An ice dancer with her arms stretched out starts into a spin with an angular velocity of 1.00 rad/s. Her moment of inertia with her arms stretched out is 2.48 kg m2. What is the increase in her rotational kinetic energy when she pulls in her arms to make her moment of inertia 1.40 kg m2?
Question
An ice dancer with her arms stretched out starts into a spin with an angular velocity of 1.00 rad/s. Her moment of inertia with her arms stretched out is 2.48 kg m2. What is the increase in her rotational kinetic energy when she pulls in her arms to make her moment of inertia 1.40 kg m2?
Solution
The rotational kinetic energy of an object can be calculated using the formula:
KE = 0.5 * I * ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
First, calculate the initial kinetic energy when her arms are stretched out:
KE_initial = 0.5 * 2.48 kg m^2 * (1.00 rad/s)^2 = 1.24 J
Next, calculate the final kinetic energy when she pulls in her arms:
KE_final = 0.5 * 1.40 kg m^2 * (1.00 rad/s)^2 = 0.70 J
Finally, subtract the initial kinetic energy from the final kinetic energy to find the increase in kinetic energy:
ΔKE = KE_final - KE_initial = 0.70 J - 1.24 J = -0.54 J
However, energy cannot be negative. The negative sign indicates that the kinetic energy has actually decreased, not increased. This makes sense because by pulling her arms in, the ice dancer decreases her moment of inertia, which decreases her kinetic energy if her angular velocity remains constant. So, there is no increase in her rotational kinetic energy, it actually decreases by 0.54 J.
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