13.5. VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION

The speed of a particle v in uniform circular motion is its angular speed ω times the radius of the circle A.

v = ω A (13.8)

In uniform circular motion, a particle moves around a circular path at a constant speed. The speed (v) can be calculated using the formula where 'v' is equal to the angular speed (ω) multiplied by the radius of the circle (A). Angular speed is a measure of how fast the particle is moving around the circle.

Think of a car driving around a circular track at a steady speed. The further the car is from the center, the greater the distance it covers in the same amount of time, which is essentially what's described by the equation for speed in circular motion.

From the geometry of Fig. 13.11, it is clear that the velocity of the projection particle P' at time t is

v(t) = – ωA sin (ωt + φ) (13.9)

When examining the projection of a particle moving in a circle (like our earlier example), we can determine its instantaneous velocity. The formula given shows that the velocity depends on the sine of the phase (ωt + φ). The negative sign indicates that as the particle moves to the right in its oscillation, the velocity is treated as moving in the negative x-direction.

Imagine a child on a swing. When the swing reaches the highest point on one side, the speed is zero because it pauses before swinging back. As it comes down and passes through the middle, it’s moving the fastest, which is similar to how the particles in SHM behave as they oscillate.

We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. 13.4) with respect of t:

d( )dv(t) = x tt(13.10)

This chunk reveals that we don’t need to rely purely on geometric understanding to derive the instantaneous velocity. By taking the derivative of the displacement function directly with respect to time (denoted as t), we can find the velocity function. This shows a fundamental concept in physics that relates to how calculus connects motion with change over time.

Consider a car's speedometer, which tells you how fast you are going at a specific moment. Similar to how that speedometer measures the car's instantaneous speed by looking at the changes in position over time, we derive velocity from displacement as a function of time.

The instantaneous acceleration of the projection particle P' is then

a(t) = – ω²A cos (ωt + φ) = – ω²x(t) (13.11)

Here, we see that acceleration is also described as a function of the cosine of the phase (ωt + φ). Like velocity, acceleration also has a negative sign indicating that it is always directed towards the center of the oscillation, which is important for restoring the system back to equilibrium.

Think of a rubber band. When you stretch it, you feel a force pulling it back toward the center. This restoring force is what causes the rubber band to oscillate back and forth, just like how acceleration in SHM always pulls the particle towards the mean position.

For simplicity, let us put φ = 0 and write the expression for x(t), v(t), and a(t):

x(t) = A cos ωt, v(t) = –ωA sin ωt, a(t) = –ω²A cos ωt

By assuming the phase constant φ equals zero, we simplify our equations for displacement, velocity, and acceleration in simple harmonic motion. This helps to visualize the relationship among these variables, which fluctuate sinusoidally over time.

If you were to watch a pendulum swinging back and forth on a clock without any initial movement to the left or right, you can see how displacement, speed, and the direction of the force (acceleration) all change smoothly and in a periodic way.

With respect to displacement plot, velocity plot has a phase difference of π/2 and acceleration plot has a phase difference of π.

In SHM, the displacement, velocity, and acceleration don't peak at the same time. The velocity graph peaks a quarter cycle (or π/2 radians) ahead of displacement, and acceleration peaks half a cycle (or π radians) ahead of displacement. This phase difference is crucial in understanding the dynamics of the oscillating system.

Picture a dancer performing a routine. The dancer moves in a flowing manner (displacement), then shifts into a leap (velocity), and finally lands softly (acceleration). The sequence and timing of their movements illustrate how these concepts relate but are not simultaneous.

--