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Answer :
Final answer:
To find the angle that the trapeze cord makes with the vertical at the instant the trapeze is 0.750 m below its initial height, we can use the concept of conservation of energy. Initially, the aerialist has potential energy only, which is converted to kinetic energy at the lowest point of the swing. Using the equations for potential energy and kinetic energy, we can equate the two expressions and solve for the speed at the lowest point. We can then use the sine function to find the angle that the cord makes with the horizontal, and subtract it from 90° to find the angle with the vertical.
Explanation:
To find the angle that the trapeze cord makes with the vertical at the instant the trapeze is 0.750 m below its initial height, we can use the concept of conservation of energy. We know that at any point in the swing, the total mechanical energy (potential energy + kinetic energy) remains constant. Initially, the aerialist has potential energy only, which is converted to kinetic energy at the lowest point of the swing.
The potential energy at the initial height is given by the formula PE = mgh, where m is the mass of the aerialist, g is the acceleration due to gravity, and h is the vertical distance. At the lowest point, the potential energy is zero, and all the energy is converted to kinetic energy given by KE = (1/2)mv^2, where v is the speed at the lowest point.
Since the total mechanical energy remains constant, we can equate the two expressions for energy:
mgh = (1/2)mv^2
Canceling the mass and rearranging the equation, we get:
gh = (1/2)v^2
Since v^2 = 2gh, we can substitute this back into the equation and solve for v:
v = sqrt(2gh)
Next, we need to find the angle that the cord makes with the horizontal at the lowest point. We can use the sine function to do this. The vertical distance at the lowest point is given as 0.750 m, and the length of the cord is 8 m. Therefore, the sine of the angle is 0.750/8. Evaluating this expression, we find that the angle is approximately 5.7°. Finally, to find the angle that the cord makes with the vertical at the lowest point, we can subtract this angle from 90°. Therefore, the angle that the cord makes with the vertical at the instant the trapeze is 0.750 m below its initial height is approximately 84.3°.
Learn more about Conservation of energy here:
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