What is the mass of the counterweight that must be used in the following case:In stage production of Peter Pan, the 57 kg actress playing Peter has to fly in vertically, and to be in time with the...

What is the mass of the counterweight that must be used in the following case:

In stage production of Peter Pan, the 57 kg actress playing Peter has to fly in vertically, and to be in time with the music, she must be lowered, starting from rest, a distance of 3.4 m in 2 s.  Backstage, a smooth surface sloped at 60 degrees supports a counterweight of mass m.  What is the mass of the counterweight that must be used?  The pulley is of negligible mass and is frictionless.  The acceleration of gravity is 9.81m/s^2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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In the stage production of Peter Pan, the 57 kg actress playing Peter has to fly in vertically. The actress is lowered starting from rest, a distance of 3.4 m in 2 s.  This is acheived by the use of counter weight of mass m kept backstage on a smooth surface at a slope of 60 degrees. The mass of the counterweight has to be determined.

It is assumed that the pulley is of negligible mass and frictionless and acceleration due to gravity is 9.81 m/s^2.

The actress playing Peter is lowered from rest and moves 3.4 m in 2 s. If the acceleration acting on her is a, 3.4 = (1/2)*a*4

=> a = 3.4*2/4 = 1.6 m/s^2.

The counterweight kept backstage has a force acting on it due to gravity. This can be divided into two components, one perpendicular to the surface it is placed on and one parallel to the surface. As there is no friction, only the parallel force needs to be considered. This is equal to m*9.81*sin 60.

57*9.81 - m*9.81*sin 60 = 57*1.6

=> m*9.81*sin 60 = 57*9.81 - 57*1.6

=> m*9.81*sin 60 = 467.97

=> m = 467.97/(9.81*sin 60)

=> m = 55.08

The mass of the counterweight is 55.08 kg.

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