Question

The cable cars in San Francisco are pulled along their tracks by an underground steel cable that moves along at $9.5mph.$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 $100ft$during its lifetime. To keep the tension constant, the cable passes around a $1.5m$diameter “tensioning pulley” that rolls back and forth on rails, as shown inFIGURE$100ft$. A $2000kg$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?

Short answer

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

Step-by-step solution

Given information

Speed of the steel cable$=9.5mph$

The length of a cable stretch$=100ft$

The cable passes around the diameter$=1.5m$

Mass of the cable$=2000kg$

Explanation

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

$F=mg$.....................(1)

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

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

Since the frictional force$=0$

Substituting the values in the equation$\left(1\right)$

$$\begin{gathered} F=\left(2200\right)\left(9.8\right) \\ F=19600N \end{gathered}$$

Substituting the values in the equation $\left(2\right)$

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

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