在本文中，我們將研究如何解決垂直圓周運動問題。解決這些問題的原理與解決向心加速度和向心力問題的原理相同。與水平圓不同的是，作用在垂直圓上的力隨著它們的轉動而變化。我們將考慮兩種情況下物體在垂直圓中移動：物體以恆定速度移動和物體以不同速度移動。

勻速運動物體垂直圓周運動問題的求解

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

物體在底部時張力最大。這是繩子最容易斷裂的地方。

如何求解變速物體的垂直圓周運動問題

對於這些情況，我們考慮物體繞圓運動時能量的變化。在頂部，物體具有最大的勢能。當物體下落時，它失去了勢能，勢能被轉換成動能。這意味著物體下降時會加速。

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

接下來，我們來看看底部繩子的張力。在這裡，向心力是向上的。然後我們有

. Substituting  , we get  .

進一步簡化，我們得到：

.

垂直圓周運動問題–示例

頭頂上擺著一桶桶水

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.

同樣的原理也被用來防止物體在經歷“一圈一圈”的運動時墜落，例如過山車和空中表演，特技飛行員駕駛飛機垂直旋轉，飛機到達頂端時“倒轉”。

例1

倫敦眼是地球上最大的摩天輪之一。它的直徑為120米，每30分鐘旋轉一圈。假設它以恆定的速度移動，找到

a） 重量為65公斤的乘客所受的向心力

b） 當乘客位於圓圈頂部時，座椅產生的反作用力

c） 當乘客位於圓圈底部時，座椅產生的反作用力

How to Solve Vertical Circular Motion Problems – Example 1

註意：在這個例子中，反作用力變化很小，因為角速度很慢。但是，請註意，用於計算頂部和底部反作用力的表示式是不同的。這意味著當涉及較大的角速度時，反作用力會有很大的不同。最大的反作用力出現在圓的底部。

Vertical Circular Motion Problems – Example – The London Eye

例2

用一根0.70米長的繩子把一袋0.80公斤重的麵粉垂直地繞著一個圈擺動。當袋子繞著圓圈移動時，它的速度是不同的。

a） 結果表明，3.2 m s-1的最小速度足以使包保持在圓形軌道上。

b） 當袋子在圓的頂端時，計算繩子的張力。

c） 當繩子從頂部向下移動65度角時，求袋子的速度。

How to Solve Vertical Circular Motion Problems – Example 2