a) 6.25 rad/s
The law of conservation of angular momentum states that the angular momentum must be conserved.
The angular momentum is given by:
[tex]L=I\omega[/tex]
where
I is the moment of inertia
[tex]\omega[/tex] is the angular speed
Since the angular momentum must be conserved, we can write
[tex]L_1 = L_2\\I_1 \omega_1 = I_2 \omega_2[/tex]
where we have
[tex]I_1 = 2.25 kg m^2[/tex] is the initial moment of inertia
[tex]\omega_1 = 5.00 rad/s[/tex] is the initial angular speed
[tex]I_2 = 2.25 kg m^2[/tex] is the final moment of inertia
[tex]\omega_2[/tex] is the final angular speed
Solving for [tex]\omega_2[/tex], we find
[tex]\omega_2 = \frac{I_1 \omega_1}{I_2}=\frac{(2.25 kg m^2)(5.00 rad/s)}{1.80 kg m^2}=6.25 rad/s[/tex]
b) 28.1 J and 35.2 J
The rotational kinetic energy is given by
[tex]K=\frac{1}{2}I\omega^2[/tex]
where
I is the moment of inertia
[tex]\omega[/tex] is the angular speed
Applying the formula, we have:
- Initial kinetic energy:
[tex]K=\frac{1}{2}(2.25 kg m^2)(5.00 rad/s)^2=28.1 J[/tex]
- Final kinetic energy:
[tex]K=\frac{1}{2}(1.80 kg m^2)(6.25 rad/s)^2=35.2 J[/tex]
