Find A Model For Simple Harmonic Motion If The Position At $t=0$ Is 5 Centimeters, The Amplitude Is 5 Centimeters, And The Period Is 4 Seconds.A. D = 4 Sin ⁡ ( 5 T D=4 \sin (5 T D = 4 Sin ( 5 T ]B. D=5 \sin \left(\frac{\pi}{2} T\right ]C. $d=5 \cos

Introduction to Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where an object oscillates about a fixed point, known as the equilibrium position. This motion is characterized by a restoring force that is proportional to the displacement of the object from its equilibrium position. SHM is a fundamental concept in physics and is commonly observed in various natural phenomena, such as the motion of a pendulum, the vibration of a spring, and the oscillation of a mass on a string.

Key Parameters of Simple Harmonic Motion

To describe SHM, we need to consider several key parameters, including:

• Amplitude: The maximum displacement of the object from its equilibrium position.
• Period: The time taken by the object to complete one oscillation or cycle.
• Frequency: The number of oscillations or cycles per second, measured in hertz (Hz).
• Angular frequency: The rate of change of the phase angle with respect to time, measured in radians per second (rad/s).

Equation of Simple Harmonic Motion

The equation of SHM can be expressed in the form:

$$d=A \sin (\omega t+\phi)$$

where:

• $d$ is the displacement of the object from its equilibrium position.
• $A$ is the amplitude of the motion.
• $\omega$ is the angular frequency of the motion.
• $t$ is the time.
• $\phi$ is the phase angle, which represents the initial displacement of the object from its equilibrium position.

Finding the Model for Simple Harmonic Motion

Given the position at $t=0$ is 5 centimeters, the amplitude is 5 centimeters, and the period is 4 seconds, we can find the model for SHM using the following steps:

1. Determine the angular frequency: The angular frequency $\omega$ is related to the period $T$ by the equation:

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

Substituting the given period $T=4$ seconds, we get:

$$\omega=\frac{2\pi}{4}=\frac{\pi}{2}$$

2. Determine the phase angle: The phase angle $\phi$ can be found using the initial displacement $d_0$ and the amplitude $A$:

$$\phi=\sin^{-1}\left(\frac{d_0}{A}\right)$$

Substituting the given initial displacement $d_0=5$ centimeters and amplitude $A=5$ centimeters, we get:

$$\phi=\sin^{-1}\left(\frac{5}{5}\right)=\sin^{-1}(1)=\frac{\pi}{2}$$

3. Write the equation of SHM: Now that we have the angular frequency $\omega$ and phase angle $\phi$, we can write the equation of SHM:

$$d=A \sin (\omega t+\phi)$$

Substituting the values of $A$, $\omega$, and $\phi$, we get:

$$d=5 \sin \left(\frac{\pi}{2} t+\frac{\pi}{2}\right)$$

Simplifying the Equation

To simplify the equation, we can use the trigonometric identity:

$$\sin (A+B)=\sin A \cos B + \cos A \sin B$$

Applying this identity to the equation, we get:

$$d=5 \sin \left(\frac{\pi}{2} t\right) \cos \left(\frac{\pi}{2}\right) + 5 \cos \left(\frac{\pi}{2} t\right) \sin \left(\frac{\pi}{2}\right)$$

Since $\cos \left(\frac{\pi}{2}\right)=0$ and $\sin \left(\frac{\pi}{2}\right)=1$, the equation simplifies to:

$$d=5 \sin \left(\frac{\pi}{2} t\right)$$

Conclusion

In this article, we have discussed the concept of simple harmonic motion and its key parameters. We have also derived the equation of SHM using the given parameters and simplified it to obtain the final model. The correct model for SHM is:

$$d=5 \sin \left(\frac{\pi}{2} t\right)$$

This equation describes the motion of an object with an amplitude of 5 centimeters, a period of 4 seconds, and an initial displacement of 5 centimeters.

Introduction

Simple harmonic motion (SHM) is a fundamental concept in physics that describes the motion of an object that oscillates about a fixed point. In our previous article, we discussed the key parameters of SHM and derived the equation of SHM using the given parameters. In this article, we will answer some frequently asked questions about SHM to help you better understand this concept.

Q&A

Q1: What is simple harmonic motion?

A1: Simple harmonic motion is a type of periodic motion where an object oscillates about a fixed point, known as the equilibrium position. This motion is characterized by a restoring force that is proportional to the displacement of the object from its equilibrium position.

Q2: What are the key parameters of simple harmonic motion?

A2: The key parameters of SHM are:

• Amplitude: The maximum displacement of the object from its equilibrium position.
• Period: The time taken by the object to complete one oscillation or cycle.
• Frequency: The number of oscillations or cycles per second, measured in hertz (Hz).
• Angular frequency: The rate of change of the phase angle with respect to time, measured in radians per second (rad/s).

Q3: How do I determine the angular frequency of an object in simple harmonic motion?

A3: The angular frequency $\omega$ is related to the period $T$ by the equation:

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

You can use this equation to determine the angular frequency of an object in SHM.

Q4: How do I determine the phase angle of an object in simple harmonic motion?

A4: The phase angle $\phi$ can be found using the initial displacement $d_0$ and the amplitude $A$:

$$\phi=\sin^{-1}\left(\frac{d_0}{A}\right)$$

You can use this equation to determine the phase angle of an object in SHM.

Q5: What is the equation of simple harmonic motion?

A5: The equation of SHM can be expressed in the form:

$$d=A \sin (\omega t+\phi)$$

where:

• $d$ is the displacement of the object from its equilibrium position.
• $A$ is the amplitude of the motion.
• $\omega$ is the angular frequency of the motion.
• $t$ is the time.
• $\phi$ is the phase angle, which represents the initial displacement of the object from its equilibrium position.

Q6: How do I simplify the equation of simple harmonic motion?

A6: You can simplify the equation of SHM by using the trigonometric identity:

$$\sin (A+B)=\sin A \cos B + \cos A \sin B$$

Applying this identity to the equation, you can simplify it to obtain the final model.

Q7: What is the correct model for simple harmonic motion?

A7: The correct model for SHM is:

$$d=5 \sin \left(\frac{\pi}{2} t\right)$$

This equation describes the motion of an object with an amplitude of 5 centimeters, a period of 4 seconds, and an initial displacement of 5 centimeters.

Conclusion

In this article, we have answered some frequently asked questions about simple harmonic motion to help you better understand this concept. We have discussed the key parameters of SHM, determined the angular frequency and phase angle, and simplified the equation of SHM to obtain the final model. We hope this article has been helpful in your understanding of SHM.

Additional Resources

If you want to learn more about simple harmonic motion, we recommend the following resources:

• Textbooks: "Physics for Scientists and Engineers" by Paul A. Tipler and Gene Mosca, "Physics" by John D. Cutnell and Kenneth W. Johnson.
• Online resources: Khan Academy, MIT OpenCourseWare, Physics Classroom.
• Videos: Crash Course Physics, 3Blue1Brown, Physics Girl.

We hope this article has been helpful in your understanding of simple harmonic motion. If you have any further questions or need additional resources, please don't hesitate to ask.