According to Newton\'s Second Law of Motion (see Section 4.8 for a further discu

Question

According to Newton's Second Law of Motion (see Section 4.8 for a further discussion), if an object with mass m is suspended from a spring attached to the ceiling, then the motion of the object is governed by the equation mx + ax + kx = 0. In this equation, x(t) is object s distance from its rest (equilibrium) position at lime t seconds. Distance below the rest position is considered positive, while distance above is considered negative. Also, a is a constant representing the air resistance and/or friction present in the system and k is the spring constant describing the "give" in the spring. (Recall that mass = weight/g, where g is the gravitational constant-32 ft/sec^2 or 9.8 m/sec^2.) Use this equation to do Problems 4-7. An object of mass 4 slugs (= 128 lbs/32 ft/sec^2) is suspended from a spring having spring constant 64 lbs/ft. The object is started in motion, with no initial velocity, by pulling it 6 inches (watch the units!) below the equilibrium position and then releasing it. If there is no air resistance, find a formula for the position of the object at any time t > 0.

Explanation / Answer

Since , there is no air resistance , the equation of motion becomes

mx'' + kx = 0

Analysing the initial situation , the object is suspended freely to and extended position

Plugging the given values of variables

=> 4x'' + 64x = 0

=> x'' + 16x = 0

Let x = ert

=> ert( r2 + 16 ) = 0

=> r2 + 16 = 0

=> r = +4i and -4i

Therefore , the solution of the equation x'' + 16x = 0 becomes

=> x(t) = c1 cos(4t) + c2 sin(4t)

=> x'(t) = -4c1 sin(4t) + 4c2 cos(4t)

Plugging the initial conditions x(0) = 0.5ft and x'(0) = 0

=> x(0) = c1 = 0.5 and x'(0) = 4c2 = 0 or c2 = 0

Plugging the above values of c1 and c2

=> x(t) = 0.5 cos(4t)

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