摩擦角(angle of friction)和休止角(angle of repose)的區別

摩擦角和休止角是機械工程中出現的兩個角。休止角在土力學中也有應用。摩擦角和休止角的主要區別在於，摩擦角定義為物體剛開始運動時，法向反作用力與法向反作用力和摩擦力的合力之間的夾角，而休止角定義為使物體從斜面上滑下的斜面的最小角度。我們稍後將看到這兩個定義是等價的。...

主要區別-摩擦角與休止角

摩擦角和休止角是機械工程中出現的兩個角。休止角在土力學中也有應用。摩擦角和休止角的主要區別在於，摩擦角定義為物體剛開始運動時，法向反作用力與法向反作用力和摩擦力的合力之間的夾角，而休止角定義為使物體從斜面上滑下的斜面的最小角度。我們稍後將看到這兩個定義是等價的。

什麼是摩擦角(angle of friction)？

摩擦角是指法向反作用力與法向反作用力和摩擦力的合力之間的夾角。讓我們首先探討這個定義，並嘗試用公式來表示摩擦角。

Defining angle of friction

The diagram above shows a block being pulled against a rough surface. If the force is **all, the static friction ( ) between the surfaces would prevent the block from moving. If you slowly increase the pull, would also increase, balancing the horizontal forces and still preventing the block from moving. However, at one point, the block would just begin to move. At this point, the static friction has reached the upper limit, and we can describe this maximum friction as where is the coefficient of friction and is the normal reaction force.

The vector diagram on the right shows the sum of and . From the definition we had above, is the angle of friction. Using trigonometry, we can express this angle as . i.e., the angle of friction between two surfaces is equal to the inverse tangent of the coefficient of friction between those two surfaces.

什麼是休止角(angle of repose)？

休止角是一個錶面可以從水平面傾斜的最大角度，這樣它上面的物體就可以停留在錶面上而不向下滑動。再一次，讓我們先看看形勢圖。

Defining Angle of Repose

When the object is just about to move, the size of friction is given by . The object is also in equilibrium (the object is about to move, but it is not moving yet!) so taking the forces acting along the plane, we can say,

垂直於平面的分力，

現在，我們取這些表示式的比率：

Using the trigonometric identity , and cancelling out common factors. we get . So, we get:

For angle of friction , we had and we see that this is the same expression that we get for angle of repose as well. Therefore, angle of friction is equal to angle of repose.

在土力學中，休止角是指一堆土的最淺角度，它使土顆粒開始下落。螞蟻在建造陷阱時會利用這個角度（見下文）：

Antli*** make these sand traps to catch ants and other **all insects. The sides of these traps are at the angle of repose. When ants wander into the pit, the disturbance causes the sand to begin rolling down, making it difficult for the ant to escape. Then the antlion, which had been hiding at the bottom of the trap, comes out and catches the ant.