在学习如何求向心加速度之前，让我们先看看什么是向心加速度。我们将从向心加速度的定义开始。向心加速度是物体以恒定速度沿圆周运动时切向速度的变化率。向心加速度总是指向圆形路径的中心，因此被称为向心加速度，在拉丁语中是“寻找中心”的意思。在本文中，我们将研究如何求物体的向心加速度。

如何推导向心加速度表达式

匀速圆周运动的物体正在加速。这是因为加速度涉及到速度的变化。由于速度是一个矢量，它要么随速度大小的变化而变化，要么随速度方向的变化而变化。即使我们例子中的物体保持相同的速度，速度的方向也在改变，因此物体在加速。

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.

由于圆周运动物体的速度总是与圆相切，这意味着加速度总是垂直于物体运动的方向。这也是为什么这个加速度不能改变物体速度大小的原因。

如何求向心加速度

现在我们有了方程，我们将看到如何在涉及圆周运动的不同情况下找到向心加速度。

例1

地球的半径是6400公里。求一个人站在地面上，由于地球绕地轴自转而产生的向心加速度。

How to find Centripetal Acceleration – Example 1

例2

骑自行车的人骑着一辆半径为0.33米的自行车。如果车轮以恒定速度转动，则在附着在自行车轮胎上的砂粒上找到向心加速度，该砂粒以4.1 m s-1的速度移动。

How to find Centripetal Acceleration – Example 2

根据牛顿第二定律，向心加速度必须伴随着作用在圆轨迹中心的合力。这个力叫做向心力。

如何计算向心力