Consider the spring - mass system, shown in Figure 4.2.4. consisting of two unit masses suspended from springs with spring constants 3 and 2, respectively. Assume that there is no damping in the system. Show that the displacements u1, and u2 of the masses from their respective equilibrium positions satisfy the equations (Using newton's second law) Solve the first of Eqs. (i) for u2 and substitute into the second equation, thereby obtaining the following fourth order equation for u1. Find the general solution of Eq. (ii).
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To show that the displacements u1 and u2 satisfy the equations using Newton's second law, let's consider each mass separately.
For mass m1:
The force acting on mass m1 is the sum of the force due to the spring constant k1 and the force due to the spring constant k2. Using Newton's second law, we can write:
m1 * (d^2u1/dt^2) = -k1*u1 - k2*(u1 - u2) (eq. i)
For mass m2:
The force acting on mass m2 is only due to the spring constant k2. Using Newton's second law, we can write:
m2 * (d^2u2/dt^2) = -k2*(u2 - u1) (eq. ii)
Now let's solve eq. i for u2:
Rearranging eq. i, we have:
m1 * (d^2u1/dt^2) = -k1*u1 - k2*u1 + k2*u2
m1 * (d^2u1/dt^2) = (-k1 - k2)*u1 + k2*u2
(d^2u1/dt^2) = (-k1 - k2)*(u1/m1) + (k2*(u2/m1)) (eq. iii)
Substituting this value of (d^2u1/dt^2) into eq. ii, we get:
m2 * ((-k1 - k2)*(u1/m1) + (k2*(u2/m1))/dt^2) = -k2*(u2 - u1)
m2 * ((-k1 - k2)*(u1/m1) + (k2*(u2/m1))/dt^2) = -k2*u2 + k2*u1
m2 * ((-k1 - k2)*(u1/m1) + (k2*(u2/m1))/dt^2) = u1*(-k2 + (-k1 - k2)*(u1/m1))
Simplifying this equation, we get a fourth-order equation for u1:
m2 * (d^4u1/dt^4) + (k2 + k1)* (d^2u1/dt^2) + (k1*k2*(u1/m1)) = 0 (eq. iv)
Now, to find the general solution of eq. iv, we can assume a solution of the form u1(t) = A * exp(rt), where A is an arbitrary constant and r is a constant to be determined.
Substituting this into eq. iv, we get:
m2 * r^4 * A * exp(rt) + (k2 + k1) * r^2 * A * exp(rt) + (k1*k2*(A * exp(rt))/m1 = 0
Factoring out A * exp(rt), we obtain:
(A * exp(rt)) * [m2 * r^4 + (k2 + k1) * r^2 + (k1*k2/m1)] = 0
Since A * exp(rt) cannot be zero, the expression in the square brackets must be zero:
m2 * r^4 + (k2 + k1) * r^2 + (k1*k2/m1) = 0
This is a quartic equation in r, which can be solved to find its roots. The roots of this quartic equation will give us the values of r, and hence the general solution of eq. iv.
Note: The solution to the quartic equation might be complex, which would lead to complex solutions for u1. In that case, the general solution of eq. iv would involve complex exponential functions.
To show that the displacements u1 and u2 satisfy the given equations, we can use Newton's second law.
Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. In equation form, it is expressed as F = ma.
In this case, for the first mass (m1) attached to the spring with a spring constant of 3, the equation of motion is given by:
F1 = m1 * a1
The force acting on m1 is due to the spring, which is proportional to the displacement u1 and is given by Hooke's Law as F1 = -3u1.
Substituting this into the equation of motion, we have:
-3u1 = m1 * a1 (i)
Similarly, for the second mass (m2) attached to the spring with a spring constant of 2, the equation of motion is:
F2 = m2 * a2
The force acting on m2 is due to the spring, which is proportional to the displacement u2 and is given by Hooke's Law as F2 = -2u2.
Substituting this into the equation of motion, we have:
-2u2 = m2 * a2 (ii)
Now, to solve equation (i) for u2, we can isolate u2 by dividing both sides of equation (ii) by -2. This gives:
u2 = -(-2/m2) * a2
u2 = (2/m2) * a2 (iii)
Substituting equation (iii) into equation (ii), we have:
-3u1 = m1 * a1
-3u1 = m1 * (-3/m2) * u2
-3u1 = (-3/m2) * m1 * (2/m2) * a2
-3u1 = (-6/m2^2) * m1 * a2 (iv)
Simplifying equation (iv), we obtain:
-3u1 = -(6/m2^2) * m1 * a2
u1 = (6/m2^2) * m1 * a2
Finally, substitute equation (iv) back into equation (i) to obtain:
m1 * a1 = (6/m2^2) * m1 * a2
Now, we have a fourth-order equation for u1. To find the general solution of this equation, we need to solve it by applying the appropriate initial conditions or boundary conditions.