Prove that in a Simple Harmonic Motion, the acceleration is proportional to the distance of the body from equilibrium, and it's always directed towards its equilibrium position,
To prove that in Simple Harmonic Motion (SHM), the acceleration of a body is proportional to its distance from the equilibrium position and directed towards that position, we can start by considering the general properties of SHM.
### Derivation:
1. **Definition of SHM**:
In SHM, the displacement \( x \) of the body from its equilibrium position is a function of time that can typically be represented as:
\[
x(t) = A \cos(\omega t + \phi)
\]
where:
- \( A \) is the amplitude (maximum displacement),
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.
2. **Velocity**:
The velocity \( v \) of the particle can be found by differentiating the displacement \( x(t) \) with respect to time \( t \):
\[
v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi)
\]
3. **Acceleration**:
To find the acceleration \( a \), differentiate the velocity \( v(t) \):
\[
a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}
\]
Differentiating \( x(t) \):
\[
a(t) = \frac{d}{dt}[-A\omega \sin(\omega t + \phi)]
= -A\omega^2 \cos(\omega t + \phi)
\]
Now, using the original displacement equation \( x(t) = A \cos(\omega t + \phi) \):
\[
a(t) = -\omega^2 x(t)
\]
4. **Proportionality and Direction**:
The equation \( a(t) = -\omega^2 x(t) \) shows that:
- The acceleration \( a \) is proportional to the displacement \( x \) from the equilibrium position (since \( -\omega^2 \) is a constant).
- The negative sign indicates that the acceleration is directed towards the equilibrium position when the displacement \( x \) is positive (the body is on one side of the equilibrium) and vice versa.
### Conclusion:
Thus, we have shown that the acceleration \( a \) in SHM is given by:
\[
a = -\omega^2 x
\]
This confirms that:
- The acceleration is directly proportional to the distance \( x \) from the equilibrium position,
- The acceleration is always directed towards the equilibrium position (since it is negative when \( x \) is positive, and positive when \( x \) is negative).
Therefore, both statements regarding acceleration in Simple Harmonic Motion are proven to be true.