Question

(a)A climber whose mass is 55 kg hangs motionless from a rope. What is the tension in the rope?

(b)Later, a different climber whose mass is 88 kg hangs from the same rope. Now what is the tension in the rope?

(c)Compare the physical state of the rope when it supports the heavier climber to the state of the rope when it supports the lighter climber. Which statements about the physical state of the rope are true? Check all that apply. (1) Because the same rope is used, the tension in the rope must be the same in both cases. (2) The interatomic bonds in the rope are stretched more when the rope supports the heavier climber than when the rope supports the lighter climber. (3) The rope is slightly longer when it supports the heavier climber than when it supports the lighter climber.

Short answer

(a) The tension in the rope when a 55 kg climber hangs motionless from it is 539 N.

(b) The tension in the rope when a 88 kg climber hangs motionless from it is 862.4 N.

(c) (2)The interatomic bonds in the rope are stretched more when the rope supports the heavier climber than when the rope supports the lighter climber.

Step-by-step solution

Given data

Mass of climber hanging from the rope in the first case = 55 kg.

Mass of climber hanging from the rope in the first case = 88 kg.

Tension in rope

The tension in a rope when a massMhangs motionless in it can be written as,

T=Mg (I)

Here gthe acceleration due to gravity of value is given as,

$\mathbf{g}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$

Determining the tension in the rope in the first case

a)

From equation (I), the tension in the rope in the first case can be calculated as,

$\begin{array}{rcl}\mathrm{T}& =& 55\mathrm{kg}\times 9.8\mathrm{m}/{\mathrm{s}}^2\\ & =& 539.\left(1\mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2\times \frac{1\mathrm{N}}{1\mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2}\right)\\ & =& 539\mathrm{N}\end{array}$

Thus, the tension is 539 N.

Determining the tension in the rope in the second case

b)

From equation (I), the tension in the rope in the second case can be calculated as,

$\begin{array}{rcl}\mathrm{T}& =& 88\mathrm{kg}\times 9.8\mathrm{m}/{\mathrm{s}}^2\\ & =& 862.4.\left(1\mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2\times \frac{1\mathrm{N}}{1\mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2}\right)\\ & =& 862.4\mathrm{N}\end{array}$

Thus, the tension is 862.4 N.

Comparing the physical states of the rope in the two cases

As obtained above, the tension in the rope is different in the two cases. In the second case, more force is applied to the rope, which increases the interatomic distance of the rope.

Thus, (2) the interatomic bonds in the rope are stretched more when the rope supports the heavier climber than when the rope supports the lighter climber is correct.