text: # Equations for SHM
  
  It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!
  
  However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.
  
  Key Concepts
  
  By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.
  
  The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.
  
  Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.
  
  Essentials
  
  Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.
  
  This can be used to show that $$a=-\omega^2x$$.
  
  Summary
  
  The equation for the displacement of a body executing SHM is either:
  
  • $$x = x_o\sin(2πft)$$
  • $$x = x_o\cos(2πft)$$
  
  Test Yourself
  
  Use quizzes to practise application of theory.
  
  Exam-style Questions
explanation: The provided text is an educational excerpt that explains the key concepts and equations related to simple harmonic motion (SHM). It describes the relationship between displacement, velocity, and acceleration in SHM and provides the equations for displacement. The text also includes a brief section titled "Test Yourself," which suggests that there are quizzes and exam-style questions available for practice, but no specific questions or answers are provided in the text.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Equations for SHM
  
  It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!
  
  However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.
  
  Key Concepts
  
  By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.
  
  The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.
  
  Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.
  
  Essentials
  
  Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.
  
  This can be used to show that $$a=-\omega^2x$$.
  
  Summary
  
  The equation for the displacement of a body executing SHM is either:
  
  • $$x = x_o\sin(2πft)$$
  • $$x = x_o\cos(2πft)$$
  
  Test Yourself
  
  Use quizzes to practise application of theory.
  
  Exam-style Questions
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an educational excerpt that explains the key concepts and equations related to simple harmonic motion (SHM). It describes the relationship between displacement, velocity, and acceleration in SHM and provides the equations for displacement. The text also includes a brief section titled "Test Yourself," which suggests that there are quizzes and exam-style questions available for practice, but no specific questions or answers are provided in the text.
  
  Label: no-question