A figure skater spins on the ice, holding two weights a distance d from the axis about which he is rotating. At a certain time, he pulls these weights toward his body as shown, causing his rotational inertia to decrease by a factor of 2. How does this affect the total kinetic energy of the skater?
To understand how the change in rotational inertia affects the total kinetic energy of the skater, we can analyze the concept of conservation of angular momentum.
Angular momentum is the product of rotational inertia and rotational velocity. According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless an external torque acts on it.
When the figure skater pulls the weights towards his body, he decreases his rotational inertia by a factor of 2. Since his total angular momentum must remain constant, the decrease in rotational inertia must be compensated by an increase in rotational velocity.
Considering the equation for angular momentum (L = Iω), where L is the angular momentum, I is the rotational inertia, and ω is the angular velocity, we can deduce that if I decreases by a factor of 2, ω must increase by a factor of 2 to keep the angular momentum constant.
Now, let's consider the relationship between kinetic energy and angular velocity. The formula for rotational kinetic energy is K = (1/2)Iω^2. As we already determined that ω doubled, the new kinetic energy can be expressed as K' = (1/2)(I/2)(2ω)^2 = (1/2)(I/2)(4ω^2) = 2(1/2)Iω^2 = 2K.
Therefore, pulling the weights towards his body and reducing the figure skater's rotational inertia by a factor of 2 increases his rotational velocity by a factor of 2, leading to a doubling of his kinetic energy.