Learning Goal To derive the formulas for the major characteristics of motion as

Question

Learning Goal To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions A block of mass m is attached to a spring whose spring constant is k. The other end of the spring is fixed so that when the spring is unstretched, the mass is located at z (Figure 1). Assume that the +x direction is to the right. The mass is now pulled to the right a distance A beyond the equilibrium position and released, at time t 0, with zero initial velocity Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is alt)t) and its solution, which provides the equation for r(t),i r(t) Acos(w)

Explanation / Answer

Part A -

For the block to come back to its original position, first time -

we should have the condition at time t1 -

cos[sqrt(k/m)*t1] = 0

=> sqrt(k/m)*t1 = pi/2

=> t1 = (pi/2)*[1 / sqrt(k/m)] = (pi/2)*sqrt(m/k)

Part B -

The expression for the position of the body is -

x(t) = A*cos[sqrt(k/m)*t]

to determine the velocity, differentiate with respect to time t -

we have -

v(t) = dx(t) / dt = A*sqrt(k/m)*-sin[sqrt(k/m)*t] = - A*sqrt(k/m)*sin[sqrt(k/m)*t]

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