What is the ratio of the ice dancer's moment of inertia while she is spinning with her hands stretched out to when her hands are folded in?
The ratio of the ice dancer's moment of inertia when her hands are stretched out to when her hands are folded in is approximately 3.34.
Calculating Moment of Inertia Ratio
Moment of Inertia: The moment of inertia of an object is a measure of its resistance to changes in its rotational motion. It depends on the distribution of mass around the axis of rotation.
The ice dancer makes a spin move in two modes - one with her hands stretched out and one with her hands folded in. In the first mode (hands stretched out), she spins at 2.9 revolutions per second, while in the second mode (hands folded in), she spins at 9.7 revolutions per second.
Using the law of conservation of angular momentum, we can calculate the ratio of the ice dancer's moment of inertia in these two modes. The angular momentum is given by the equation: L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity.
Let's denote the moment of inertia in the first mode as I1 and in the second mode as I2. We are given that the angular velocities in mode 1 and mode 2 are 2.9 and 9.7 revolutions per second, respectively.
The ratio of angular velocities is given as ω1/ω2 = I2/I1. Substituting the values, we get 2.9/9.7 = I2/I1. Solving for the moment of inertia ratio, we find I1/I2 = 9.7/2.9 ≈ 3.34.
Therefore, the ratio of the ice dancer's moment of inertia while her hands are stretched out to when her hands are folded in is approximately 3.34. This indicates that her moment of inertia is larger when her hands are stretched out compared to when they are folded in.