# natural period of vibration for the system - positive and negative work

A 10kg block is suspended from a cable wrapped around a 5kg disk, as shown in the figure. If the spring has a stiffness constant of k=200N/m, determine the natural period of vibration for the system

The power energy of the system is given by

Why is the work produced by the spring positive and the work produced by the weight negative?

Since the spring is opposing the motion and the weight of the block "helps" the motion, I thought the spring did negative work and the weight did positive work.

When you compress the spring by $$dx$$ the energy stored in the spring is given by:

$$E_s = \frac{1}{2} k \cdot (dx)^2$$

By raising to the square power, the energy quantity becomes positive and that means that energy is stored in the system. I.e. however you compress the spring energy is always stored in the system.

On the other hand, when you displace a mass by the same amount dx, the energy in the system is not the same in either direction. Going up means that you are storing energy (in this convention positive), while going down means that the potential energy of the system lowers (therefore negative). This is shown by the :

$$E_p = -m\cdot g \cdot dx$$

Note: the acceleration of gravity in the above formula should have a negative sign (if x is positive upwards).

• Nice, Got it. Thanks Feb 9 at 19:30