Chapter 5.3, Problem 42SSC

To determine

To Compute: The tension in the rope, just after applying a force at the midpoint at right angle to the rope.

Explanation of Solution

Given info:

Tension in the rope, T

Angle made by rope from horizontal $=\theta$

Formula used:

Newton’s second law:

  $F_{net}=ma$

Where,

  $m=$ Mass.

  $a=$ Acceleration.

At equilibrium, acceleration is zero, therefore:

  $F_{net}=0$

Calculation:

The free body diagram:

  

In practical cases, when a car accidentally stuck in mud, then knowledge of mechanics is used to understand the problem.

As shown in figure, one end of the rope is bound with a tree and other end is with the car, then by pulling the rope with a force F at right angle to the rope at its midpoint, huge tension T compared to the applied force F exists in the rope.

The horizontal components of the tension would cancel out to each other while the vertical components of the tension would give net force in downward direction.

Applying Newton’s second law in equilibrium:

  $\begin{array}{c}{F}_{net}=0\\ F-2T\sin \theta =0\\ F=2T\sin \theta \\ T=\frac{F}{2\sin \theta }\end{array}$

Since θ is small, value of T will be very large in compare to F and will be effective enough to free the stuck car.

Conclusion:

Tension on the rope is T, after applying force F at right angle to the rope if the rope makes an angle θ with the horizontal diraction, then the value of tension: $T=\frac{F}{2\sin \theta }$