A mass hanging from the ceiling

[Lisa...]

Hey there!

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?

 

[Astronuc]

Is u(t) velocity (speed), i.e. does u(t) = \dot{x}(t), or is u(t) = displacement (from equilibrium), which is the difference in position?

Based on u(0) = A, it would appear to be displacement.

 

[Lisa...]

Yes, it is the displacement... I forgot to add that... sorry!

 

[Astronuc]

Well u(t) = x(t) - x_o, i.e. displacement is simply the difference between two positions or locations. u(t) is the displacement, x(t) is the position at time t, and x_o is the initial position, which could be zero in some reference frame, e.g. position of spring unloaded or its equilibrium position - where kx = mg for example.

Now, since u(0) is not zero, the solution for x(t) cannot be sin \omegat. because sin (0) = 0, and the initial displacement is nonzero.

The velocity is simply the rate of change of position, \frac{dx(t)}{dt}, and since u(t) = x(t) - x_o, and du/dt = dx/dt.

As for potential energy - I expect it refers to mechanical potential energy which is related to the spring, or rather spring's displacement. One is given,
U(X)= -integral F(x) dx, and F= -kx

Maximum kinetic energy obviously occurs where the velocity is maximum.

See some references on simple harmonic motion -

http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html

http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html

http://theory.uwinnipeg.ca/physics/shm/node1.html - SHM