How does displacement vary with time in SHM?

In Simple Harmonic Motion (SHM), displacement varies sinusoidally with time.

In more detail, Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement. It is characterised by its amplitude, frequency, and phase. The displacement from the equilibrium position as a function of time can be described by a sine or cosine function, depending on the initial conditions.

The equation for displacement in SHM is given by x = A cos(wt + φ), where x is the displacement, A is the amplitude (maximum displacement), w is the angular frequency, t is the time and φ is the phase constant. The angular frequency w is related to the period of oscillation T by the equation w = 2π/T. The phase constant φ depends on the initial conditions of the motion.

The displacement x varies between -A and +A. When the object is at maximum displacement, the velocity is zero. As the object moves towards the equilibrium position, the velocity increases. At the equilibrium position, the velocity is maximum and the displacement is zero. As the object moves past the equilibrium position towards the other extreme, the velocity decreases until it reaches zero at maximum displacement. This process then repeats.

The graph of displacement versus time for SHM is a sinusoidal curve. The displacement is positive for half of the period, and negative for the other half. The points of maximum and minimum displacement correspond to the peaks and troughs of the curve, respectively. The period of the motion is the time taken for one complete cycle of the motion, and is the same as the period of the sinusoidal curve.

In summary, in Simple Harmonic Motion, the displacement varies sinusoidally with time, with the displacement being a maximum at the extremes of the motion and zero at the equilibrium position. The velocity is zero at the extremes and maximum at the equilibrium position. The displacement as a function of time can be described by a sine or cosine function, depending on the initial conditions.