# If the mean temperature of the earth increased significantly, how this would affect the earth's angular velocity?

I was wondering, if somehow the temperature of the earth increased singificantly, how would this affect the period of rotation and angular velocity. I mean the earth would expand and this would also affect the moment of inertia right? How could I connect this increase in temperature with the period?

• Are you asking about an increase in temperature for the bulk material of the earth, or an increase in temperature of the surface of the earth? Dec 17, 2020 at 4:36

Assuming that angular momentum was conserved throughout this expansion process then

$$I_i \omega_i = I_f \omega_f$$

where $$I_i$$, $$I_f$$ are the initial and final moments of inertia, and $$\omega_i$$, $$\omega_f$$ are the initial and final angular velocities. Now since

$$\omega = \frac{ 2 \pi}{T}$$

where $$T$$ is the period for one revolution, you can write

$$2 \pi \frac{I_i}{T_i} = 2 \pi \frac{I_f}{T_f}$$

or

$$\frac{I_i}{T_i} = \frac{I_f}{T_f}$$

which relates the system before and after the temperature change causing this expansion.

• but you have not taken into account the amount of energy change.. Dec 17, 2020 at 8:35
• I tried to think of a way to work that into the equations but it would take more time than I have. It is involved. I will get back to it soon though. Dec 17, 2020 at 8:40
• Is it wrong to asume that the initial energy is equal to the final energy, where E_i=(1/2)*I_iw_i^2 and E_f=(1/2)*I_fw_f^2 - C_pMΔT ? Dec 17, 2020 at 9:49
• in this case I mean that the earh's temperature has decreased Dec 17, 2020 at 9:50