Normal Force and Acceleration down the slope

[lazyguy91]

Homework Statement

2. Homework Equations + questions
1) Normal Force
when considering mg as the vertical force,
mg = N cos θ
however, when considering N as the vertical force,
N = mg cos θ
note: θ is the same due to opposite angle

which equation is correct?

2) Acceleration down the slope due to gravity (frictionless)
a = g sin θ
why can't I use the originally given θ?
e.g. sin θ = g/a
a = g/sin θ
why must I use the derived θ instead of the one between the horizontal and the inclined place?

where are my mistakes? Please reply as soon as possible. Thanks

 

[jegues]

I don't know what your trying to do with all those equations and such, just try to understand the normal force.

The normal force is always going to be perpendicular to the plane on which the object lies. (Your drawing depicts this accurately)

We know that since its on an inclined plane its not going to simply just be N = mg(flat plane), its going to be some variation of that because of the angle of the plane. So how do we figure this out?

Well just simply draw a force triangle and solve it. You're going to have mg going straight down as the hypotenuse and the other two sides will be determined based on the angle of your plane. ( mgcos(theta) and mgsin(theta) ) [ You should find that mgcos(theta) is in the opposite direction of the normal force ]

As for part 2), you can also answer this once you've drawn your force triangle, mgsin(theta) should be acting parallel to the plane (depending on how you've defined your x and y axis).

 

[kerimek]

I appreciate your curiosity and attention to detail in your homework. Let me address your questions one by one.

1) Normal Force: Both equations are correct, as they are simply different ways of expressing the same relationship between the normal force and the weight of an object. In the first equation, the normal force is calculated using the weight (mg) and the angle between the weight and the normal force (cos θ). In the second equation, the weight is calculated using the normal force and the same angle (cos θ). Both equations give the same result, so you can use whichever one you find more convenient in a given situation.

2) Acceleration down the slope: The reason you cannot use the originally given θ is because it is the angle between the inclined plane and the horizontal, not the angle between the slope and the vertical. In order to calculate the acceleration down the slope, you need to use the angle between the slope and the vertical, which is the same as the angle between the weight and the normal force (since they are perpendicular). This is why we use the derived θ, which is the same as the one between the weight and the normal force.

I hope this helps clarify any confusion you may have had. Keep up the good work in your studies!