A Buoy Floating In The Ocean Is Bobbing In Simple Harmonic Motion With An Amplitude Of 6 Ft And A Period Of 7 Seconds. Its Displacement \[$d\$\] From Sea Level At Time \[$t = 0\$\] Seconds Is 0 Ft, And Initially, It Moves Upward. (Note

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Introduction

Simple harmonic motion (SHM) is a fundamental concept in physics that describes the motion of an object that oscillates about a fixed point, known as the equilibrium position. In this article, we will explore the motion of a buoy floating in the ocean, which is an excellent example of SHM. We will analyze the displacement of the buoy from sea level as a function of time and derive the equation of motion.

Understanding Simple Harmonic Motion

Simple harmonic motion is characterized by a periodic motion about a fixed point, with the object oscillating between two extreme positions. The motion is sinusoidal in nature, with the object moving in a circular path. The key parameters that describe SHM are:

  • Amplitude: The maximum displacement of the object from the equilibrium position.
  • Period: The time taken by the object to complete one oscillation.
  • Frequency: The number of oscillations per second, which is the reciprocal of the period.
  • Angular frequency: The rate of change of the angular displacement, which is related to the frequency.

Equation of Motion

The equation of motion for SHM is given by:

d(t)=Asin⁡(ωt+ϕ)d(t) = A \sin(\omega t + \phi)

where:

  • d(t) is the displacement of the object from the equilibrium position at time t.
  • A is the amplitude of the motion.
  • ω is the angular frequency, which is related to the frequency by the equation ω = 2πf.
  • φ is the phase angle, which represents the initial displacement of the object from the equilibrium position.

Buoy's Displacement from Sea Level

In this problem, the buoy's displacement from sea level at time t = 0 seconds is 0 ft, and initially, it moves upward. This means that the phase angle φ is 0, and the equation of motion becomes:

d(t)=Asin⁡(ωt)d(t) = A \sin(\omega t)

where A is the amplitude of the motion, which is given as 6 ft.

Angular Frequency

The period of the motion is given as 7 seconds. We can use this information to find the angular frequency ω:

ω=2πT\omega = \frac{2\pi}{T}

where T is the period of the motion.

Substituting the values, we get:

ω=2π7\omega = \frac{2\pi}{7}

Equation of Motion for the Buoy

Now that we have the angular frequency ω, we can write the equation of motion for the buoy:

d(t)=6sin⁡(2π7t)d(t) = 6 \sin\left(\frac{2\pi}{7}t\right)

This equation describes the displacement of the buoy from sea level as a function of time.

Graphical Representation

To visualize the motion of the buoy, we can plot the displacement d(t) as a function of time t. The resulting graph will be a sinusoidal curve, with the buoy oscillating between two extreme positions.

Conclusion

In this article, we have analyzed the motion of a buoy floating in the ocean, which is an excellent example of simple harmonic motion. We have derived the equation of motion for the buoy and discussed the key parameters that describe SHM. The equation of motion provides a mathematical description of the displacement of the buoy from sea level as a function of time. This article has demonstrated the importance of SHM in understanding the motion of objects in various fields, including physics, engineering, and mathematics.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Further Reading

  • Simple Harmonic Motion by Khan Academy
  • Simple Harmonic Motion by MIT OpenCourseWare
  • Simple Harmonic Motion by Physics Classroom
    A Buoy Floating in Simple Harmonic Motion: Q&A =====================================================

Introduction

In our previous article, we explored the motion of a buoy floating in the ocean, which is an excellent example of simple harmonic motion (SHM). We derived the equation of motion for the buoy and discussed the key parameters that describe SHM. In this article, we will answer some frequently asked questions about SHM and the buoy's motion.

Q: What is simple harmonic motion?

A: Simple harmonic motion is a type of periodic motion where an object oscillates about a fixed point, known as the equilibrium position. The motion is sinusoidal in nature, with the object moving in a circular path.

Q: What are the key parameters that describe SHM?

A: The key parameters that describe SHM are:

  • Amplitude: The maximum displacement of the object from the equilibrium position.
  • Period: The time taken by the object to complete one oscillation.
  • Frequency: The number of oscillations per second, which is the reciprocal of the period.
  • Angular frequency: The rate of change of the angular displacement, which is related to the frequency.

Q: What is the equation of motion for SHM?

A: The equation of motion for SHM is given by:

d(t)=Asin⁡(ωt+ϕ)d(t) = A \sin(\omega t + \phi)

where:

  • d(t) is the displacement of the object from the equilibrium position at time t.
  • A is the amplitude of the motion.
  • ω is the angular frequency, which is related to the frequency by the equation ω = 2πf.
  • φ is the phase angle, which represents the initial displacement of the object from the equilibrium position.

Q: How do you find the angular frequency of an object in SHM?

A: The angular frequency ω can be found using the equation:

ω=2πT\omega = \frac{2\pi}{T}

where T is the period of the motion.

Q: What is the significance of the phase angle in SHM?

A: The phase angle φ represents the initial displacement of the object from the equilibrium position. It can be used to determine the initial position of the object.

Q: Can you provide an example of SHM in real-life?

A: Yes, the motion of a buoy floating in the ocean is an excellent example of SHM. The buoy oscillates about a fixed point, known as the equilibrium position, and its motion is sinusoidal in nature.

Q: How do you graph the displacement of an object in SHM?

A: To graph the displacement of an object in SHM, you can plot the displacement d(t) as a function of time t. The resulting graph will be a sinusoidal curve, with the object oscillating between two extreme positions.

Q: What are some applications of SHM in real-life?

A: SHM has numerous applications in real-life, including:

  • Mechanical systems: SHM is used to describe the motion of mechanical systems, such as pendulums and springs.
  • Electrical systems: SHM is used to describe the motion of electrical systems, such as LC circuits.
  • Optical systems: SHM is used to describe the motion of optical systems, such as mirrors and lenses.

Conclusion

In this article, we have answered some frequently asked questions about simple harmonic motion and the buoy's motion. We have discussed the key parameters that describe SHM, the equation of motion, and some applications of SHM in real-life. This article has demonstrated the importance of SHM in understanding the motion of objects in various fields, including physics, engineering, and mathematics.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Further Reading

  • Simple Harmonic Motion by Khan Academy
  • Simple Harmonic Motion by MIT OpenCourseWare
  • Simple Harmonic Motion by Physics Classroom