A gymnast is swinging on a high bar. The distance between his waist and the bar is 1.1 m, as the drawing shows. At the top of the swing his speed is momentarily zero. Ignoring friction and treating the gymnast as if all of his mass is located at his waist, find his speed at the bottom of the swing.
• 2.6 m/s
• 6.6 m/s
• 3.6 m/s
• 5.6 m/s
• 6.0 m/s
To solve this problem, we can use the principle of conservation of mechanical energy. At the top of the swing, the gymnast will have gravitational potential energy, and at the bottom of the swing, this potential energy will be converted into kinetic energy.
The gravitational potential energy at the top of the swing can be calculated using the formula:
PE = mgh
Where:
m = mass of the gymnast
g = acceleration due to gravity (9.81 m/s^2)
h = vertical distance between the waist and the bar (1.1 m)
Since his speed is momentarily zero at the top of the swing, his initial kinetic energy is zero. Therefore, the total mechanical energy at the top of the swing is equal to the gravitational potential energy.
At the bottom of the swing, the gravitational potential energy is converted into kinetic energy. The kinetic energy can be calculated using the formula:
KE = 0.5mv^2
Where:
v = speed at the bottom of the swing
Setting the initial gravitational potential energy equal to the final kinetic energy:
mgh = 0.5mv^2
Substitute in the given values:
m * 9.81 * 1.1 = 0.5 * m * v^2
Solve for v:
9.81 * 1.1 = 0.5 * v^2
v^2 = 10.791
v = √10.791
v ≈ 3.28 m/s
Therefore, the gymnast's speed at the bottom of the swing is approximately 3.28 m/s. Since this value is not provided in the answer choices, we can conclude that the closest option is 3.6 m/s.