If an undamped spring-mass system with a mass that weighs 6 lb and a spring constant 1 lb/in is suddenly set in motion at t = 0 by an external force of 2 cos 7t lb, determine the position of the mass at any time. (Use g = 32 ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)

U(t) = (48/45)(cos 7t - cos 8t) seconds

Explanation:

Weight = 6lb

K = 1lb/in or 12lb/ft

U'(0) = 0 ft

U''(0) = 0 ft/s

F(t) = 2 cos 7t lb

We know that, w = mg

Thus, m = w/g = 6/32 = 3/16 lb²/ft

Thus, the initial value problem cam be written as;

3u'' + 192u = 48 cos 7t

Since, u'(0) = 0 ft and u''(0) = 0 ft/s, the corresponding homogenous equation is;

3u'' + 192u = 0

With a characteristic of ;

3p² + 192p = 0

So, p = ±√(192/3)i = ±√64i = ±8i

This is a pair of conjugate roots and thus the solution is;

u(t) = c1cos 8t + c2sin8t

And the particular solution can be written as;

Y(t) = A cos 7t + B sin 7t

Now, let's plug Y(t) into the initial value problem;

3Y''(t) + 192Y(t) = 2 cos 7t

3(-49Acos 7t - 49Bsin 7t) + 192(A cos 7t + B sin 7t) = 48 cos 7t

Thus;

-147ACos 7t - 147BSin 7t + 192A cos 7t + 192B sin 7t = 48 cos 7t ;

45A Cos 7t + 45B Sin 7t = 48 cos 7t

By inspection,A = 48/45 and B=0

Hence, the particular solution is;

Y(t) = (48/45)cos 7t

The general solution will now be;

U(t) = Uc(t) + Y(t)

U(t) = c1 cos8t + c2 sin8t + (48/45)cos 7t

Mow, let's find c1 and c2 from the conditions in the initial value problem.

Thus ;

At u(0) =0;

0 = c1 cos 0 + c2 sin0 + (48/45)cos 0

So, c1 = 48/45

Also, at u'(0) = 0;

0 = - 8c1 sin 0 + 8c2 cos 0 - (7)(48/45)sin 0

8c2 = 0 and c2 = 0

Thus, the general solution of u(t) is;

U(t) = (48/45)( cos 7t - cos 8t) seconds

Quickly moving neutrons coming out of the reaction are slowed down by water. The water heats up and turns into steam. The steam turns the turbine and produces electricity.

Explanation: Hope that helps :D