Question

Tarzan, who weighs $688N,$ swings from a cliff at the end of a vine $18m$ long (Fig. 8-40). From the top of the cliff to the bottom of the swing, he descends by $3.2m.$ The vine will break if the force on it exceeds $950N.$(a) Does the vine break? (b) If no,what is the greatest force on it during the swing? If yes, at what angle with the vertical does it break ?

Answer 1

(a) To find out whether or not the vine breaks, it is sufficient to examine it at the moment Tarzan swings through the lowest point, which is when the vine — if it didn't break — would have the greatest tension. Choosing an upward positive, Newton's second law leads to

$T-mg=m\frac{v^2}{r}$

where $r=18.0m$ and $m=W/g=688/9.8=70.2kg.$ We find the $v^2$ from energy conservation (where the reference position for the potential energy is at the lowest point).

$mgh=\frac{1}{2}m{v}^2\Rightarrow v^2=2gh$

where $h=3.20m.$ Combining these results, we have
$T=mg+m\frac{2gh}{r}=mg(1+\frac{2h}{r})$

which yields $933N.$ Thus, the vine does not break.

(b) Rounding to an appropriate number of significant figures, we see the maximum tension is roughly $9.3\times {10}^{2}N.$