In this lesson, we will learn:
- Vibrating and oscillating systems
- Hooke’s law
- How to find the amount of energy stored in a vibrating system?
- There are many examples of vibrating objects around us, an object attached to the end of spring, guitar strings, ruler held firmly at the end of the table.
- During vibration (oscillation) the object moves back and forth about a fixed position called “equilibrium position”.
- Let’s consider a mass vibrating at the end of a uniform spring.;
(a) The mass is at its equilibrium position, there is no force exerted.
(b) The mass is oscillating; the restoring force tries to pull the mass back to its equilibrium position.
(c) The mass is oscillating; the restoring force tries to push the mass back to its equilibrium position.
- The magnitude of the restoring force is directly proportional to the displacement ;
- The direction of the restoring force is always opposite to the displacement which is indicated by a minus sign in the equation.
: Force exerted by the spring on the mass
: Spring constant
- As we know the mechanical energy of a system is the sum of kinetic and potential energies.
- In the case of the spring-mass system, the potential energy would be in the form of elastic potential energy in the spring which is calculated using the following equation;
Therefore; the mechanical energy of the system is;
(a) At the extreme points where the mass stops momentarily to change the direction;
(amplitude, maximum displacement)
(b) At the equilibrium position the mass moves with maximum velocity;
From (1) and (2);
(c) At intermediate points, the energy is a part kinetic and part potential;
- From the above equation we can find velocity as the function of position;
This gives the velocity of the object at any position.