Answer :
Answer:
(a). The angular speed of the rotating is 33.8 rad/s.
(b). The speed of its center is 2.7 m/s.
Explanation:
Given that,
Radius = 8.00 cm
Mass = 0.180 kg
Height = 75.0 m
We need to calculate the angular speed of the rotating
Using conservation of energy
[tex]\dfrac{1}{2}I\omega_{1}^2+\dfrac{1}{2}mv_{1}^{2}+mgh_{1}=\dfrac{1}{2}I\omega_{2}^2+\dfrac{1}{2}mv_{2}^{2}+mgh_{2}[/tex]
Here, initial velocity and angular velocity are equal to zero.
[tex]mgh_{1}=\dfrac{1}{2}I\omega_{2}^2+\dfrac{1}{2}mv_{2}^{2}+mgh_{2}[/tex]
[tex]mg(h_{1}-h_{2})=\dfrac{1}{2}I\omega_{2}^2+\dfrac{1}{2}mv_{2}^{2}[/tex]
[tex]mgH=\dfrac{1}{2}mr^2\omega_{2}^2+\dfrac{1}{2}m(r\omega_{2})^2[/tex]
Here, [tex]H = h_{1}-h_{2}[/tex]
[tex]gH=r^2\omega_{2}^2[/tex]
[tex]\omega_{2}^2=\dfrac{gH}{r^2}[/tex]
[tex]\omega_{2}=\sqrt{\dfrac{9.8\times75.0\times10^{-2}}{(8.00\times10^{-2})^2}}[/tex]
[tex]\omega_{2}=33.8\ rad/s[/tex]
The angular speed of the rotating is 33.8 rad/s.
(b). We need to calculate the speed of its center
Using formula of speed
[tex]v=r\omega[/tex]
Put the value into the formula
[tex]v=8.00\times10^{-2}\times33.8[/tex]
[tex]v=2.7\ m/s[/tex]
Hence, (a). The angular speed of the rotating is 33.8 rad/s.
(b). The speed of its center is 2.7 m/s.
Answer
given,
radius of the hoop = 8 cm = 0.08 m
mass = 0.18 kg
hoop descended to 75 cm = 0.75 m
conservation of energy
[tex]\dfrac{1}{2}I_1\omega_1^2 + \dfrac{1}{2}mv_1^2 + mgh_1 = \dfrac{1}{2}I_2\omega_2^2 + \dfrac{1}{2}mv_2^2 + mgh_2[/tex]
here [tex]\omega_1 = v_1 = 0[/tex]
[tex] mgh_1= \dfrac{1}{2}I_2\omega_2^2 + \dfrac{1}{2}mv_2^2 + mgh_2[/tex]
[tex] mg(h_2-h_1)= \dfrac{1}{2}I_2\omega_2^2 + \dfrac{1}{2}mv_2^2 [/tex]
[tex]h_2 - h_1 = h[/tex]
[tex] mgh= \dfrac{1}{2}I_2\omega_2^2 + \dfrac{1}{2}mv_2^2 [/tex]
[tex] mgh= \dfrac{1}{2}mr^2\omega_2^2 + \dfrac{1}{2}m(r\omega_2)^2 [/tex]
[tex]r^2\omega_2^2 = gh[/tex]
[tex]\omega_2 = \dfrac{\sqrt{gh}}{r}[/tex]
[tex]\omega_2 = \dfrac{\sqrt{9.8\times 0.75}}{0.08}[/tex]
[tex]\omega_2 = 33.89\ rad/s[/tex]
b) [tex]v = r \omega[/tex]
= 33.89 x 0.08
v = 2.704 m/s