An Equation for Simple Harmonic Motion of a Spring

Mathispower4u
5 Oct 201207:50
EducationalLearning
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TLDRThis educational video script demonstrates the concept of simple harmonic motion using a spring-mass system. An object attached to a spring is initially displaced 20 cm from its equilibrium position and released, completing one oscillation in four seconds. The script guides through graphing the cosine function to model the motion, determining the amplitude and angular frequency, and calculating the position of the object at a specific time (1.2 seconds). It concludes with finding the frequency of oscillation, emphasizing the importance of understanding the properties of trigonometric functions in physics.

Takeaways
  • πŸ˜€ The video demonstrates an example of simple harmonic motion with a spring and an object.
  • πŸ” The object is initially pulled down 20 centimeters from its equilibrium position and released to oscillate.
  • ⏱ The time taken for one complete oscillation of the object is four seconds.
  • πŸ“‰ The displacement graph of the object's motion is modeled using a cosine function.
  • πŸ“ At time t=0, the object is 20 cm below the equilibrium point, which is the starting point of the graph.
  • πŸ”„ After two seconds, the object reaches its maximum height of 20 cm above the equilibrium point, completing half of the oscillation.
  • πŸ“ˆ At one second, the object is at the equilibrium point, halfway between the maximum and minimum displacement.
  • πŸ“Š The function graphed resembles a cosine wave, starting at a minimum rather than the midline.
  • πŸ“ The amplitude 'A' of the function is determined to be -20 cm, reflecting the object's initial displacement.
  • πŸ”’ The value of 'B' in the cosine function is calculated to be Ο€/2, based on the period of the oscillation.
  • πŸ“˜ The final equation modeling the object's position at any time 't' is given by "d(t) = -20 * cos(Ο€/2 * t)".
  • πŸ•’ The position of the object at 1.2 seconds is calculated to be approximately 6.18 cm above the midline.
  • πŸ” The frequency of the oscillation is determined to be 1/4 oscillation per second, which is the reciprocal of the period.
Q & A

  • What is the initial displacement of the object from its equilibrium position?

    -The object is initially pulled down a distance of 20 centimeters from its equilibrium position.

  • What is the time period of one complete oscillation of the spring?

    -The time for one complete oscillation is four seconds.

  • What is the amplitude of the simple harmonic motion described in the script?

    -The amplitude of the motion is 20 centimeters, as the object moves 20 cm above and below the equilibrium point.

  • How does the object's displacement at time t=0 compare to the equilibrium position?

    -At time t=0, the object is 20 centimeters below the equilibrium position, which is the displacement's minimum value.

  • What trigonometric function is used to model the simple harmonic motion of the spring?

    -The cosine function is used to model the simple harmonic motion of the spring.

  • Why is the cosine function chosen over the sine function for this model?

    -The cosine function is chosen because the object starts at a minimum rather than at the midline, and using the sine function would require a phase shift.

  • What is the value of 'A' in the equation modeling the spring's motion?

    -The value of 'A' is -20, as the graph is reflected across the x-axis due to the object starting at a minimum at time t=0.

  • How is the value of 'B' in the equation determined?

    -The value of 'B' is determined by the period of the function. Since the period is 4 seconds, B is calculated as 2Ο€ divided by the period, which gives B = Ο€/2.

  • What is the equation that models the position of the object at any time 't'?

    -The equation modeling the position of the object at time 't' is d(t) = -20 * cos(Ο€/2 * t).

  • What is the position of the object at 1.2 seconds into the motion?

    -At 1.2 seconds, the position of the object is approximately 6.18 centimeters above the midline.

  • How is the frequency of the oscillation calculated?

    -The frequency is the reciprocal of the period. Since the period is 4 seconds, the frequency is 1/4 oscillations per second.

Outlines
00:00
πŸ“š Introduction to Simple Harmonic Motion of a Spring

This paragraph introduces the concept of simple harmonic motion using a spring system. An object is attached to a spring and pulled down 20 centimeters from its equilibrium position before being released. The time for one complete oscillation is noted as four seconds. The process of modeling this motion begins by plotting key positions at different times: at t=0, the object is 20 cm below the equilibrium; at t=2 seconds, it reaches its highest point 20 cm above; and at t=4 seconds, it returns to the starting position. The graph of the function is then discussed, and it's established that the cosine function will be used to model the motion without any phase or vertical shifts due to the symmetry of the motion around the equilibrium point.

05:06
πŸ“ˆ Deriving the Equation for Simple Harmonic Motion

The second paragraph delves into the specifics of deriving the equation for the simple harmonic motion of the spring system. The amplitude 'A' and angular frequency 'B' of the cosine function are determined based on the motion's characteristics. The amplitude is identified as -20 cm, reflecting the motion's symmetry and the starting position. The angular frequency 'B' is calculated using the period of the motion, which is four seconds, resulting in B = Ο€/2. The equation modeling the position of the object at any time 't' is then formulated as d(t) = -20 * cos(Ο€/2 * t). The paragraph concludes by applying this equation to find the object's position at 1.2 seconds, which is approximately 6.18 cm above the midline, and by explaining how to determine the frequency of the motion, which is the reciprocal of the period, yielding a frequency of 1/4 oscillations per second.

Mindmap
Keywords
πŸ’‘Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force on the object is directly proportional to the displacement from an equilibrium position, and it acts in the opposite direction. In the context of the video, SHM is exemplified by the motion of a spring with an attached object. The object moves back and forth around its equilibrium position, which is a fundamental concept in physics and is the central theme of the video.
πŸ’‘Spring
A spring in this context refers to an elastic object that stores potential energy when it is stretched or compressed and releases this energy when it is allowed to return to its original shape. The spring is a key component in demonstrating SHM, as it provides the restoring force that propels the attached object through its oscillating motion, as described in the video script.
πŸ’‘Equilibrium Position
The equilibrium position is the point around which an object in SHM oscillates. It is the position where the net force on the object is zero, and it is typically the center of the object's motion. In the video, the equilibrium position is the central point from which the spring is displaced and to which it returns during its oscillation.
πŸ’‘Displacement
Displacement refers to the change in position of an object from its initial position. In the context of SHM, displacement is measured from the equilibrium position and is a crucial parameter in determining the object's state during its motion. The script describes the object being pulled down 20 centimeters from its equilibrium position, which sets the stage for the SHM.
πŸ’‘Oscillation
Oscillation is the repetitive back-and-forth motion of an object around an equilibrium position. It is characterized by a regular periodicity. In the video, the term is used to describe the motion of the spring with the attached object, which completes one full cycle of motion in four seconds.
πŸ’‘Frequency
Frequency is the number of oscillations or cycles that occur per unit of time. It is the reciprocal of the period and is measured in hertz (Hz). In the video, the frequency of the spring's oscillation is calculated by taking the reciprocal of the period, which is four seconds, resulting in a frequency of 1/4 Hz.
πŸ’‘Amplitude
Amplitude is the maximum displacement of the object from its equilibrium position. It is a measure of the extent of the oscillation and is related to the energy of the system. In the script, the amplitude is given as 20 centimeters, which is the distance the object is initially pulled down from the equilibrium position.
πŸ’‘Period
The period of an oscillating system is the time taken for one complete cycle of motion. It is a fundamental property of periodic phenomena and is used to characterize the rate of oscillation. In the video, the period is stated as four seconds, which is the time it takes for the spring to complete one full oscillation.
πŸ’‘Cosine Function
The cosine function is a trigonometric function that is used to model periodic phenomena such as SHM. It has a wave-like pattern that repeats every 2Ο€ radians. In the video, the cosine function is chosen to model the SHM of the spring because the motion starts at a minimum displacement, resembling the cosine wave's starting point.
πŸ’‘Phase Shift
A phase shift refers to a horizontal displacement of a function along the time axis, which affects the starting point of the wave. In the context of SHM, a phase shift can be necessary if the function being used does not naturally align with the observed motion. However, in the video, it is noted that no phase shift is required because the cosine function is used, and the motion starts at a minimum, aligning with the natural phase of the cosine wave.
πŸ’‘Trigonometric Functions
Trigonometric functions, such as sine and cosine, are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are particularly useful in modeling periodic phenomena like SHM. In the video, the cosine function is specifically chosen to represent the SHM of the spring, as it naturally fits the observed motion without the need for additional transformations.
Highlights

Introduction to modeling simple harmonic motion of a spring with an object attached.

Object is initially pulled down 20 cm from its equilibrium position and released.

Time for one complete oscillation is four seconds.

Displacement of the object at time t=0 is -20 cm from equilibrium.

Object reaches a height 20 cm above equilibrium after two seconds.

Object returns to the starting position after four seconds.

Graphing the function to determine the equation of the motion.

The function resembles a cosine function starting at a minimum.

No phase shift or vertical translation needed for the graph.

Amplitude 'A' is determined to be -20 cm based on the motion's reflection.

Calculating the value of 'B' using the period of the function.

Equation of the function is derived as "y = -20 cos(\frac{\pi}{2}t)".

Position at 1.2 seconds is calculated using the derived equation.

Frequency is determined as the reciprocal of the period.

Frequency of the oscillation is 1/4 oscillation per second.

Conclusion of the example with a summary of the findings.

Transcripts

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Simple Harmonic MotionSpring PhysicsOscillation AnalysisCosine FunctionAmplitude CalculationTime PeriodFrequency DeterminationGraphical ModelingEducational ContentPhysics Tutorial