Here is the missing information.
An exhausted bicyclist pedal somewhat erraticaly when exercising on a static bicycle. The angular velocity of the wheels takes the equation ω(t)=at − bsin(ct) for t≥ 0, where t represents time (measured in seconds), a = 0.500 rad/s2 , b = 0.250 rad/s and c = 2.00 rad/s .
Answer:
0.793 rad
Explanation:
From the given question:
The angular velocity of the wheel is expressed by the equation:
[tex]\omega (t) =\dfrac{d\theta}{dt}[/tex]
The angular velocity of the wheels takes the description of the equation ω(t)=at−bsin(ct)
SO;
[tex]\dfrac{d \theta}{dt} = at - b \ sin \ ct[/tex]
dθ = at dt - (b sin ct) dt
Taking the integral of the above equation; we have:
[tex]\int \limits^{\theta}_{0} \ d \theta = \int \limits ^{t=2}_{0} at \ dt - (b \ sin \ ct) \dt[/tex]
[tex][\theta] ^{\theta}_{0} = a \bigg [\dfrac{t^2}{2} \bigg]^2_0 - \bigg[ -\dfrac{b}{c} \ cos \ ct \bigg] ^2_0[/tex]
where;
a = 0.500 rad/s2 ,
b = 0.250 rad/s and
c = 2.00 rad/s
[tex]\theta = (0.500 \ rad/s^2 ) \bigg [\dfrac{(2s)^2}{2} \bigg] - \bigg[ -\dfrac{0.250 \ rad/s}{2.00 \ rad/s} \ cos \ (2.00 \ rad/s )( 2.00 \ s) \bigg] - \bigg [ \dfrac{0.250 \ rad/s}{2.00 \ rad/s}\bigg ] cos 0^0[/tex]
[tex]\mathbf{\theta = 0.793 \ rad}[/tex]
Hence, the angular displacement after two seconds = 0.793 rad
