Uniform circular motion angle relationship

[syllll_213]

Homework Statement

When reading the textbook, I was told the below marked angles are equal to each other, but I am not sure why because I am only familiar with the complementary and supplementary relationships of angles found in parallel line in math, but unsure how we derive this conclusion. For this free body diagram, I am assuming the same rule is used if we draw a line from the negative end of the x-axis to the force exerted by the weight?

Relevant Equations

angle = angle

 

[Lnewqban]

Note that the lines in question remain perpendicular to each other, rather than perpendicular, for any value of the angles that are equal.

Please, see:
https://www.mathsisfun.com/geometry/parallel-lines.html

 

[syllll_213]

Hi, I am struggling to understand your response here. Should I focus on perpendicular lines or parallel? The link direct me to parallel line rules but I struggle to find any lines that have parallel relationship in the diagram.

Note that the lines in question remain perpendicular to each other, rather than perpendicular, for any value of the angles that are equal.

Please, see:
https://www.mathsisfun.com/geometry/parallel-lines.html

 

[anuttarasammyak]

Here is the geometry.

 

[Lnewqban]

Hi, I am struggling to understand your response here. Should I focus on perpendicular lines or parallel? The link direct me to parallel line rules but I struggle to find any lines that have parallel relationship in the diagram.

You can also use the concept of similar triangles (which happen to be right-angled triangles in this case).

https://www.mathsisfun.com/geometry/triangles-similar-finding.html

 

[erobz]

You can also make this construction:

 

[kuruman]

Hi, I am struggling to understand your response here. Should I focus on perpendicular lines or parallel? The link direct me to parallel line rules but I struggle to find any lines that have parallel relationship in the diagram.

A useful theorem in geometry states that "Two angles with mutually perpendicular sides are equal." The free body diagram of a block on an incline on the right shows two such angles $\theta.$ The black side of one angle is perpendicular to red side of the other. If you decrease the angle of the incline to zero, the red line that is perpendicular to the incline will be along the vertical and the other angle $\theta$ will also be zero.