Child on a swing Physics problem

Question

A 350-N child is in a swing that is attached to a pair of ropes 2 m long. Derive an algebraic expression for the the
gravitational potential energy of the child–Earth system as a function of the rope length and the angle of the swing relative to the child'slowest position for the following times.

 

(a) when the ropes are horizontal.
(b) when the ropes make a 27.0° angle with the vertical.
(c) when the child is at the bottom of the circular arc.

 

For part a) I done U_{g =} mgy  where mg = 350N. The child must be 2m above  the ground when the rope is horizontal.

 

U_g=350N x 2m = 700 J

 

I have no idea how to do part b) or c) providing a) is correct.  I know you will need to use trig for part b but I don't know what to do or why

Answer 1

Hi Ian!

 

I think you are right on with Part A, except that technically the child is 2 m above his or her lowest point (which should be pretty near the ground for a typical swing).  This implies a (reasonable) h = 0 level at the child's lowest point.

 

In Part (c), it looks like the child is meant to be at his or her lowest point, at the bottom of the circular path followed by the child.  Since that was implicitly defined to be h = 0, that would give a direct answer for the child's gravitational potential energy at that point.  If you are still not sure how to get it after thinking of it that way, we can definitely talk about it further.

 

For Part (b), you are right that we need trig, because we need to calculate the vertical distance.  This may be tricky to see without a diagram, but the key element here is that the length L of the ropes supporting the swring are always the same, no matter where the seat is.  When the swing is at the bottom, all of that length L is directed toward the bottom.  When the ropes make an angle θ with the vertical, the L is directed along the line at θ from the vertical.  At that point, part of the length L is in the vertical direction, and part is in the horizontal direction.  The difference between L and the part of L in the vertical direction at the angle θ will be the vertical distance between where the swing is at its lowest point (h = 0) and where it is at θ degrees with the vertical.

 

If you draw a diagram, you may be able to see that -- for an angle θ defined with respect to the vertical, the horizontal and vertical components of the length L make a right triangle, and the vertical component side is adjacent to the angle θ (while length L is the hypotenuse).  You may then be able to write an expression for the vertical component in terms of L, θ, and a trig function.  The difference between the length of that vertical component and the total length L (since all of L is in the vertical when the swing is at the bottom) will be the vertical height of the child above the h = 0 level at angle θ.

 

I hope that may get some things started for you, but, if not, just let me know, and we can talk about it further.  Good luck!