Question

Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting a force of 1420 N on the rope, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 81.9 kg and the vine is 5.2 m long.

Answer 1

Answer:

Angular velocity:

$w=1.2/s$

Linear velocity:

$v=6.24m/s$

Explanation:

When he swings he es ina a circular movement with radius 5.2m. To maintain this circular movement at any time there must be total force on Tarzan equal to the centripetal force:

$F_{cent}=m*r*w^{2}$

where m is his mass, r the radius and w the angular velocity. At the lowest point on Tarzan there are two forces acting, gravity and the rope tension, this rope tension equals the force on Tarzan arms:

$F_{tot}=T-W$

Where T is the tension on his arms, and W the weigth calculated as:

$W=m*g$

Where g is the acceleration of gravity. replacing this and remembering that the total force must equal the centripetal force:

$F_{tot}=T-m*g$

$F_{cent}=m*r*w^{2}=T-m*g$

Solving for w:

$w=\sqrt{\frac{T/m-g}{r}}=\sqrt{\frac{1420N/81.9kg-9.8m/s^{2}}{5.2m}}=1.2/s$

To get the linear velocity at this point:

$v=w*r=1.2/s*5.2m=6.24m/s$