在本文中，我们将研究如何解决垂直圆周运动问题。解决这些问题的原理与解决向心加速度和向心力问题的原理相同。与水平圆不同的是，作用在垂直圆上的力随着它们的转动而变化。我们将考虑两种情况下物体在垂直圆中移动：物体以恒定速度移动和物体以不同速度移动。

匀速运动物体垂直圆周运动问题的求解

If an object is traveling at a c***tant speed in a vertical circle, then the centripetal force on the object, remains the same. For example, let’s think of an object with mass that is swung about in a vertical circle by attaching it to a string of length . Here, then, is also the radius for the circular motion. There will be a tension always acting along the string, pointed towards the center of the circle. But the value of this tension will c***tantly vary, as we will see below.

Vertical Circular Motion of an Object at C***tant Speed v

Let us c***ider the object when it is at the top and the bottom of its circular path. Both the object’s weight, , and the centripetal force (pointed at the centre of the circle) remain the same.

How to Solve Vertical Circular Motion Problems – C***tant Speed Object Tension at Top and Bottom

物体在底部时张力最大。这是绳子最容易断裂的地方。

如何求解变速物体的垂直圆周运动问题

对于这些情况，我们考虑物体绕圆运动时能量的变化。在顶部，物体具有最大的势能。当物体下落时，它失去了势能，势能被转换成动能。这意味着物体下降时会加速。

Suppose an object attached to a string moves in a vertical circle with varying speed such that, at the top the object has just enough speed to maintain its circular path. Below, we will derive expressi*** for this object’s minimum speed at the top, the maximum speed (when it is at the bottom) and the tension of the string when it is at the bottom.

At the top, the centripetal force is downward and   . The object will have just enough speed to maintain its circular path if the string is just about to go slack when it is at the top. For this case, the tension of the string is almost 0. Inserting this into the centripetal force equation, we will have . Then, .

When the object is at the bottom, its kinetic energy is greater. The gain in kinetic energy is equal to the loss in potential energy. The object falls through a height of  when it reaches the bottom, so the gain in kinetic energy is . Then,

.

Since our  , we have

接下来，我们来看看底部绳子的张力。在这里，向心力是向上的。然后我们有

. Substituting  , we get  .

进一步简化，我们得到：

.

垂直圆周运动问题–示例

头顶上摆着一桶桶水

A bucket of water can be swung overhead without the water falling down if it is moved at a large enough speed. The weight  of the water is trying to pull the water down; however, the centripetal force   is trying to keep the object in the circular path. The centripetal force itself is composed of the weight plus the normal reaction force acting on the water. Water will stay on the circular path as long as .

How to Solve Vertical Circular Motion Problems – Swinging a Bucket of Water

If the speed is low, such that  , then not all of the weight is “used up” to create the centripetal force. The downwards acceleration is greater than the centripetal acceleration, and so the water will fall down.

同样的原理也被用来防止物体在经历“一圈一圈”的运动时坠落，例如过山车和空中表演，特技飞行员驾驶飞机垂直旋转，飞机到达顶端时“倒转”。

例1

伦敦眼是地球上最大的摩天轮之一。它的直径为120米，每30分钟旋转一圈。假设它以恒定的速度移动，找到

a） 重量为65公斤的乘客所受的向心力

b） 当乘客位于圆圈顶部时，座椅产生的反作用力

c） 当乘客位于圆圈底部时，座椅产生的反作用力

How to Solve Vertical Circular Motion Problems – Example 1

注意：在这个例子中，反作用力变化很小，因为角速度很慢。但是，请注意，用于计算顶部和底部反作用力的表达式是不同的。这意味着当涉及较大的角速度时，反作用力会有很大的不同。最大的反作用力出现在圆的底部。

Vertical Circular Motion Problems – Example – The London Eye

例2

用一根0.70米长的绳子把一袋0.80公斤重的面粉垂直地绕着一个圈摆动。当袋子绕着圆圈移动时，它的速度是不同的。

a） 结果表明，3.2 m s-1的最小速度足以使包保持在圆形轨道上。

b） 当袋子在圆的顶端时，计算绳子的张力。

c） 当绳子从顶部向下移动65度角时，求袋子的速度。

How to Solve Vertical Circular Motion Problems – Example 2