Answer:
a) 627.84 Joules
b) 117.72 Joules
c) 1255.68 Joules
Explanation:
(See figure 1)
The gravitational potential energy relative to the child’s lowest position is:
[tex] U_{g}=mgh [/tex] (1)
with h the vertical distance of the swing from the lowest position, m the mass of the child and g the acceleration of gravity.
a) When the ropes are horizontal, the swing is at 1.60 m from the lowest position, so by (1):
[tex] U_{g}=(40)(9.81)(1.60)=627.84\,J[/tex]
b) When the ropes make a 36.0◦ angle with the vertical, the swing is at a distance d from the lowest position, we should use trigonometric relations to find that distance. By figure 2 we have a right triangle with adjacent side [tex] 1.60 - d [/tex] and hypotenuse 1.60, so using the trigonometric relation [tex]\cos(36)=\frac{(1.60-d)}{1.60} [/tex] we can solve for d:
[tex] d=1.60-1.60*\cos(36)\approx0.30\,m [/tex]
Using d on (1):
[tex] U_{g}=(40)(9.81)(0.30)=117.72\,J [/tex]
c) Note that at the bottom of the circular arc the distance of the swing relative to the lowest position is two times the length of the rope, so h=3.20 m, using this on (1):
[tex]U_{g}=(40)(9.81)(3.20)=1255.68\,J [/tex]
