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I'm having a few problems with a classical model of a mass hanging from the ceiling by a string (Yeah my teacher did not explain a hell of a lot on this subject). Could anyone of you please help me to solve them? I'd appreciate it a hell of a lot!

The force on the mass is given by Hooke's Law: F= -kx

At t=0 the mass has a displacement A. The movement of the mass is described by Newton's second law: F= ma = mx'' (x''= d^2x/dt^2).

Therefore the differential equation is d^2x/dt^2= -kx/m

Now x1(t)= C sin(wt) and x2(t)= C cos (wt) are two solutions that satisfy the differential equation.

Questions:

1) The boundary condition is given by u(0)=A. Determine which function, x1 or x2 describes the problem and determine the constant C.

2) Determine the velocity of the mass as a function of t.

The energy of the mass consists of two parts: the potential and kinetic energy. The potential energy of a mass in one dimension is given by:

U(X)= -integral F(x) dx

3) What is the potential energy of the mass? When is the potential energy at a maximum?

4) Determine the maximum kinetic energy the mass can have. Is the total energy of this system conserved?

5) What is the power of the mass?

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# A mass hanging from the ceiling

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