A

Step 1: The tension in the string when the mass rotates is given by the formula: T = m r \omega^2, where m is the mass, r is the length of the string, and \omega is the angular velocity.
Step 2: Initially, the tension is T_0 = m r_0 \omega_0^2.
Step 3: When the length of the string and angular velocity are both doubled, we have:

• New length of the string: r = 2r_0
• New angular velocity: \omega = 2\omega_0

Step 4: Substitute these values into the tension formula:
T_new = m (2r_0) (2\omega_0)^2 = m (2r_0) (4\omega_0^2) = 8 m r_0 \omega_0^2 = 8 T_0.
Therefore, the new tension in the string is 8T_0.