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 maximum speed with which any part of his body moves?
When the figure skater pulls the two weights towards his body, causing his rotational inertia to decrease by a factor of 2, the law of conservation of angular momentum comes into play. According to this law, the product of rotational inertia and angular velocity remains constant unless an external torque is applied.
Initially, the figure skater has a certain rotational inertia, I_init, and angular velocity, ω_init. After pulling the weights towards his body, his rotational inertia decreases to I_final = I_init/2.
To find out how this affects the maximum speed with which any part of his body moves, we need to analyze the relationship between rotational inertia and angular velocity.
Angular momentum (L) is defined as the product of rotational inertia (I) and angular velocity (ω): L = I * ω.
Since we have a decrease in rotational inertia, we'll need to determine how the angular velocity changes to maintain constant angular momentum.
Initial angular momentum (L_init) = I_init * ω_init
Final angular momentum (L_final) = I_final * ω_final
According to the law of conservation of angular momentum, L_init = L_final.
Therefore, I_init * ω_init = I_final * ω_final
Dividing both sides by I_final, we get:
ω_final = (I_init * ω_init) / I_final
Since I_final = I_init/2, we can substitute this into the equation:
ω_final = (I_init * ω_init) / (I_init/2)
ω_final = 2 * ω_init
This equation tells us that when the rotational inertia decreases by a factor of 2, the final angular velocity doubles compared to the initial angular velocity.
The maximum speed with which any part of the figure skater's body moves is directly proportional to the angular velocity. Therefore, as the final angular velocity doubles, the maximum speed with which any part of his body moves also doubles.
To understand how pulling the weights affects the maximum speed at which any part of the skater's body moves, we need to consider the concept of conservation of angular momentum. Angular momentum is the rotational equivalent of linear momentum and is defined as the product of rotational inertia (also known as moment of inertia) and angular velocity.
According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless an external torque is applied. In other words, the sum of the initial angular momentum and the angular momentum due to any external torques acting on the system is conserved.
In this scenario, the skater starts with a certain rotational inertia (I_initial) when he is spinning without pulling the weights. When he pulls the weights toward his body, his rotational inertia decreases by a factor of 2, resulting in a new rotational inertia (I_final = I_initial/2).
Since the skater is not applying any external torque to the system, the conservation of angular momentum tells us that the initial angular momentum is equal to the final angular momentum:
I_initial * ω_initial = I_final * ω_final
Here, ω_initial is the initial angular velocity (rate of rotation) of the skater, and ω_final is the final angular velocity.
Now, let's consider the maximum speed with which any part of the skater's body moves. This occurs when the skater's body is fully extended. At this point, the maximum linear speed (v) of any part of his body can be related to the angular velocity (ω) by the formula:
v = ω * r,
where r is the distance of the body part from the axis of rotation.
To determine the effect of changing rotational inertia on the maximum speed, we can compare the initial and final situations:
For the initial situation:
v_initial = ω_initial * r
For the final situation:
v_final = ω_final * r
Since we know that I_final = I_initial/2, we can rewrite ω_initial and ω_final in terms of I_initial and I_final:
ω_initial = L_initial / I_initial
ω_final = L_initial / I_final,
where L_initial represents the initial angular momentum.
Substituting these values in the equations for v_initial and v_final, we have:
v_initial = (L_initial/I_initial) * r
v_final = (L_initial/(I_initial/2)) * r.
Now, let's compare the two equations:
v_final/v_initial = ((L_initial/(I_initial/2)) * r) / ((L_initial/I_initial) * r)
= (2 * L_initial * r) / (L_initial * r)
= 2.
This means that the final maximum speed (v_final) is 2 times the initial maximum speed (v_initial). In other words, pulling the weights toward his body decreases the rotational inertia by a factor of 2, resulting in the skater being able to increase his maximum speed by a factor of 2.
Therefore, pulling the weights toward his body increases the maximum speed with which any part of his body moves by a factor of 2.