In the study of Simple Harmonic Motion (SHM), it's important to know how amplitude and velocity are linked.

SHM is when an object moves back and forth around a central point.

• Amplitude is the farthest distance the object moves from that central point.
• Velocity is how fast the object is moving and in which direction.

How to Describe SHM with Math

We can describe how far an object moves in SHM with this formula:

$$x(t) = A \cos(\omega t + \phi)$$

Here’s what the symbols mean:

• ( A ): This is the amplitude (the maximum distance from the center).
• ( \omega ): This is the angular frequency, which is related to how long it takes to complete one cycle.
• ( \phi ): This is the phase constant. It shows where the object is at the start (when time is 0).

To find the velocity ( v(t) ) of the object, we take the first derivative (which just means how the distance changes over time):

$$v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi)$$

How Amplitude Affects Velocity

1. Maximum Velocity: The fastest speed happens when the object is at the central point (where it isn't displaced). We can calculate this maximum speed ( v_{\text{max}} ) with:

$$v_{\text{max}} = A \omega$$

This means that the maximum speed increases if the amplitude ( A ) increases, as long as the angular frequency ( \omega ) stays the same.

2. Velocity Changes with Displacement: The speed doesn't stay the same. It’s zero when the object is at its maximum distance (either ( |x| = A ) or ( |x| = -A )). The velocity changes as the object moves toward the center.

3. Impact of Angular Frequency: Angular frequency ( \omega ) affects how fast the object moves. It can be calculated like this:

$$\omega = \frac{2\pi}{T}$$

Here, ( T ) is how long one full cycle takes (the period). With a constant amplitude, a higher angular frequency makes the maximum velocity larger. This shows that both amplitude and frequency are important in understanding how velocity behaves in SHM.

Quick Summary

• Direct Proportionality: Maximum velocity is directly connected to the amplitude: ( v_{\text{max}} = A \omega ).
• Velocity Changes: Velocity goes up and down, being zero at the highest and lowest points, and the fastest at the center.
• Angular Frequency’s Role: If we increase the angular frequency while keeping the amplitude the same, the maximum velocity goes up.

Conclusion

In simple terms, changes in amplitude can greatly affect how fast an object moves in a Simple Harmonic Motion system. Knowing the equations for SHM helps students understand how things like amplitude and angular frequency work together in these movements. As the amplitude increases, the maximum velocity increases too, which shows how these basic physics ideas are connected.