Proof needed to show that an angle remains constant along a line

In summary: I guess one could calculate it by trigonometry, but it would require more information than the OP has given.In summary, the conversation discusses a problem with a knife sharpener where the angle of the sharpening media needs to remain constant along the length of the blade. The sharpener consists of a clamp, base rod, pivot point, and guide rod. The diagrams and images provided show that the sharpening angle remains constant due to the fact that the arc described by the guide rod lies in a plane, and the straight line of the blade also lies in that plane. This allows for a consistent angle to be maintained despite the curve of the knife's edge.
  • #1
clayallison
5
0
This is my problem:

On my knife sharpener, there is a Clamp that holds the blade vertically, edge up. At the base of the clamp and perpendicular to the blade, there is a Base Rod with a Pivot Point that slides along the Base Rod to establish an angle relative to the blade. The third side of the triangle is created by the sharpening media sliding on a Guide Rod, anchored at the Pivot Point. T triangle could be described as follows: Side A = height of the Clamp + height of blade above the clamp. Side B = Distance along Base Rod between the Pivot Point and Side A. We can calculate the angle AC knowing that angle AB is a right angle and knowing the length of Side A and Side B. Where I run into problems is proving that the angle of the sharpening media will remain constant along the length of the blade. The diagrams below help illustrate the issue.

[PLAIN][PLAIN]http://www.wickededgeusa.com/graphics/Constant [Broken][/URL][/URL] Angle- pt 1.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/graphics/Constant [Broken][/URL][/URL] Angle- pt 2.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/graphics/Constant [Broken][/URL][/URL] Angle- pt 3.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/graphics/Constant [Broken][/URL][/URL] Angle- pt 4.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/graphics/Constant [Broken][/URL][/URL] Angle- pt 5.png[/PLAIN] [Broken]

In the diagrams, the Clamp is 6" and the distance of the Pivot Point from the clamp is 6". The height of the knife above the clamp is 1". I can see that the sharpening medium stays in the same plane relative to the knife, I just can't remember a way to prove it.

Any help would be most appreciated.
 
Last edited by a moderator:
Mathematics news on Phys.org
  • #2
Another couple of pictures to illustrate the problem:

[PLAIN]http://www.wickededgeusa.com/photos/angle-change-at-0-inches-sm.jpg [Broken]
[PLAIN]http://www.wickededgeusa.com/photos/angle-change-at-5-inches-sm.jpg [Broken]

If it helps to see the sharpener in action, you can see it here: http://www.wickededgeusa.com" [Broken]
***I'm not trying to promote the product just supplying the link if seeing the sharpener will help with the math.
 
Last edited by a moderator:
  • #3
Since there are no replies yet, I'm thinking of offering a prize. Any interest?
 
  • #4
It might help for clarity's sake if you labeled your diagram with what side A is, what the pivot point is etc. It's not really clear to me what it is you're trying to solve (perhaps because of my unfamiliarity with knife sharpeners)
 
  • #5
The 40.601 degree angle with identical (6.000) leg lengths has me baffled.
 
  • #6
Office_Shredder said:
It might help for clarity's sake if you labeled your diagram with what side A is, what the pivot point is etc. It's not really clear to me what it is you're trying to solve (perhaps because of my unfamiliarity with knife sharpeners)

Thank you, sorry for the lack of clarity. I've been out of school and away from maths so long that I'm sure my question seems a mess. Here are a few more images, labeled a little more clearly.

[PLAIN][PLAIN]http://www.wickededgeusa.com/blog/images/constant [Broken][/URL][/URL] angle 1.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/blog/images/constant [Broken][/URL][/URL] angle 2.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/blog/images/constant [Broken][/URL][/URL] angle 5.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/blog/images/constant [Broken][/URL][/URL] angle 4.png[/PLAIN] [Broken]
[PLAIN][PLAIN]http://www.wickededgeusa.com/blog/images/constant [Broken][/URL][/URL] angle 3.png[/PLAIN] [Broken]

The stone is shown moving in relation to the blade, the plane staying constant, keeping the sharpening angle constant.
 
Last edited by a moderator:
  • #7
Chronos said:
The 40.601 degree angle with identical (6.000) leg lengths has me baffled.

That was 6" for the height of the clamp + 1" for the height of the knife, making the overall length of the side 7".

Again, sorry for the messy first posting. I had forgotten to pull out the dimensions since they were arbitrary.
 
  • #8
Simple. The arc described by your "guide rod" lies in a plane. The "pivot point" lies in that plane and the straight line, which in this case is the edge of your knife blade, lies in that plane as well. Since planes are flat, at least in Newtonian physics they are, last time I checked, the angle between the blade and the plane must remain constant. I hope you are now unbaffled!
 
  • #9
SingleNote said:
the straight line, which in this case is the edge of your knife blade
Where does the OP say the edge of the knife is straight? Isn't that exactly the problem? Most knives have curved edges. Unfortunately I can't seem to view the diagrams, but the only way I can imagine that the sharpener could maintain a constant angle to the plane of the blade is if it has some fancy device for detecting the curve of the knife and adjusting the distance to the pivot accordingly. And I very much doubt it has that.
Rather, I should think it depends on the fact that the distance from pivot to knife edge is large compared with the variation in blade height, so although the angle changes it won't be by much.
 

1. What does it mean for an angle to remain constant along a line?

When an angle remains constant along a line, it means that the angle formed by two intersecting lines or line segments remains the same regardless of any changes in the length or position of the lines.

2. Why is it important to prove that an angle remains constant along a line?

Proving that an angle remains constant along a line is important because it provides evidence that the lines are parallel or perpendicular. It also helps in solving geometric problems and making accurate calculations.

3. What are the different methods used to prove that an angle remains constant along a line?

There are various methods used to prove that an angle remains constant along a line, including the use of congruent triangles, alternate interior angles theorem, vertical angles theorem, and the sum of angles in a triangle theorem.

4. Can an angle remain constant along a curved line?

No, an angle cannot remain constant along a curved line. This is because the curvature of the line results in changing angles formed by intersecting lines or line segments.

5. How can we apply the concept of an angle remaining constant along a line in real-life situations?

The concept of an angle remaining constant along a line is commonly used in fields such as engineering, architecture, and navigation. It helps in designing structures, creating accurate maps, and determining the direction and distance between two points.

Similar threads

Replies
2
Views
2K
  • Classical Physics
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
6
Views
3K
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Special and General Relativity
Replies
6
Views
3K
  • Mechanics
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
16
Views
3K
Replies
3
Views
13K
  • Special and General Relativity
2
Replies
40
Views
2K
Back
Top