Answer to Question #165132 in Physics for Brooke C

Question

A 13.5-kg monkey is hanging by one arm from a branch and swinging on a vertical circle. As an approximation, assume a radial distance of 97.1 cm is between the branch and the point where the monkey's mass is located. As the monkey swings through the lowest point on the circle, it has a speed of 1.96 m/s. Find (a) the magnitude of the centripetal force acting on the monkey and (b) the magnitude of the tension in the monkey's arm.

Answer 1

The centripetal acceleration of the monkey at that point is given as follows:

$$a = \dfrac{v^2}{R}$$

where $v = 1.96 m/s$ is its speed, and $R = 97.1cm = 0.971m$ is the radus of the circle. The

centripetal force is then (according to the second Newton's law):

$$F = ma\\ F = \dfrac{mv^2}{R}\\ F= \dfrac{13.5kg\cdot (1.96 m/s)^2}{0.971}\approx 53.41N$$

Here $m = 13.5kg$ is the mass of the monkey.

This centripetal force is the result of two opposite forces: the gravity force, that pulls the monkey downward, and the tension force, that pulls it upward. Since the centripetal force is also directed upward, obtain:

$$F = T-F_g$$

where $T$ is the tension force, and $F_g = mg$ is the gravity force. Expressing the tension, obtain:

$$T = F + F_g\\ T = F + mg\\$$

where $g = 9.81N/kg$ is the gravitational acceleration. Thus, obtain:

$$T = 53.41N +13.5kg\cdot 9.81N/kg = 185.85N$$

Answer. a) 53.41N, b) 185.85 N.