Simple inclined plane reaction question, I need a debate solved

[TyronLab]

Homework Statement

Two bodies are acting on one another on an inclined surface. Assuming a constant vertical input force and the lower body being vertically fixed in place on the right hand side, what will the angle of the inclined surface affect. In essence, will the vertical component of the reaction force on the supporting body's inclined face be constant with the angular reaction force increasing as the inclined angle theta increases, or will the reaction force be constant and its vertical component decrease as the angle theta increases?

The Attempt at a Solution

The two proposed solutions are mentioned above. We've got a debate running in our office about which one is correct, and I would like a solution if possible.

Thanks!

Oh and this is my first post.

 

[PhanthomJay]

Sorry, you both lose.

Your diagram shows a reaction force perpendicular to the incline. It neglects to show the friction reaction force 'nmj acts parallel to the incline.

 

[TyronLab]

That diagram is just a simplification. The actual product is a cylinder with a chamfer cut out of it on the outer edge, and the supported body is a plate with a hole in the centre with a corresponding angle chamfered out of it.

I've uploaded a new attachment which represents a section view of the part we're actually concerned with.

So friction in this case isn't important, as the upper body is supported by the lower one without friction.

I hope this diagram is more explanatory.

 

[PhanthomJay]

That diagram is just a simplification. The actual product is a cylinder with a chamfer cut out of it on the outer edge, and the supported body is a plate with a hole in the centre with a corresponding angle chamfered out of it.

I've uploaded a new attachment which represents a section view of the part we're actually concerned with.

So friction in this case isn't important, as the upper body is supported by the lower one without friction.

I hope this diagram is more explanatory.

Oh OK, I get it. From equilibrium considerations in the vertical direction, the vertical component of the reaction force on the supporting body's inclined face will be constant, and the normal reaction force perpendicular to the incline will be increasing as the inclined angle theta increases. As the inclined angle decreases, the vertical component of the reaction force still says the same, but the normal reaction force then decreases.

So as I see it, whoever chose option 1 wins the debate.

 

[rezeew]

I would approach this problem by first understanding the basic principles of inclined planes. An inclined plane is a simple machine that reduces the amount of force needed to lift an object by increasing the distance over which the force is applied. In this case, the input force is acting vertically and the inclined surface is acting as the ramp, reducing the force needed to lift the lower body.

Now, let's consider the two proposed solutions. The first solution suggests that the vertical component of the reaction force on the supporting body's inclined face will be constant with the angular reaction force increasing as the inclined angle theta increases. This means that as the angle increases, the reaction force will be distributed more horizontally, resulting in a smaller vertical component. The second solution proposes that the reaction force will be constant and its vertical component will decrease as the angle theta increases. This means that as the angle increases, the reaction force will remain the same, but its vertical component will decrease due to the increased horizontal component.

To determine which solution is correct, we need to look at the forces acting on the lower body. The only external force acting on the body is the input force, which is constant. The reaction force is an internal force, which means it is dependent on the angle of the inclined surface. As the angle increases, the reaction force will be distributed more horizontally, resulting in a smaller vertical component. This supports the first solution.

Furthermore, we can use trigonometry to calculate the vertical and horizontal components of the reaction force. As the angle increases, the vertical component decreases while the horizontal component increases. This supports the first solution as well.

In conclusion, based on the principles of inclined planes and the analysis of the forces acting on the lower body, the first solution is correct. As the angle of the inclined surface increases, the vertical component of the reaction force on the supporting body's inclined face will decrease while the angular reaction force will increase. This is due to the distribution of the reaction force becoming more horizontal as the angle increases. I hope this helps to settle the debate in your office.