Sunday, November 25, 2012

The Physics of a Trapeze

Trapeze artists gained popularity by performing at the circus, swinging high above the ground.  The trapeze and trapeze artist can be modeled as a simple pendulum because the artist is basically a mass at the end of a string, which in this case is the trapeze made of a horizontal bar hanging by two cables.  I assumed that the trapeze artist doesn’t move his body, but hangs straight in line with the middle of the bar.  I wanted to determine the maximum speed that the trapeze artist moves.  Air resistance was ignored.

I estimated the mass of the trapeze artist to be 65 kg and the mass of the horizontal bar to be 4 kg (the mass of the cables is negligent).  Therefore, the mass of the trapeze system is 69 kg.  The cables connecting the horizontal bar to the ceiling are 4.2 m long and the trapeze artist is 1.6 m tall.  The trapeze artist starts by standing on a platform.  I assumed that the platform was 4.5 meters above the floor.  The man is modeled as a point mass and his center of mass is assumed to be half of his body length.  Therefore, the distance of the trapeze system is 4.5 m + (1.6 m/2) = 5.3 m from the floor.

The trapeze artist starts by standing on a platform at the highest point of his swing.  Therefore, while standing on the platform, at rest, the trapeze artist has his maximum potential energy.

PE = mgh = (69 kg)(9.8 m/s2)(5.3 m) = 3.6 x 103 J

When the trapeze artist swings to the lowest height in his swing, he is at his maximum speed and his maximum kinetic energy.  The trapeze artist is at his lowest potential energy here.  I assumed that there are no nonconservative forces.  I also assumed that the trapeze system was now at a height of 4.6 m from the floor.

One trick performed on a swinging trapeze, which is a trapeze in which the trapeze artist starts from rest and swings the trapeze himself, instead of creating the swing by walking off the high platform (this is a flying trapeze), is the stand seats-off.  In this trick, the trapeze artist first stands upright on the bar, swings, and eventually finishes hanging upside down from the bar with his feet.  I assumed the angle of the trapeze artist’s swing to be 50°.  To determine the maximum speed at which the trapeze artist swings, I used the same formula as above.  This time the height is the height of the “pendulum” or the trapeze system, which is the length of the cables plus half of the performer’s height (4.2 m + (1.6 m/2) = 5.0 m.  The performer starts from rest.

Now I want to find the maximum velocity of the trapeze artist after he starts falling.  I estimated that the performer fell 2.2 m.  First, I have to find the potential energy before the performer falls.  I ignored the mass of the bar.
PE = mgh = (65 kg)(9.8 m/s2)(5.0m – (5.0 m * cos(50°))) = 1137 J
The PE at the bottom of the fall is:
PE = mgh = (65 kg)(9.8 m/s2)(-2.2 m) =  -1401 J
The PE is negative because the man is falling down.

The trapeze artist has a large increase in velocity when he performs this trick.


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