在學習如何求向心加速度之前，讓我們先看看什麼是向心加速度。我們將從向心加速度的定義開始。向心加速度是物體以恆定速度沿圓周運動時切向速度的變化率。向心加速度總是指向圓形路徑的中心，因此被稱為向心加速度，在拉丁語中是“尋找中心”的意思。在本文中，我們將研究如何求物體的向心加速度。

如何推導向心加速度表示式

勻速圓周運動的物體正在加速。這是因為加速度涉及到速度的變化。由於速度是一個向量，它要麼隨速度大小的變化而變化，要麼隨速度方向的變化而變化。即使我們例子中的物體保持相同的速度，速度的方向也在改變，因此物體在加速。

To find this acceleration, we c***ider the motion of the object during a very short time . On the diagram below, the object has moved through an angle during the period .

How to find Centripetal Acceleration – Deriving Centripetal Acceleration

The velocity change during this time is given by . This is shown by the grey arrows in the vector triangle drawn on top right. With the blue arrows, we have placed and in a different arrangement to get the same . The reason why I have drawn the second diagram the blue vectors is because this is how the vectors are actually directed at, at the two different times c***idered on the diagram on left. Since the velocity vectors are always at a tangent to the circle, it then follows that the angle between the vectors and is also .

Since we’re c***idering a very **all interval of time, the distance traveled by the object during time is almost a straight line. This distance, along with the radii, is shown on the red triangle.

The blue triangle of velocity vectors and the red triangle of lengths are similar triangles. We already saw that they both contain the same angle . Next, we realize that they are both isosceles triangles. On the red triangle, the sides attached to the angle are both , the size of the radius.

On the blue triangle, the lengths of the sides attached to the angle represent the magnitudes of velocities and . Since the object is traveling at c***tant velocity, . This means the blue triangle is isoceles as well, and so the blue and red triangles are indeed similar.

If we take , then we can use the similarity of triangles to say,

.

The magnitude of acceleration can be given by . Then, we can write,

. Since ,

Since we found when we looked at finding angular speed, we can also write this acceleration as

We can also show that the direction of this acceleration, which is in the direction of , is directed towards the centre of the circle. C***equently, this acceleration is called centripetal acceleration because it is always pointing to the centre of the circular path.

由於圓周運動物體的速度總是與圓相切，這意味著加速度總是垂直於物體運動的方向。這也是為什麼這個加速度不能改變物體速度大小的原因。

如何求向心加速度

現在我們有了方程，我們將看到如何在涉及圓周運動的不同情況下找到向心加速度。

例1

地球的半徑是6400公里。求一個人站在地面上，由於地球繞地軸自轉而產生的向心加速度。

How to find Centripetal Acceleration – Example 1

例2

騎腳踏車的人騎著一輛半徑為0.33米的腳踏車。如果車輪以恆定速度轉動，則在附著在腳踏車輪胎上的砂粒上找到向心加速度，該砂粒以4.1 m s-1的速度移動。

How to find Centripetal Acceleration – Example 2

根據牛頓第二定律，向心加速度必須伴隨著作用在圓軌跡中心的合力。這個力叫做向心力。

如何計算向心力