(Solved): 2. (a) A complex dynamic mechanism has an angular displacement equation as the following: sin()+t ...

2. (a) A complex dynamic mechanism has an angular displacement equation as the following: $\sin (?)+t^2{?}^3?\cos t=2?$. Find the expression for the angular velocity $\stackrel{?}{?}$. [7 marks] (b) A motor under load generates heat internally at a constant rate of $\mathrm{H}$ and radiates heat in accordance with Newton's law of cooling at a rate $k?$ where $k$ is a constant and $?$ is the temperature difference. The tempelature of the motor is given by the following differential equation: $\frac{d?}{dt}=H?k?$ find the time-dependant equation of the temperature $?(t)$. [10 marks] (c) Given that $\frac{?f}{?x}=2x?y\sin (xy)+3x^2{y}^2$ and $\frac{?f}{?y}=?x\sin (xy)+2x^{3}y?1$, find $f(x,y)$.