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6.2: Slipping vs Tipping

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    50595
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    Imagine a box sitting on a rough surface as shown in the figure below. Now imagine that we start pushing on the side of the box. Initially the friction force will resist the pushing force and the box will sit still. However, as we increase the force pushing the box one of two things will occur.

    1. The pushing force will exceed the maximum static friction force and the box will begin to slide across the surface (slipping).
    2. Or, the pushing force and the friction force will create a strong enough couple that the box will rotate and fall on its side (tipping).
    Free body diagram showing a tall box sitting on a flat surface as a pushing force is applied to its left side. Arrows indicate how this box can go through one of two experiences: the pushing force causes it to slide towards the right, or the pushing force creates a net moment that tips the box towards the right.
    Figure \(\PageIndex{1}\): As the pushing force increases on the box, it will either begin to slide along surface (slipping) or it will begin to rotate (tipping).

    When we look at cases where either slipping or tipping may occur, we are usually interested in finding which of the two options will occur first. To determine this, we usually determine both the pushing force necessary to make the body slide and the pushing force necessary to make the body tip over. Whichever option requires less force is the option that will occur first.

    Determining the Force Required to Make an Object "Slip":

    A body will slide across a surface if the pushing force exceeds the maximum static friction force that can exist between the two surfaces in contact. As in all dry friction problems, this limit to the friction force is equal to the static coefficient of friction times the normal force between the body. If the pushing force exceeds this value then the body will slip.

    Free body diagram of a tall box sitting on a flat surface, with a rightwards push force applied to its left side. A combination of words and equations states that the maximum possible friction force is equal in magnitude to the coefficient of static friction times the magnitude of the normal force, and that the box will begin to slide if the magnitude of the pushing force is greater than the magnitude of the max possible static friction force.
    Figure \(\PageIndex{2}\): If the pushing force exceeds the maximum static friction force \((\mu_s * F_N)\) then the body will begin to slide.

    Determining the Force Required to Make an Object "Tip":

    The normal forces that support bodies are distributed forces. These forces will not only prevent the body from accelerating into the ground due to gravitational forces, but they can also redistribute themselves to prevent a body from rotating when forces cause a moment to act on the body. This redistribution will result in the equivalent point load for the normal force shifting to one side or the other. A body will tip over when the normal force can no longer redistribute itself any further to resist the moment exerted by other forces (such as the pushing force and the friction force).

    Free body diagram of a tall box sitting on a flat surface, experiencing a pushing force of increasing magnitude on its left side. As the magnitude of the push force increases, the arrows that represent the distributed normal force on the box shifts from evenly distributed across the entire box bottom (at zero push force) to being concentrated in the right end of the box bottom. The arrow representing the equivalent point load for the normal force shifts increasingly rightwards on the box bottom. The leftwards-pointing arrow that represents the friction force increases in magnitude.
    Figure \(\PageIndex{2}\): At rest (A) the normal force is a uniformly distributed force on the bottom of the body. As a pushing force is applied (B) the distributed normal force is redistributed, moving the equivalent point load to the right. This creates a couple between the gravity force and the normal force that will counter the couple exerted by the pushing force and the friction force. If the pushing force becomes large enough (C), the couple exerted by the gravitational force and the normal force will be unable to counter the couple exerted by the pushing force and the friction force.

    The easiest way to think about the shifting normal force and tipping is to imagine the equivalent point load of the distributed normal force. As we push or pull on the body, the normal force will shift to the left or right. This normal force and the gravitational force create a couple that exerts a moment. This moment will be countering the moment exerted by the couple formed by the pushing force and the friction force.

    Because the normal force is the direct result of physical contact, we cannot shift the normal force beyond the surfaces in contact (i.e., the edge of the box). If countering the moment exerted by the pushing force and the friction force requires shifting the normal force beyond the edge of the box, then the normal force and the gravity force will not be able to counter the moment and as a result the box will begin to rotate (i.e., tip over).

    Free body diagram of a tall box on a flat surface, with a pushing force applied at its left edge. The horizontal distance between the box's center of mass and its right edge (where the vector normal force is placed) is labeled x, and the vertical distance from the surface to the location where the push is applied is labeled h. A combination of words and equations states that if the moment created by the push and friction forces (each calculated by multiplying the magnitude of the push force by h) exceeds the moment created by the gravitational and normal forces (each calculated by multiplying the magnitude of the gravitational force by x), the box will begin to tip to the right.
    Figure \(\PageIndex{3}\): The body will tip when the moment exerted by the pushing and friction forces exceeds the moment exerted by the gravity and normal forces. For impending motion, the normal force will be acting at the very edge of the body.
    Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/M2OZOkgVRBQ.

    Example \(\PageIndex{1}\)

    The box shown below is pushed as shown. If we keep increasing the pushing force, will the box first begin to slide or will it tip over?

    A 60-kg box 3 meters wide and 4 meters tall on a flat surface experiences a pushing force applied on its left side, at the point 3 meters above the surface. Coefficient of static friction between the box and surface is given as 0.62.
    Figure \(\PageIndex{4}\): problem diagram for Example \(\PageIndex{1}\). A box on a flat surface, with a coefficient of static friction of 0.62 between the two surfaces, experiences a pushing force at a point on its left side.
    Solution
    Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/TeYgkfd4rTA.

    Example \(\PageIndex{2}\)

    What is the maximum value of \(d\) that will allow the box to slide along the surface before tipping?

    A 60-kg box 3 meters wide and 4 meters tall on a flat surfaces experiences a pushing force on its left side. The distance between the flat surface and the point on the box's side where the push is applied is labeled as d. The coefficient of static friction between the box and the surface is given as 0.62.
    Figure \(\PageIndex{5}\): problem diagram for Example \(\PageIndex{2}\). A box on a flat surface, with a coefficient of static fricton of 0.62 between the two surfaces, experiences a pushing force at a point of unknown height (\(d\) on its left side.
    Solution
    Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/STDWwJsng6k.

    This page titled 6.2: Slipping vs Tipping is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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