# A mass hanging from the ceiling

1. Nov 6, 2005

### 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?

2. Nov 6, 2005

### Astronuc

Staff Emeritus
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.

3. Nov 6, 2005

### Lisa...

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

4. Nov 6, 2005

### Astronuc

Staff Emeritus
Well u(t) = x(t) - xo, 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 xo 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 $\omega$t. 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) - xo, 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