Why does a=g*sin(theta) on an inclined plane

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Discussion Overview

The discussion revolves around the equation for acceleration on an inclined plane, specifically examining why the acceleration is expressed as a=g*sin(θ) and exploring alternative formulations such as a=g/sin(θ). Participants analyze the geometric relationships involved and the implications of different angles in their reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asserts that the acceleration on an inclined plane is a=g*sin(θ) but questions whether it could also be expressed as a=g/sin(θ), leading to confusion about the relationships between the variables.
  • Another participant clarifies that when resolving gravitational acceleration into components, g represents the hypotenuse of a triangle formed by the forces acting on the object.
  • A subsequent reply points out that the two formulas proposed by the first participant use different angles, suggesting that both could be correct under specific conditions but should be expressed with proper angle definitions.
  • Another participant challenges the validity of both formulas, arguing that they cannot simultaneously represent the same acceleration and concludes that the correct expression remains a=g*sin(θ).
  • One participant suggests that the confusion may arise from mixing different triangles and encourages drawing a diagram to clarify the forces and accelerations involved.
  • A later post offers a resource link for further understanding of forces acting on objects on inclined planes.
  • Another participant provides a general principle about vector resolution, stating that a resolved vector is always the hypotenuse of the corresponding triangle.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the alternative formulations for acceleration on an inclined plane. There is no consensus on the correctness of the proposed equations, and the discussion remains unresolved.

Contextual Notes

Participants highlight potential confusion stemming from the use of different angles in their formulations and the need for clear geometric representation. The discussion does not resolve the mathematical relationships or assumptions underlying the different perspectives.

Jow
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I understand the derivation that on an inclined plane of angle θ, the acceleration of the object on the plane, parallel to the plane, is a=g*sinθ. However, I was just thinking about it, and should it not be a=g/sinθ ?

I got this because sinθ = g/a. θ is the angle between the ramp and the earth, g is the acceleration straight downwards (the opposite side from θ) and a is the hypotenuse of the triangle.
Rearranging, a=g/sinθ.

I feel like something must be wrong, but I can't see what it is.
 
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When g is resolved into components, it represents the hypotenuse of the triangle.
 
The problem is the two formulas you have created (a=g/sinx and a=gsinx), use different angles. The first formula you created uses the angle between the ramp and the earth. The second one uses the angle between the vertical axis and the direction of acceleration. So both formulas are correct but they should be written as a=g/sinx and a=gsiny where y=90-x
 
TysonM8 said:
The problem is the two formulas you have created (a=g/sinx and a=gsinx), use different angles. The first formula you created uses the angle between the ramp and the earth. The second one uses the angle between the vertical axis and the direction of acceleration. So both formulas are correct but they should be written as a=g/sinx and a=gsiny where y=90-x
This is clearly incorrect. If you substitute y = 90-x into the second equation, you get a=g cosx. The acceleration cannot simultaneously be equal to g/sinx and gcos x, since cosx≠1/sinx. In fact, both these equations are wrong. The acceleration (in the absence of friction) is a=gsinx.

Chet
 
Jow said:
I got this because sinθ = g/a. θ is the angle between the ramp and the earth, g is the acceleration straight downwards (the opposite side from θ) and a is the hypotenuse of the triangle.
Rearranging, a=g/sinθ.

I feel like something must be wrong, but I can't see what it is.
g is not opposite to the angle theta.
It may be that you are mixing triangles. The triangle made by the components of the accelerations with the triangle made by the inclined planer itself.
Draw a diagram of the forces (or accelerations) and you will see.
 
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Here's a trick: Whenever a vector is resolved into orthogonal components, the vector is always the hypotenuse of the triangle.
 

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