Simple Harmonic Motion(SHM) - Graphs of Position, Velocity, and Acceleration

Summary

In this video, Mr. P. explores the relationship between simple harmonic motion (SHM) and circular motion. The lesson dives into position, velocity, and acceleration equations and their graphs, illustrating how the vertical motion of a dot on a rotating marker cap mirrors SHM. The students discuss phase constants, angular frequency, and the connection between tangential and vertical components of motion. Through interactive graphing, they uncover how the slope of position relates to velocity and how the slope of velocity correlates with acceleration. The session concludes with a hands-on explanation of these concepts to clarify any confusion.

Takeaways

• 😀 The video demonstrates the relationship between simple harmonic motion (SHM) and circular motion using a yellow marker cap on a turntable.
• 😀 The position equation for SHM is derived as position = amplitude * cosine(angular frequency * time), and the phase constant (phi) is assumed to be 0 for simplicity.
• 😀 The velocity equation for SHM is velocity = -amplitude * sine(angular frequency * time), representing the vertical component of the tangential velocity in circular motion.
• 😀 The acceleration equation for SHM is acceleration = -amplitude * angular frequency squared * cosine(angular frequency * time), representing the vertical component of the centripetal acceleration in circular motion.
• 😀 The velocity in SHM reaches maximum magnitude at the equilibrium position and is zero at the maximum displacement positions (1 and 3).
• 😀 The acceleration in SHM is maximum at the maximum displacement positions (1 and 3) and zero at the equilibrium position (2).
• 😀 The velocity and acceleration in SHM are both derived as the slope of the position and velocity graphs, respectively, using the magic tangent line finder technique.
• 😀 The position graph and the velocity graph are related, where the slope of the position graph gives the velocity at each point in time.
• 😀 Similarly, the velocity graph and the acceleration graph are related, where the slope of the velocity graph gives the acceleration at each point in time.
• 😀 The explanation of SHM uses a visual and interactive approach, involving both graphical representation and real-world examples like a mass-spring system and circular motion on a turntable.

Q & A

What is the relationship between simple harmonic motion and circular motion?

Simple harmonic motion is directly related to circular motion, where the vertical component of the circular motion's position corresponds to the position of the object undergoing simple harmonic motion. The velocity and acceleration in simple harmonic motion can also be viewed as the vertical components of the tangential velocity and centripetal acceleration in circular motion.

What does the phase constant (phi) represent in the equation for simple harmonic motion?

The phase constant (phi) represents the phase shift of the curve along the horizontal axis. It determines the starting position of the oscillating object, which can be different from the typical starting point of a cosine function. In this script, phi is assumed to be 0 for simplicity.

What does angular frequency (omega) represent, and how is it derived?

Angular frequency (omega) represents how fast the object oscillates in simple harmonic motion, defined as 2π times the frequency of oscillation. It is derived from the equation for angular velocity, where angular velocity equals angular displacement over change in time, and considering a full revolution takes a period (T), omega is 2π times the frequency.

Why does the position equation for simple harmonic motion take the form of a cosine function?

The position equation follows a cosine function because of the way the object's displacement changes as a function of time, which is similar to the vertical projection of an object in uniform circular motion. The cosine function is used because, in the context of this experiment, the motion is considered to start after a 90-degree rotation of the turntable.

How does the velocity in simple harmonic motion relate to the velocity in circular motion?

The velocity in simple harmonic motion is the vertical component of the tangential velocity of an object moving in circular motion. The velocity graph in simple harmonic motion is derived from the rate of change of the position function with respect to time, and it is represented by the negative sine function of angular frequency times time.

What is the significance of the velocity having a value of 0 at positions 1 and 3 in the graph?

The velocity reaches 0 at positions 1 and 3 because these positions correspond to the maximum displacement points in the simple harmonic motion. At these points, the object changes direction, meaning the instantaneous velocity is momentarily zero before it accelerates back toward the equilibrium position.

What is the relationship between acceleration and velocity in simple harmonic motion?

In simple harmonic motion, acceleration is the derivative of velocity with respect to time. The acceleration graph is also a sinusoidal curve, with its maximum magnitude occurring at the positions of maximum displacement (positions 1 and 3), and it is 0 at the equilibrium position (position 2).

How can the derivative be used to explain the transition between the position, velocity, and acceleration graphs?

By taking the derivative of the position equation with respect to time, you get the velocity equation. Similarly, the derivative of the velocity equation gives the acceleration equation. The slopes of the position, velocity, and acceleration graphs can be used to understand how each graph evolves over time.

What role does the 'magic tangent line finder' play in understanding the graphs?

The 'magic tangent line finder' is used to determine the slope of the position and velocity graphs at any given point. The slope of the position graph gives the velocity, and the slope of the velocity graph gives the acceleration. This tool helps to visually relate the derivatives of each graph to one another.

Why is it important to study the graphs of position, velocity, and acceleration in simple harmonic motion?

Studying these graphs helps us understand the dynamic relationship between position, velocity, and acceleration in simple harmonic motion. The graphs make it clear how the object’s motion changes over time and demonstrate the periodic nature of the oscillation. They also reveal key characteristics such as the maximum displacement, velocity, and acceleration at specific points in the motion.