Maximum angle of inclined plane before falling off the plane

In summary, the conversation revolves around determining the maximum angle between the ground and an inclined plane that an object of given mass, height, and coefficient of friction can remain in contact with the plane without sliding or toppling. It is suggested that, for sliding, the tangent of the angle should not exceed the coefficient of static friction, while for toppling, the vertical line through the center of mass should not go outside the base.
  • #1
Sinnaro
2
0
I've been thinking about this for a model I'm devising. Assuming that you have an object of mass m, height h, coefficient of friction u, how large can you make the angle between the ground and the inclined plane. Otherwise, at what angle does the torque from the center of mass of the object overcome the force that is keeping the object on the inclined plane?
 
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  • #2
Welcome to PF!

If the incline is completely flat and the coefficient of friction is for static friction, then you can find the largest angle for which the mass won't slide by noting that the maximum force along the slide is usually modeled simply as the normal force times the coefficient, and you can equate that with the force from gravity down the incline, that is,

[tex] \mu F_n = \mu mg \cos(\alpha) = F_g = mg \sin(\alpha) \ \Rightarrow \ \tan(\alpha) = \mu [/tex]

where [itex]\alpha[/itex] is the angle and [itex]\mu[/itex] the coefficient of static friction.
 
  • #3
Welcome to PF!

Hi Sinnaro! Welcome to PF! :wink:

I'm not sure whether you're talking about sliding (which as Filip Larsen :smile: says depends on whether the tangent exceeds the coefficient of static friction), or toppling.

If it's toppling, then all that matters is whether a vertical line through the centre of mass goes outside the base. :wink:
 
  • #4
To clarify: it is assumed that the mass is sliding down the plane. I'm looking for the angle (as the angle approaches 90 degrees) for which the mass will no longer be in contact with the plane (falls off).

Picture of what I'm talking about. Assume that the mass looks similar to my drawing (tall vertical height with wide base):

http://i.imgur.com/kIwSt.png
 
  • #5


The maximum angle of an inclined plane before an object falls off can be calculated using the formula tanθ = u, where θ is the angle of the incline and u is the coefficient of friction. This formula assumes that the object is in equilibrium and the force of gravity is balanced by the normal force from the inclined plane.

To determine the maximum angle, we can use the concept of torque. Torque is the force that causes an object to rotate, and it is calculated by multiplying the force applied by the distance from the pivot point. In this case, the pivot point is the point where the object is in contact with the inclined plane.

The torque from the center of mass of the object is equal to the weight of the object multiplied by the perpendicular distance from the pivot point to the center of mass. This torque must be balanced by the frictional force acting on the object, which is equal to the coefficient of friction multiplied by the normal force from the inclined plane.

If we set these two torques equal to each other and solve for the angle, we get the maximum angle of the inclined plane before the object falls off, which is given by the formula θ = tan^-1(u). This means that the maximum angle will depend on the coefficient of friction, with a higher coefficient allowing for a steeper incline.

In summary, the maximum angle of an inclined plane before an object falls off is determined by the coefficient of friction between the object and the inclined plane. This can be calculated using the formula θ = tan^-1(u), where u is the coefficient of friction. This concept can be applied to various situations, such as designing ramps or determining the stability of objects on an incline.
 

1. What is the maximum angle of an inclined plane before an object falls off?

The maximum angle of an inclined plane before an object falls off is known as the critical angle. This angle is dependent on the coefficient of friction between the object and the inclined plane, as well as the weight and shape of the object itself.

2. How is the maximum angle of an inclined plane calculated?

The maximum angle of an inclined plane can be calculated using the formula tanθ = μ, where θ is the angle of inclination and μ is the coefficient of friction. This equation assumes that the object is on the verge of slipping, known as the critical state.

3. What factors affect the maximum angle of an inclined plane?

The maximum angle of an inclined plane is affected by the coefficient of friction, the weight and shape of the object, and the surface texture of both the object and the inclined plane. The angle of inclination also plays a significant role in determining the maximum angle.

4. Can the maximum angle of an inclined plane be greater than 90 degrees?

No, the maximum angle of an inclined plane cannot be greater than 90 degrees. This is because an object on an inclined plane with an angle greater than 90 degrees would be facing downwards, and the force of gravity would pull the object off the plane.

5. How does the maximum angle of an inclined plane affect the stability of an object?

The maximum angle of an inclined plane is directly related to the stability of an object. If the angle of inclination is too steep, the object is more likely to slip or fall off the plane. If the angle is too shallow, the object may not have enough friction to remain in place. The maximum angle of an inclined plane must be carefully considered to ensure the stability of the object.

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