Question

(a) Evaluate $\int (y+yzcos(xyz))dx+(x^2+xzcos(xyz))dy+(z+xycos(xyz))dz$ along the elipse $x=2cos(\Theta ), y=3sin(\Theta ), z=1, 0\leq \Theta \leq 2\pi$

(b) A mass that weights 12 [lb] stretches a spring 6 [in]. The system is acted on by an external force of [lb]. At $t=0$ he mass is pulled up 5 [in] above its equilibrium position and then released. Write an initial value problem describing this spring-mass system. Determine the position of the massat any time. Determine the first two times at which the velocity of the mass is zero.

Answer 1

(b) In a standard form, we have

m u'' + _$\gamma$ u' + k u = F (t)                                                       { eq.1 }

where, m = mass of an object

_$\gamma$ = damping coefficient

k = spring constant

Weight of an object is given by ; w = m g $\Leftrightarrow$ m = [(12 lb) / (32 ft/s²)]

m = (3/8) slugs

A spring constant is given by ;   k = F / s = [(12 lb) / (6 in)]

k = [(12 lb) / (0.5 ft)]

k = 24 lb/ft

Since, there is no damping. The mass is pulled up 5 in above its equilibrium position and then released.

u (t) = 5 in

u (0) = (5 / 12) ft

Inserting the value of 'm', ' _$\gamma$ ', 'k' and F(t) in eq.1 & we get

(3/8) u'' + (0) u' + (24) u = (6) sin (8 t)

(3/8) u'' + (24) u = (6) sin (8 t)

(3) u'' + (192) u = (48) sin (8 t)

u'' + (64) u = (16) sin (8 t)

The characteristic equation will be given by -

m² + (64) = 0

m = _$\pm$ 8 i

And

u_c = c₁ cos (8 t) + c₂ sin (8 t)