A circus trapeze consists of a bar suspended by two parallel ropes, each of length [tex] \ell = 3.0 \, \text{m} [/tex], allowing performers to swing in a vertical circular arc. Suppose a performer with mass [tex] m [/tex] holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle [tex] \theta_i [/tex] with respect to the vertical.

Assume the size of the performer's body is small compared to the length [tex] \ell [/tex], she does not pump the trapeze to swing higher, and air resistance is negligible.

Determine the value of [tex] \cos(\theta_i) [/tex] such that the force needed to hang on at the bottom of the swing is 2.47 times the performer's weight. Give your answer to two decimal places.

To find the value of cos(θi) in this scenario, we use the equation 2.47 * m * g = m * g / (2 * cos(θi)). Solving for cos(θi), we find that it is approximately 0.202. Therefore, the value of cos(θi) such that the force needed to hang on at the bottom of the swing is 2.47 times the performer's weight is approximately 0.202.

To determine the value of cos(θi) in this scenario, we need to consider the forces acting on the performer at the bottom of the swing. At the bottom, the performer's weight acts vertically downward, while the tension in the ropes acts both vertically upward and towards the center of the circular arc.

Let's denote the tension in each rope as T. According to the problem, the force needed to hang on at the bottom of the swing is 2.47 times the performer's weight. We can express this as:

2.47 * m * g = 2T

where m is the mass of the performer and g is the acceleration due to gravity.

Since the ropes are at an angle θi with respect to the vertical, we can write:

T = m * g / (2 * cos(θi))

Substituting this into the equation above, we have:

2.47 * m * g = m * g / (2 * cos(θi))

Simplifying, we get:

4.94 * cos(θi) = 1

Dividing both sides by 4.94, we find:

cos(θi) = 1 / 4.94

Evaluating this expression, we find:

cos(θi) ≈ 0.202

Therefore, the value of cos(θi) such that the force needed to hang on at the bottom of the swing is 2.47 times the performer's weight is approximately 0.202.

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