For The Simple Harmonic Motion Equation $d=2 \sin \left(\frac{\pi}{3} T\right$\], What Is The Maximum Displacement From The Equilibrium Position?Answer Here: $\qquad$

Simple harmonic motion is a type of periodic motion where the acceleration of the object is directly proportional to the displacement from its equilibrium position. This type of motion is commonly observed in various physical systems, such as a pendulum, a spring-mass system, and a simple harmonic oscillator. In this article, we will focus on the simple harmonic motion equation and determine the maximum displacement from the equilibrium position.

The Simple Harmonic Motion Equation

The simple harmonic motion equation is given by:

$$d = 2 \sin \left(\frac{\pi}{3} t\right)$$

where $d$ is the displacement from the equilibrium position, and $t$ is time.

Understanding the Equation

To understand the equation, let's break it down into its components. The displacement $d$ is given by the sine function, which oscillates between $-1$ and $1$. The amplitude of the sine function is $2$, which means that the maximum displacement from the equilibrium position is $2$ units.

Maximum Displacement from Equilibrium Position

To determine the maximum displacement from the equilibrium position, we need to find the maximum value of the sine function. The maximum value of the sine function occurs when the argument of the sine function is equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$.

In this case, the argument of the sine function is $\frac{\pi}{3} t$. To find the maximum value of the sine function, we need to find the value of $t$ that makes the argument equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$.

Finding the Maximum Displacement

To find the maximum displacement, we need to find the value of $t$ that makes the argument of the sine function equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$. We can do this by setting the argument equal to $\frac{\pi}{2}$ and solving for $t$:

$$\frac{\pi}{3} t = \frac{\pi}{2}$$

Solving for $t$, we get:

$$t = \frac{3}{2}$$

Now that we have found the value of $t$, we can substitute it into the equation to find the maximum displacement:

$$d = 2 \sin \left(\frac{\pi}{3} \cdot \frac{3}{2}\right)$$

Simplifying the equation, we get:

$$d = 2 \sin \left(\frac{\pi}{2}\right)$$

The maximum value of the sine function is $1$, so the maximum displacement from the equilibrium position is:

$$d = 2 \cdot 1$$

$$d = 2$$

Conclusion

In this article, we have determined the maximum displacement from the equilibrium position for the simple harmonic motion equation $d=2 \sin \left(\frac{\pi}{3} t\right)$. We found that the maximum displacement is $2$ units.

Key Takeaways

• The simple harmonic motion equation is given by $d = 2 \sin \left(\frac{\pi}{3} t\right)$.
• The maximum displacement from the equilibrium position is determined by finding the maximum value of the sine function.
• The maximum value of the sine function occurs when the argument of the sine function is equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$.
• The maximum displacement from the equilibrium position is $2$ units.

Frequently Asked Questions

• What is the simple harmonic motion equation?
• The simple harmonic motion equation is given by $d = 2 \sin \left(\frac{\pi}{3} t\right)$.
• What is the maximum displacement from the equilibrium position?
• The maximum displacement from the equilibrium position is $2$ units.
• How do I determine the maximum displacement from the equilibrium position?
• To determine the maximum displacement from the equilibrium position, you need to find the maximum value of the sine function.