A mountain climber, in the process of crossing between two cliffs by a rope, pauses to rest. She weighs 520 N. As the drawing shows, she is closer to the left cliff than to the right cliff, with the result that the tensions in the left and right sides of the rope are not the same. Find the tensions in the rope to the left and to the right of the mountain climber.
Whats the answer
To find the tensions in the rope to the left and right of the mountain climber, we need to consider the forces acting on the climber.
Let's label the left tension in the rope as T_left and the right tension as T_right.
We know that the climber is at rest, so the forces acting on her must be balanced. In other words, the sum of the vertical forces and the sum of the horizontal forces must be equal to zero.
In the vertical direction, there are two forces: the weight of the climber acting downward (520 N) and the tension in the rope pulling upward. Since the climber is at rest, the sum of these forces in the vertical direction is zero:
520 N - T_left - T_right = 0
In the horizontal direction, there are no forces acting, so the sum of the horizontal forces is zero. This means that the tension in the rope to the left is equal to the tension in the rope to the right:
T_left = T_right
Now we have two equations:
520 N - T_left - T_right = 0
T_left = T_right
We can solve this system of equations to find the values of T_left and T_right.
Adding the second equation to the first equation gives:
520 N - T_left - T_right + T_left = 0
520 N - T_right = 0
Simplifying further:
T_right = 520 N
Since T_right = T_left, we have:
T_left = 520 N
Therefore, the tensions in the rope to the left and right of the mountain climber are both 520 N.