How do you derive the equation for SHM?

The equation for Simple Harmonic Motion (SHM) is derived from Newton's second law of motion and Hooke's law.

To derive the equation for Simple Harmonic Motion (SHM), we start with Newton's second law of motion, which states that the force acting on an object is equal to its mass times its acceleration (F = ma). In the case of SHM, the force acting on the object is the restoring force, which always acts in the opposite direction to the displacement and is directly proportional to it. This relationship is described by Hooke's law, which states that the force required to extend or compress a spring by some distance is proportional to that distance (F = -kx), where k is the spring constant and x is the displacement.

Combining these two laws, we get ma = -kx. We can rearrange this equation to get a = -k/m * x. This equation tells us that the acceleration of the object is directly proportional to the displacement and always directed towards the equilibrium position, which is the defining characteristic of SHM.

However, acceleration is also the second derivative of the position with respect to time (a = d²x/dt²). Substituting this into our equation, we get d²x/dt² = -k/m * x. This is a second order differential equation, and its solution gives us the equation for SHM: x(t) = A cos(wt + φ), where A is the amplitude of the motion, w is the angular frequency, and φ is the phase constant.

The angular frequency w is related to the mass of the object and the spring constant by the equation w = sqrt(k/m). This tells us that the frequency of the motion is determined by the stiffness of the spring and the mass of the object. The phase constant φ determines the starting point of the motion.

IB Physics Tutor Summary: To derive the equation for Simple Harmonic Motion (SHM), we use Newton's second law (force = mass × acceleration) and Hooke's law (force = -spring constant × displacement). Combining these, we get an equation showing acceleration is proportional to displacement but in the opposite direction. Solving this gives the SHM equation, with motion dependent on the object's mass and the spring's stiffness.