Find Constraint Equation for Block on Wedge

In summary, the constraint equation for the acceleration of a block on a wedge can be written as $$h - y = (x - X) \tan{\theta}$$ where X is the distance to the edge of the wedge. However, if X is measured to the center of mass of the wedge, the equation becomes $$h - y = (X - x) \tan{\theta}$$ and the second derivatives of X and X_cm must be equal.
  • #1
PFuser1232
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Suppose we want to find the constraint equation relating the acceleration of a block on a wedge to the acceleration of the wedge (see attached images). We can write:
$$h - y = (x - X) \tan{\theta}$$
What if ##X## was measured to the centre of mass of the wedge for instance? Wouldn't the correct equation be:
$$h - y = (X - x) \tan{\theta}$$
 

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  • #2
MohammedRady97 said:
Suppose we want to find the constraint equation relating the acceleration of a block on a wedge to the acceleration of the wedge (see attached images). We can write:
$$h - y = (x - X) \tan{\theta}$$
I see how you got this and it makes sense.

MohammedRady97 said:
What if ##X## was measured to the centre of mass of the wedge for instance? Wouldn't the correct equation be:
$$h - y = (X - x) \tan{\theta}$$
I do not see how you got that. You still need the sides of the same triangle, you've just redefined what X is. So you need to rewrite your first equation using your new X. For example, if ## X_{cm} = X + a##, then your formula would become ##h - y = (x - X_{cm} + a) \tan{\theta}##.
 
  • #3
Doc Al said:
I see how you got this and it makes sense.I do not see how you got that. You still need the sides of the same triangle, you've just redefined what X is. So you need to rewrite your first equation using your new X. For example, if ## X_{cm} = X + a##, then your formula would become ##h - y = (x - X_{cm} + a) \tan{\theta}##.

I just realized the error in my reasoning. This makes sense now. Since ##X## and ##X_{cm}## differ by a constant, their second derivatives should be equal. We chose ##X## to be the distance to the edge of the wedge to simplify the trigonometry required, right?
 
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  • #4
MohammedRady97 said:
We chose XX to be the distance to the edge of the wedge to simplify the trigonometry required, right?
Right.
 
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What is a block on a wedge?

A block on a wedge is a simple mechanical system consisting of a block placed on top of a wedge-shaped object. The block is typically resting on the inclined surface of the wedge, and the two objects are in contact with each other.

What is the purpose of finding the constraint equation for a block on a wedge?

The constraint equation for a block on a wedge is used to describe the motion of the block on the incline of the wedge. It takes into account the angle of the wedge, the mass of the block, and the effects of gravity and friction on the system. This equation is important in understanding the dynamics of the system and predicting its behavior.

How is the constraint equation for a block on a wedge derived?

The constraint equation for a block on a wedge is derived using principles of classical mechanics, specifically the laws of motion and the concept of forces. By analyzing the forces acting on the block and wedge, and taking into account the geometry of the system, the constraint equation can be derived.

What factors can affect the constraint equation for a block on a wedge?

The constraint equation for a block on a wedge is affected by several factors, including the angle of the wedge, the mass of the block, the coefficient of friction between the two objects, and the forces acting on the system such as gravity and applied forces.

How is the constraint equation for a block on a wedge used in real-world applications?

The constraint equation for a block on a wedge is used in various engineering and physics applications, such as designing and analyzing machines and structures that use inclined planes, such as ramps and conveyor belts. It can also be used in understanding the motion of objects on inclined surfaces, such as cars driving up a hill or objects sliding down a slope.

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