In this article, we will look at how to calculate work done. Work is done when the point of application of a force moves along the force’s line of action. The line of action of the force is a line drawn through the point of application in the direction of the force. Suppose a c***tant force acts on an object and causes the object to move through a displacement  in a direction parallel to its line of action, as shown below:

How to Calculate Work Done – Displacement Parallel to Line of Action

Then, the work done is given by,

现在考虑力与物**移成一定角度的情况。例如，把割草机推到这里，割草机沿着地面移动，尽管力与地面成一定角度。在这里，只有平行于地面的分力在做功。为了计算在这些情况下所做的功，我们把位移方向上的力的分量乘以位移。

How-to-Calculate-Work-Done – Displacement at an Angle to Line of Action

也就是说，

这实际上是力向量和位移向量之间的标量积的定义。因此，我们也可以将所做的工作写成：

注意，所做的功是一个标量。它有焦耳（J）的国际单位制。当力的作用点在位移方向上的分力为1N时，位移为1m，则功为1焦耳。

If the force is not c***tant but rather a function of position , then the work done by the force to move the object from a position to another position is given by,

So if a force vs. distance graph is drawn, the work done to move the object from to is equal to the area under that graph between and .

How to Calculate Work Done – Force vs Distance Graph

如前所述，所做的功不是矢量量。然而，我们通常把作用在物**移相反方向的分力所做的功取为负。

能量是做功的能力，做功传递能量。测量能量的单位也是焦耳。

如何计算已完成工时-示例

例1

在一个操场上，一个坐在玩具车里的孩子被她的朋友向前拉，她的朋友用60牛顿的力沿着玩具车上的绳子拉着玩具车。绳子与地面成35度角。计算孩子的朋友将孩子向前拉20米所做的功。

We have   ,   and   .

  .

例2

一辆重达1500公斤的汽车的发动机突然停止运转。汽车行驶160米，13秒后完全停下。假设停车的唯一动力是汽车轮胎和路面之间的摩擦力，找出这个力需要做多少功才能使汽车完全停车。

For the car, we have  , , , .

To find the acceleration, we use (which is an altered form of the more familiar ). Then,

.

The size of average friction is then,

.

由于摩擦力作用于160 m的距离，所做的功为：

.

例3

地球和太阳之间的引力是3.5×1022牛顿。如果太阳和地球之间的距离是1.59×1011米，计算太阳使地球在其轨道上运行一整年所做的功。假设地球绕太阳的轨道是圆的。

This is a trick question! It would seem that, to find the work done, you first find the total length of Earth’s orbit using  , and then multiply that by the force. However, the Earth is in uniform circular motion around the Sun. Therefore, the force from the Sun is centripetal. i.e. it is always perpendicular to the Earth’s direction of motion. C***equently, at any given time the Sun has no force component along the direction of Earth’s motion. So the Sun does no work to keep the Earth in orbit. That is, the work done is 0. In general, no work needs to be done to keep an object in uniform circular motion.