A skater spins with an angular speed of 12.4 rad/s with her arms outstretched. She lowers her arms, decreasing her moment of inertia by a factor of 8.00. Ignoring friction on the skates, what will be the skater's final angular speed?
I wonder what makes this a "world" problem. The same laws of pysics apply to the nationals and regional competitions, as well as practice sessions. Do you mean a WORD problem?
When friction can be ignored, which is nearly true in figure skating if you are skating on your edge properly, then angular momentum is conserved (wh8ich means it remains the same) during spins. The angular momentum is the product of the moment of inertia and the spin rate, in radians per second.
If the skater's moment of inertia decreases by a factor of eight (which is very hard to do, but possible), then the spin rate must increase by the same factor in order to keep the angular momentum the same.
So, multiply 12.4 by 8 for the answer, in radians per second.
To find the skater's final angular speed, we need to apply the law of conservation of angular momentum. According to this law, the initial angular momentum of a system remains constant unless an external torque acts upon it.
Initially, the skater's angular momentum is given by:
L_initial = I_initial * ω_initial
Where:
L_initial is the initial angular momentum,
I_initial is the initial moment of inertia, and
ω_initial is the initial angular speed.
Let's denote the final angular momentum, moment of inertia, and angular speed with L_final, I_final, and ω_final, respectively.
Based on the law of conservation of angular momentum, we have:
L_initial = L_final
Substituting the expressions for angular momentum:
I_initial * ω_initial = I_final * ω_final
Given the angular speed of 12.4 rad/s and the factor by which the moment of inertia decreases, we can relate the initial and final moments of inertia as:
I_final = I_initial / 8.00
Substituting this expression into our conservation of angular momentum equation:
I_initial * ω_initial = (I_initial / 8.00) * ω_final
Simplifying the equation by canceling the I_initial terms:
ω_initial = ω_final / 8.00
We can rearrange this equation to solve for ω_final:
ω_final = ω_initial * 8.00
Substituting the given value of ω_initial:
ω_final = 12.4 rad/s * 8.00
Calculating the final angular speed:
ω_final = 99.2 rad/s
Therefore, the skater's final angular speed will be 99.2 rad/s.