The total number of revolutions made by the wheel is 514
Explanation:
We can solve the problem by applying the suvat equations for rotational motion, to the two different parts of the motion.
During the first part (acceleration), we have:
[tex]\omega_0 =0[/tex] (initial angular velocity)
[tex]\omega_1=58 rad/s[/tex] (final angular velocity)
[tex]t_1=10 s[/tex] (time)
So the angular displacement covered in this part is
[tex]\theta_1 = (\frac{\omega_0+\omega_1}{2})t_1 =(\frac{0+58}{2})(10)=290 rad[/tex]
In the second part, we have uniform (circular) motion, with constant angular velocity
[tex]\omega_2 = 58 rad/s[/tex]
for
t = 30 s
So the angular displacement in this part is
[tex]\theta_2 = \omega_2 t_2 = (58)(30)=1740 rad[/tex]
In the third part, we have a decelerated motion, with constant angular acceleration of
[tex]\alpha=-1.4 rad/s^2[/tex]
and initial angular velocity
[tex]\omega_2 = 58 rad/s[/tex]
while final angular velocity is
[tex]\omega_3 = 0[/tex]
So the angular displacement in this part is given by
[tex]\omega_3^2 - \omega_2^2 = 2\alpha \theta_3\\\theta_3 = \frac{\omega_3^3-\omega_2^2}{2\alpha}=\frac{0-58^2}{2(-1.4)}=1201 rad[/tex]
So the total angular displacement of the wheel is
[tex]\theta=290 + 1740 + 1201 = 3231 rad[/tex]
Converting into revolutions,
[tex]\theta=3231 rad \cdot \frac{1}{2\pi rad/rev}=514 rev[/tex]
Learn more about rotational motion:
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