A mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed

Question

A mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity $\omega_0$. If the length of the string and angular velocity are doubled, the tension in the string which was initially $T_0$ is now:

Accepted Answer

1. **Identify the Force:** For a mass moving in a horizontal circle on a frictionless surface, the tension ($T$) in the string provides the necessary centripetal force ($F_c$) [Source 61, 66].
2. **Formula:** The centripetal force in terms of angular velocity ($\omega$) and radius ($r$) is given by $F_c = m \omega^2 r$. Therefore, the initial tension is:
   $$ T_0 = m \omega_0^2 l $$
   (where $l$ is the initial length of the string) [Source 42, 61].
3. **New Conditions:**
   - New length, $l' = 2l$
   - New angular velocity, $\omega' = 2\omega_0$
4. **Calculate New Tension ($T'$):**
   $$ T' = m (\omega')^2 l' $$
   $$ T' = m (2\omega_0)^2 (2l) $$
   $$ T' = m (4\omega_0^2) (2l) $$
   $$ T' = 8 (m \omega_0^2 l) $$
   $$ T' = 8 T_0 $$

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