Question 534477: how can you determine the period of the swing in the playground?
Answer by KMST(5398)   (Show Source):
You can put this solution on YOUR website! THE PRACTICAL MEASURMENT:
If you have a playground with swingers nearby, you could time their swing cycles. If you want better precision, you could time 10 swing cycles and divide by 10.
THE THEORY:
The weight on the swing is a vertical force equal to the mass  of the swing seat plus occupant, multiplied by the acceleration of gravity  . You can consider the weight force  to be made up of a large component along the lines of the rope(s)/chain(s), plus another smaller, perpendicular force  that will try to bring the swing down to the vertical. You can visualize those forces as arrows with the proper directions, and lengths representing the magnitude of each force. When the swing is away from the vertical rest position by a distance/displacement  , the major component of the weight is fully counteracted by the pull of rope(s)/chain(s) of length  . Only the smaller component remains, unopposed, prodding the swing back towards the vertical, opposing the displacement. To keep track of the direction of force and displacement, we would use positive numbers to represent force/displacement in one direction, with negative numbers for the opposite direction, and you see that and will always have opposite signs. The large, real triangle with sides and and the imaginary one made of the arrows we use to represent the forces and  are similar triangles, and the ratios of the sides are the same, so
 and 
That force will cause an acceleration  in a direction opposing the displacement . According to Newton, the magnitude of that acceleration will equal the magnitude of the force divided by the mass being accelerated, so
and substituting the previous equation
We know that , and are periodic functions of time.
All those functions will vary over time, go from positive to negative (or vice versa) and back, and after a period  we would be back to the starting conditions and everything would repeat after that.
If we start by letting go of the swing we were holding a distance  away from the vertical, we will have  at  .
We could work a lot of math to get to the conclusion that a good function to represent the movement is
 where the trigonometric function sine is involved, with the angles measured in radians.
The period of the function would be such that
 so 
The only thing you can change is the length of the chain/rope, and the square root makes the effect of the change less marked that you could wish. Besides, a very short chain may have a much shorter period, but it is less fun. Since the theoretical length was measured to a point where all the mass was concentrated, changing the center of gravity of the swinger by standing up on the swing, or putting more weight on the feet would change the theoretical length and the period.
Using approximations,
  
I calculate  for chain/rope length in meters, and
 for chain/rope length in feet.
That would give you a period of 10.9 seconds for a 9 foot long swing chain.
If you have a playground with swingers nearby, you could compare theoretical values with real life.
DISCLAIMERS:
That is a physics problem. The solution has been worked out and described in textbooks. We have to bear in mind that science usually works with the ideal, simplified situation, like no air resistance, relatively narrow swinging amplitude, and strong swing ropes/chains that weigh nothing at all. For dealing with the real world we have engineers that end up developing empirical formulas.
Math is used to figure out a function describing the motion. Geometry, trigonometry and calculus are invoked along the way. Geometry, trigonometry and calculus are collections of conclusions discovered over centuries (millennia for geometry) that students end up remembering/memorizing so as not to have to re-discover them every time they sit down to solve a problem. Any explanation would have to be tailored to the knowledge base, interest, and attention span of the audience. I tried anyway.
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