Chapter 1.10, Problem 68E

To determine

To find: the combined lifting force applied to the door by the springs when the door is closed.

Answer to Problem 68E

The force required to make the door closed is 120 pounds.

Explanation of Solution

Given information:

A force of 15 pounds is required to stretch given spring 1 foot. Because of a pulley system, the springs stretch only one half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural lengths when the door is open.

Calculation:

  

Consider the force applied to an overhead garage door springs when the door is fully closed.

The overhead garage door has two springs, one on each side of the door and a force os 15 pounds is required to stretch each spring 1 foot.

Since the distance stretched by the spring varies directly with the force applied we can write,

  $F=kx$ , where $F$ is the force applied, $x$ is the distance stretched and $k$ is the constant of proportionality.

Substitute for $F=15$ and $x=1$ for finding $k$ .

  $\begin{array}{l}F=kx\\ 15=k\left(1\right)\\ k=15\end{array}$

The door moves a total of 8 feet when it is opened and at the time springs are not stretched. Also as springs stretch only half of the distance the door travels.

So in order to close the door, the door has to travel 8 feet and spring have to stretch 4 feet.

That is,

  $x=4$ for closing the door.

The force required for stretching on spring by 4 feet.

  $\begin{array}{l}F=kx\\ F=\left(15\right)4\text{Substitute}\text{for}x\text{and}k\\ F=60\end{array}$

Thus for two springs, force required is

  $\begin{array}{c}F=60\times 2\\ =120\end{array}$

Thus the force required to make the door closed is 120 pounds.