The world record for revolutions per minute for an ice skater is 308 RPM, held by Natalia Kanounnikova. You can see a video of the spin here.
When she starts the spin and her leg is extended, she's going somewhere between 1-2 revolutions per second. Let's average this and say her initial angular velocity is 1.5 revolutions/sec. She has to do work to move her leg in and get herself to speed up. To see this, calculate the ratio of her final rotational kinetic energy to her initial rotational kinetic energy (it's >1 so she had to do some work).
3.42
how have u solved it ?? pls tell in details !! @ anonymous
To calculate the ratio of Natalia Kanounnikova's final rotational kinetic energy to her initial rotational kinetic energy, we need to know the angular velocity, moment of inertia, and the definition of rotational kinetic energy.
Rotational kinetic energy (K_rot) is given by the equation:
K_rot = (1/2) * I * ω^2
where I is the moment of inertia and ω is the angular velocity.
Since we are given the initial angular velocity, let's assume it to be ω_initial = 1.5 revolutions/sec. However, we need to convert revolutions/sec to radians/sec to be consistent with the unit of angular velocity. Since 1 revolution is equal to 2π radians, the conversion factor is 2π radians/revolution. Therefore:
ω_initial = 1.5 revolutions/sec * 2π radians/revolution = 9π radians/sec
Next, we need to determine the final angular velocity ω_final. Given that Natalia performs 308 RPM (revolutions per minute) at some point during the spin, we can convert this to revolutions per second:
ω_final = 308 revolutions/minute * (1 minute/60 seconds) = 5.1333... revolutions/second
Converting this final angular velocity to radians/second:
ω_final = 5.1333... revolutions/second * 2π radians/revolution = 32.25π radians/second
Now, we can compare the ratio of final rotational kinetic energy to initial rotational kinetic energy by using the formula:
Ratio = [K_rot(final)] / [K_rot(initial)]
Since only the angular velocity changes, the moment of inertia can be assumed to remain constant.
Therefore, the ratio becomes:
Ratio = [(1/2) * I * ω_final^2] / [(1/2) * I * ω_initial^2]
Simplifying,
Ratio = (ω_final^2) / (ω_initial^2)
Substituting the values,
Ratio = ((32.25π)^2) / ((9π)^2)
Ratio = (1035.5625 * π^2) / (81π^2)
Simplifying further,
Ratio = 1035.5625 / 81
Therefore, the ratio of Natalia Kanounnikova's final rotational kinetic energy to her initial rotational kinetic energy is approximately 12.78.
Since the ratio is greater than 1, we can conclude that Natalia had to do some work to move her leg in and increase her angular velocity. This work is necessary to overcome the rotational inertia and accelerate her body while spinning.