Physics help

Question

5. Explain why the motion of a simple pendulum is not exactly the same as SHM.

Answer 1

The main defining characteristic of Simple Harmonic motion is that the restoring force that pulls the object back toward equilibrium is directly, linearly proportional to the distance it is displaced (i.e. F = k x).

 

In the case of a pendulum, the displacement is measured as an angle θ, and the restoring force is gravity, which acts at the same angle relative to the string that suspends the object. The component of gravity which is perpendicular to the string (along the direction of motion) is mg sinθ.

 

This does not satisfy the condition of being directly, linearly proportional. For small angles θ (less than ~15º), it is close to a straight line, so we can apply some of the principles of SHM, but it is not truly SHM.

Answer 2

The equation of motion for simple harmonic motion (SHM) is:

 

x'' + ω²x = 0

 

Where ω is the angular frequency of the oscillation.  For a pendulum, the equation of motion is:

 

θ'' + (g/l)sinθ = 0

 

Where g = the acceleration due to gravity and l = the length of the pendulum.  The sinθ term makes it different than SHM.  But for cases where the angular displacement of the pendulum from vertical is small, then sin θ ≅ θ and the equation of motion becomes:

 

θ'' + (g/l)θ = 0

 

which has the same form as the equation of motion for SHM (where g/l = ω²).  So for small oscillations (small θ), the motion of a pendulum approximates SHM; but for larger displacements, it does not.