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An acrobat, starting from rest, swings freely on a trapeze of length 3.7 m (Figure 6). If the initial angle of the trapeze is 48°, use the law of conservation of energy to determine:

(a) The acrobat's speed at the bottom of the swing.

(b) The maximum height, relative to the initial position, to which the acrobat can rise.

Answer :

The energy conservation and trigonometry we can find the results for the questions about the movement of the acrobat are;

a) The maximum speed is v = 4.89 m / s

b) The maximum height is h = 1.22 m



The energy conservation is one of the most fundamental principles of physics, stable that if there are no friction forces the mechanistic energy remains constant. Mechanical energy is the sum of the kinetic energy plus the potential energies.

Em = K + U


Let's write the energy in two points.

Starting point. Highest part of the oscillation

Em₀ = U = m g h

Final point. Lower part of the movement

[tex]Em_f[/tex] = K = ½ m v²

Energy is conserved.

Emo = [tex]Em_f[/tex]

m g h = ½ m v²

v² = 2 gh


Let's use trigonometry to find the height, see attached.

h = L - L cos θ

h = L (1- cos θ)

They indicate that the initial angle is tea = 48º and the length is L = 3.7 m, let's calculate.

h = 3.7 (1- cos 48)

h = 1.22 m

this is the maximum height of the movement.


Let's calculate the velocity.

[tex]v= \sqrt{2 \ 9.8 \ 1.22}[/tex]

v = 4.89 m / s


In conclusion using the conservation of energy and trigonometry we can find the results for the questions about the movement of the acrobat are;

a) The maximum speed is v = 4.89 m / s

b) The maximum height is h = 1.22 m

Learn more here: brainly.com/question/13010190

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Rewritten by : Barada

Final answer:

The acrobat's speed at the bottom of the swing can be determined by equating his potential energy at the top of the swing with his kinetic energy at the bottom, and the maximum height he could rise is calculated from the initial angle and length of the trapeze.

Explanation:

Using the principle of the conservation of energy, the acrobat's kinetic energy at the bottom of the swing is equal to his potential energy at the top of the swing. At the top, the potential energy is calculated by the formula Potential Energy = m * g * h, where m is his mass, g is the acceleration due to gravity, and h is the height. The height h, in this case, is obtained from the trapeze length and the initial angle, by the relationship h = 3.7m * (1 - cos(48°)).

At the bottom of the swing, the acrobat has no potential energy, so all his initial potential energy has been transformed into kinetic energy, which is calculated by the formula Kinetic Energy = 0.5 * m * v^2, where m is his mass, and v is his speed. Equating the potential energy at the top with the kinetic energy at the bottom, we obtain for the speed v the equation v = sqrt(2gh).

To calculate the maximum height too which the acrobat could rise relative to the initial position, we use the equation h = L - L*cos(theta) where L is the length of the rope and theta is the initial angle of the trapeze. Substituting the given values, we have h = 3.7m - 3.7m*cos(48°).

Learn more about Conservation of Energy here:

https://brainly.com/question/13345660

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