Step 1: Recall the formulas for linear speed and centripetal acceleration in uniform circular motion. The linear speed ($v$) is related to the angular velocity ($\omega$) and radius ($r$) by: $$v = r\omega$$ The centripetal acceleration ($a_c$) can be expressed in terms of linear speed and radius, or angular velocity and radius: $$a_c = \frac{v^2}{r} = r\omega^2$$ Step 2: Define the initial conditions. Let the initial angular velocity be $\omega_1$. Let the initial linear speed be $v_1 = V$. Let the initial radius be $r_1$. The initial centripetal acceleration is $a_{c1}$. From the formulas: $$V = r_1 \omega_1 \quad (1)$$ $$a_{c1} = r_1 \omega_1^2 \quad (2)$$ Step 3: Define the final conditions based on the problem statement. The angular velocity is doubled: $\omega_2 = 2\omega_1$. The centripetal acceleration is unchanged: $a_{c2} = a_{c1}$. Let the new linear speed be $v_2$. Let the new radius be $r_2$. From the formulas: $$v_2 = r_2 \omega_2 = r_2 (2\omega_1) \quad (3)$$ $$a_{c2} = r_2 \omega_2^2 = r_2 (2\omega_1)^2 = 4r_2 \omega_1^2 \quad (4)$$ Step 4: Use the condition that centripetal acceleration is unchanged to find the relationship between the radii. Since $a_{c1} = a_{c2}$: $$r_1 \omega_1^2 = 4r_2 \omega_1^2$$ Divide both sides by $\omega_1^2$ (assuming $\omega_1 \neq 0$): $$r_1 = 4r_2$$ This means the new radius is $r_2 = \frac{r_1}{4}$. Step 5: Substitute the relationship for the radius into the equation for the new speed. Substitute $r_2 = \frac{r_1}{4}$ into equation (3): $$v_2 = \left(\frac{r_1}{4}\right) (2\omega_1)$$ $$v_2 = \frac{2 r_1 \omega_1}{4}$$ $$v_2 = \frac{1}{2} (r_1 \omega_1)$$ Step 6: Substitute the initial speed $V$. From equation (1), we know that $V = r_1 \omega_1$. Substitute $V$ into the expression for $v_2$: $$v_2 = \frac{1}{2} V$$ The new speed will be $\boxed{\frac{V}{2}}$. That's 2 down. 3 left today — send the next one.