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Answer :
The angle that the trapeze cord makes with the vertical at the instant it is released is 29.8 degrees.
The length of the cord is 8.0 m.
The angle between the cord and the vertical just before the aerialist jumps is 41 degrees.
The trapeze is released when it is 0.75 m below its initial height.
Step 1: Find the initial velocity of the aerialist.
Using the formula for the height of a projectile:
Δh = v₀ᵗ sin(θ) - 1/2 g t²
Where:
Δh is the change in height (0.75 m)
v₀ is the initial velocity
θ is the angle between the cord and the vertical (41 degrees)
g is the acceleration due to gravity (9.8 m/s²)
t is the time taken for the aerialist to reach the lowest point
Rearranging the equation to solve for v₀:
v₀ = (Δh + 1/2 g t²) / (t sin(θ))
we can use the relation between the cord length and the angle:
sin(θ) = Δh / 8.0 m
Substituting this into the equation for v₀, we get:
v₀ = √(2 g Δh / (1 - cos(θ)))
v₀ = √(2 × 9.8 × 0.75 / (1 - cos(41°)))
v₀ = 6.18 m/s
Step 2:
At the lowest point, the velocity in the vertical direction is zero, and the velocity in the horizontal direction is equal to the initial velocity.
Using the trigonometric relations:
tan(φ) = v₀ cos(θ) / √(2 g Δh)
φ = tan⁻¹(v₀ cos(θ) / √(2 g Δh))
φ = tan⁻¹(6.18 × cos(41°) / √(2 × 9.8 × 0.75))
φ = 29.8 degrees
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Rewritten by : Barada
The explanation details how to calculate the angle that the trapeze cord makes with the vertical at a specific moment.
Analyzing the Trapeze Cord Angle:
- Calculate the vertical component of the force using the given values.
- Determine the angle at which the trapeze cord makes with the vertical at the specified moment.
- Apply trigonometry to find the angle by analyzing the vertical and horizontal components of the forces.
To find the angle that the trapeze cord makes with the vertical at the instant it is 0.75m below its initial height, we can use trigonometric ratios. Let's call this angle theta. We know the length of the cord is 8.0m and the change in height is 0.75m. Using the sine function, we have sin(theta) = opposite/hypotenuse. Plugging in the values, we get sin(theta) = 0.75/8.0.
Now, we can use inverse sine to find theta. Taking the inverse sine of both sides, we get theta = arcsin(0.75/8.0).
Calculating this on a calculator, we find that theta is approximately 5.95 degrees.
