motion and 1 contact condition equation. The unknown variables are the degrees of freedom (accelerations) $$\ddot{s}$$ and $$\ddot{\theta}$$ as well as the two contact forces $$F$$ and $$N$$. So four equations with four unknowns.

8. Special Cases - The solution is valid for as long the situation can be described by the aforementioned degrees of freedom. But consider the case where the pipes fly off the base and lose contact. Then the contact condition is no longer valid, and an additional degree of freedom $$y$$ is needed to describe the center of mass of the system.

This happens as soon as $$N<0$$. Then the problem transitions to a free body undergoing projectile motion and rotation under the influence of gravity.

• I'd appreciate a comment on the downvote considering the amount of effort put forth in this answer. – ja72 Nov 18 '19 at 14:34
• I know I am not supposed to do this here but, as I canâ€™t upvote yet, thank you! You have helped me very much! – 2 False Nov 18 '19 at 21:17