Displacement Function in Simple Harmonic Motion

The displacement function in simple harmonic motion is given by the equation D(t) = Acos(wt), where A represents the amplitude of the oscillation and w represents the angular frequency of the motion. This equation describes the position of an object undergoing simple harmonic motion at any given time, t.

Understanding Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion that occurs when a restoring force is proportional to the displacement from an equilibrium position. It is characterized by a repetitive back-and-forth motion around a stable equilibrium point. Examples of systems exhibiting SHM include a mass-spring system and a pendulum.

Displacement-Time Graph in Simple Harmonic Motion

The displacement-time graph in simple harmonic motion represents the variation in position of the object as a function of time. It provides insights into the amplitude, frequency, and phase of the oscillation.

Deriving the Displacement Function

The displacement function, D(t) = Acos(wt), can be derived using the principles of SHM. Here's a step-by-step explanation:

1. Start with the general equation for SHM: F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.
2. Apply Newton's second law of motion, F = ma, where m is the mass of the object and a is the acceleration.
3. Since acceleration is the second derivative of displacement with respect to time (a = d²x/dt²), we can rewrite the equation as m(d²x/dt²) = -kx.
4. Divide both sides of the equation by m to obtain (d²x/dt²) = -(k/m)x.
5. Notice that the left side of the equation represents the second derivative of x with respect to time, which can be written as d²x/dt² = w²x, where w is the angular frequency.
6. Substitute this expression into the equation to get w²x = -(k/m)x.
7. Simplify the equation by canceling out x on both sides: w² = -(k/m).
8. Solve for w to obtain w = sqrt(k/m).
9. The general solution to the differential equation is x = Acos(wt) + Bsin(wt), where A and B are constants.
10. Since the cosine function represents the amplitude and phase of the oscillation, we can rewrite the equation as D(t) = Acos(wt), where D(t) represents the displacement as a function of time.

Conclusion

The displacement function in simple harmonic motion is given by D(t) = Acos(wt), where A represents the amplitude of the oscillation and w represents the angular frequency. This equation describes the position of an object undergoing simple harmonic motion at any given time. It is derived from the principles of SHM, involving the relationship between the restoring force, displacement, and acceleration. The displacement-time graph provides valuable information about the characteristics of the oscillation, such as amplitude, frequency, and phase.