Conditions for Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM) is a special type of oscillatory motion where an object moves back and forth around an equilibrium position.
Restoring Force Proportional to Displacement
- For SHM to occur, the system must have a restoring force that:
- Acts in the opposite direction to the displacement.
- Is directly proportional to the displacement from the equilibrium position.
- This relationship can be expressed mathematically as: $$
F = -kx
$$ where:- $F$ is the restoring force.
- $k$ is a constant (often called the spring constant in a spring-mass system).
- $x$is the displacement from the equilibrium position.
The negative sign indicates that the force acts in the opposite direction to the displacement.
Example- In a mass-spring system, when the mass is displaced to the right, the spring exerts a force to the left, trying to bring the mass back to equilibrium.
- This force is proportional to how far the mass is stretched or compressed.
Acceleration in SHM
- The restoring force causes the object to accelerate back towards the equilibrium position.
- Using Newton’s second law, $F = ma$, we can express the acceleration as: $$
a = -\frac{k}{m}x
$$ - This shows that the acceleration is also proportional to the displacement and acts in the opposite direction.
Defining Equation for SHM: $a = -\omega^2 x$
The defining equation for SHM is:
$$
a = -\omega^2 x
$$
where:
- $a$ is the acceleration.
- $x$ is the displacement.
- $\omega$ (omega) is the angular frequency, a constant that characterizes the system.
The negative sign indicates that the acceleration is always directed opposite to the displacement, ensuring the object is pulled back towards equilibrium.
Why $a = -\omega^2 x$?
- The term $\omega^2$ is derived from the system’s properties.
