To solve this problem we will use the trigonometric concepts to find the distance h, which will allow us to find the speed of Jeff and that will finally be the variable that will indicate the total tension, since it is the variable of the centrifugal Force given in the vine at the lowest poing of the swing.
From the image:
[tex]cos (32) = \frac{(7.6-h)}{7.6}[/tex]
[tex]h = 1.1548m[/tex]
When Jeff reaches his lowest point his potential energy is converted to kinetic energy
[tex]PE = KE[/tex]
[tex]mgh = \frac{1}{2} mv^2[/tex]
[tex]v = \sqrt{2gh}[/tex]
[tex]v = \sqrt{2(9.8)(1.1548)}[/tex]
[tex]v= 4.75m/s[/tex]
Tension in the string at the lowest point is sum of weight of Jeff and the his centripetal force
[tex]T = W+F_c[/tex]
[tex]T = mg + \frac{mv^2}{r}[/tex]
[tex]T = (83)(9.8)+\frac{(9.8)( 4.75)^2}{7.6}[/tex]
[tex]T = 842.49N[/tex]
Therefore the tension in the vine at the lowest point of the swing is 842.49N
