Combined Rotation and Translation of a Rectangular Plate

In summary, you need to give the object enough linear velocity so that it will rotate around its corner. However, this can be done in any direction, and it doesn't matter how much initial velocity is required.
  • #1
xkcda
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Homework Statement
A rectangular plate of mass 10 kg is at rest.It has a length of 4 meter and width of 3 meter.What is the minimum velocity required to make it rotating freely?
Relevant Equations
##I= \frac {M(l^2+b^2)} {12}##
##E=mgh##
##E=\frac {MV^2} {2}##
I tried to solve it using the work-energy theorem.The work done to make it stand on its one vertex should be equal to the change in its kinetic energy.
WhatsApp Image 2023-06-16 at 11.06.09 AM.jpeg

I am confused what will be the value of radius here? I have seen formula of kinetic energy for rolling of circular objects.Can anyone please derive a formula for kinetic energy of a rolling rectangular plate?
 
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  • #2
xkcda said:
What is the minimum velocity required to make it rotating freely?
I have no idea what that means.
Judging from your diagram and working, the idea is to give it enough angular velocity from a position where it is lying on its long edge that it will turn right over onto its short edge. Is that it?
 
  • #3
haruspex said:
I have no idea what that means.
Judging from your diagram and working, the idea is to give it enough angular velocity from a position where it is lying on its long edge that it will turn right over onto its short edge. Is that it?
enough linear velocity
 
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  • #4
xkcda said:
enough linear velocity
No amount of linear velocity will change the orientation of the object or make it rotate freely. Can you post the exact wording of your problem?
 
  • #5
Nugatory said:
No amount of linear velocity will change the orientation of the object or make it rotate freely. Can you post the exact wording of your problem?
The term I use is tangential velocity.Rotational or circular motion may require a minimal linear velocity.For instance, the bare minimum speed necessary to finish a loop.
Note: I am unable to provide the exact wording because this problem is written in my mother tongue.
 
  • #6
I think the only interpretation is to suppose that the front edge hits a small protuberance elastically. Then the question makes sense and the problem can be solved simply using conservation of energy.
For the minimum speed the object will rise to a balance point with no motion so $$v^2 =2gh$$ at h=0.5m (the CM elevation). Only higher speeds require moment of inertia explicitly.
 
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  • #7
xkcda said:
Homework Statement: A rectangular plate of mass 10 kg is at rest.It has a length of 4 meter and width of 3 meter.What is the minimum velocity required to make it rotating freely?
I would do the calculation in reverse.
Balance the body over its corner, and let it fall over.
The tangential velocity with which the rotating corner is hitting the ground should be the minimum value that you are looking for.
 
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  • #10
xkcda said:
I can translate better than google:smile:
In terms of your algebra in post #1, the appropriate velocity is the initial instantaneous velocity of the mass centre of the plate. Note that this is not vertical.
Correspondingly, the rotation is about the fixed corner, so the radius is the distance to there.
But whether that is the velocity the original question asks for is unclear. A Google translation might help even if, overall, it is inferior to your own.
For one thing, it might produce correct English, which yours is not.
 
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Related to Combined Rotation and Translation of a Rectangular Plate

What is combined rotation and translation of a rectangular plate?

Combined rotation and translation of a rectangular plate refer to the simultaneous movement of the plate in both translational motion (moving in a straight line) and rotational motion (spinning around an axis). This type of motion is common in various engineering and physics applications, such as machinery and robotics.

How do you describe the motion mathematically?

The motion can be described using a combination of linear kinematics for translation and angular kinematics for rotation. The position of any point on the plate can be expressed as a vector sum of its translational displacement and its rotational displacement around the center of mass. Mathematically, this can be represented as r(t) = R(t) + r'(t), where R(t) is the translational component and r'(t) is the rotational component.

What are the practical applications of studying combined rotation and translation?

Understanding combined rotation and translation is crucial in fields such as mechanical engineering, robotics, aerospace, and biomechanics. For example, it helps in the design of robotic arms, the analysis of vehicle dynamics, the stability of satellites, and the study of human joint movements.

What forces and torques affect the combined motion of a rectangular plate?

The combined motion is influenced by external forces and torques acting on the plate. Forces affect the translational motion according to Newton's second law, F = ma, where F is the force, m is the mass, and a is the acceleration. Torques affect the rotational motion according to the rotational analog of Newton's second law, τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

How do you simulate combined rotation and translation of a rectangular plate?

Simulation of this combined motion can be performed using computational tools and software such as MATLAB, Simulink, or specialized physics engines. These tools allow the creation of mathematical models that incorporate the equations of motion, initial conditions, and external forces/torques. Numerical methods, such as the Runge-Kutta method, can be used to solve these equations and predict the plate's motion over time.

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