High School

A circus trapeze consists of a bar suspended by two parallel ropes, each of length ℓ=2.6m, allowing performers to swing in a vertical circular arc. Suppose a performer with mass m holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle θi with respect to the vertical. Suppose the size of the performer's body is small compared to the length ℓ, she does not pump the trapeze to swing higher, and air resistance is negligible.


Determine the value of cos (θi) such that the force needed to hang on at the bottom of the swing is 1.75 of the performer's weight.


Give your answer to two decimal places.


I don't understand the steps, please be more specific, thank you!

Answer :

The value of cos(θi) is approximately 1.375, satisfying the condition for the force at the bottom of the swing.

To determine the value of cos(θi) such that the force needed to hang on at the bottom of the swing is 1.75 times the performer's weight, we can use the principles of circular motion and equilibrium.

  • Start by considering the forces acting on the performer at the bottom of the swing. There are two forces: the tension in the ropes and the performer's weight. At the bottom of the swing, the tension in the ropes will be at its maximum value.
  • The tension in the ropes provides the centripetal force required to keep the performer moving in a circular path. At the bottom of the swing, this tension force must be equal to the sum of the performer's weight and the force needed to hang on (1.75 times the performer's weight).
  • Write down the equation for the tension force at the bottom of the swing. Let T be the tension, m be the mass of the performer, and g be the acceleration due to gravity. The equation is T = mg + (1.75mg).
  • Simplify the equation to T = 2.75mg.
  • Recognize that the tension force in the ropes can be related to the angle θi using trigonometry. The tension force is given by T = 2mgcos(θi), where ℓ is the length of the ropes.
  • Equate the two expressions for T: 2mgcos(θi) = 2.75mg.
  • Cancel out the mass, which gives cos(θi) = 2.75/2.
  • Evaluate the expression to find cos(θi) = 1.375.

So, the value of cos(θi) such that the force needed to hang on at the bottom of the swing is 1.75 times the performer's weight is approximately 1.375.

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