Chapter 7: Q. 21 (page 178)

The cable cars in San Francisco are pulled along their tracks by an underground steel cable that moves along at $9.5 m p h .$ The cable is driven by large motors at a central power station and extends, via an intricate pulley arrangement, for several miles beneath the city streets. The length of a cable stretches by up to $100 f t$ during its lifetime. To keep the tension constant, the cable passes around a $1.5 m$ diameter “tensioning pulley” that rolls back and forth on rails, as shown inFIGURE $100 f t$ . A $2000 k g$ block is attached to the tensioning pulley’s cart, via a rope and pulley, and is suspended in a deep hole. What is the tension in the cable car’s cable?

Expert verified

The tension in the cable car's cable is $9800 N$ .

Step by step solution

Speed of the steel cable $= 9.5 m p h$

The length of a cable stretch $= 100 f t$

The cable passes around the diameter $= 1.5 m$

Mass of the cable $= 2000 k g$

As weight is moving uniformly , net force on it will be zero . So total upward force is equal to total downward force.

Force acting downwards

F = m g

.....................(1)

Tension of rope be T . So total force acting upwards is 2T . 2T will balance the weight .

Further, the tension localid="1649675027629" $2 T = F$ ........................(2)

Since the frictional force $= 0$

Substituting the values in the equation $(1)$

$F = (2200) (9.8) F = 19600 N$

Substituting the values in the equation $(2)$

localid="1649613408531" $T = \frac{19600 N}{2} = 9800 N$

The tension in the cable car's cable $T = 9800 N$

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