A gymnast is swinging on a high bar. The distance between his waist and the bar is 1.15 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.
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To find the speed of the gymnast at the bottom of the swing, we can use the principle of conservation of mechanical energy. At the top of the swing, when the speed is momentarily zero, the potential energy is at its maximum. At the bottom of the swing, when the gymnast reaches the lowest point, the potential energy is zero but the kinetic energy is at its maximum.
The formula for potential energy is given by:
Potential energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
Since the distance between the waist and the bar is given as 1.15 m, the height (h) is equal to 1.15 m.
At the top of the swing, the potential energy is maximum, so it can be written as:
PE_top = m * g * h
At the bottom of the swing, the potential energy is zero, so it can be written as:
PE_bottom = 0
The formula for kinetic energy is given by:
Kinetic energy (KE) = (1/2) * mass (m) * velocity^2
Since the gymnast's mass is not given, we can cancel out the mass by taking ratios of the potential and kinetic energies.
Therefore, we have:
PE_top / PE_bottom = KE_top / KE_bottom
Substituting the values, we get:
(m * g * h) / 0 = (1/2) * m * v^2
Simplifying the equation gives:
2 * g * h = v^2
Now, we can solve for the speed (v) at the bottom of the swing by taking the square root of both sides:
v = √(2 * g * h)
Given that the acceleration due to gravity (g) is approximately 9.8 m/s^2 and the height (h) is 1.15 m, we can plug in these values to calculate the speed (v) at the bottom of the swing.
v = √(2 * 9.8 * 1.15)
v ≈ √(22.54)
v ≈ 4.74 m/s
Therefore, the speed of the gymnast at the bottom of the swing is approximately 4.74 m/s.