Older buildings often have fire escapes such as the one shown in the picture to

Question

Older buildings often have fire escapes such as the one shown in the picture to the right (which is actually on Drexel cam pus, about 300 feet from DRL). A stair is mounted on a pivot, and a cable at the end of the stair is connected via a pullev to a hanging weight. Normally, the cable holds the end of the stair off the ground which prevents trespassers from using it to get into the building. However, if a person walks out to the end of the stair it rotates down to the ground, allowing them to escape from a fire the hanging mass is important: if it is too small then the tension in the cable will not ,ETTRY be enough to support the stair, while if it is too great then the weight of the person will not be enough to make the stair rotate to the ground. ble . The value of TnT a) For this part, assume that the stair is perfectly horizontal and the cable is perfectly vertical. If the stair has a mass of 350 kg and a small adult has a mass of 45 kg, what is the acceptable range of masses for the hanging weight? You should specify both a maximum and a minimum b) In fact, in the picture shown the stair is about 7.0° below the horizontal and the cable points about 6.0° away from the vertical. Redo your calculations for part (a) using these values. Using the correct number of significant figures, does this change your final answers?

Explanation / Answer

Let L be the length of the stair

wieght 350g of the stair ats at the center i.e. at a distance of L/2 from the hinge.

Tension in the cable T acts at the end i.e at a distance L from the hinge

taking moments about the hing

T*L = 350g *L/2

T = 175 g

if the hanging mass is m then tension in the cable is mg = 175 g

hanging mass = 175 kg, this the minum mass required just to support the stair from falling down.

with the situation when the adult of 45 kg walks to the ened of the stair the stair would fall free under gravity as there is no net upward force but 45g (weight of the adult) acts down ward. Free fall under gravity is not desirable.

The tension in the cable must be a little less than the down force of the adult so that it slowly descends

The tension can be 44g more than the minimum required so that when the adult walks to the end the net downwad force is 1g so that the stair descend slowly.

The maximum mass = 175 + 44 = 219 kg

b) The stair is 70 below the horizontal, assuming the cable is vertical

distance of the COM of stair and the end of the stair from the hing are

L/2 Cos(7) and LCos(7)

Cos(7) appears in both moments and it has no effect on the cable tension or the hanging mass.

Now take the case of cable making 6o with the verticle.

Componet of tension along the verticle = T Cos(6)

T = 175g/Cos(6)  

hanging mass m = 177 kg , mimnimum

The maximum also increases by a factor of 1/Cos(6)

              = 219/Cos(6) = 220 kg

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