Figure P14.32 is the velocity-versus-time graph of a...

Key Concepts

Amplitude

The amplitude in SHM refers to the maximum displacement from the equilibrium position. In a velocity-time graph, the maximum velocity observed is related to the amplitude through the angular frequency of the oscillation. It represents the extent of the oscillation regardless of the velocity or phase.

Initial Position Determination

The initial position of an oscillating particle in SHM can be determined by combining the information about the amplitude and the phase constant. Since position in SHM is typically modeled with a cosine (or sine) function, evaluating the function at time zero using the determined phase constant provides the particle’s displacement from equilibrium at the start of the motion.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where a restoring force is proportional to the displacement and acts in the opposite direction. This results in oscillations that are sinusoidal in time, and the behavior of the system is characterized by parameters such as amplitude, angular frequency, and phase constant.

Velocity-Time Graph Analysis

A velocity-time graph in the context of SHM displays the sinusoidal variation of velocity with time. By analyzing this graph, one can determine characteristics of the motion such as maximum velocity, which relates directly to the amplitude and angular frequency, and the phase information, which provides insights into the timing of the oscillation relative to the equilibrium position.

Phase Constant

The phase constant is a parameter that defines the initial angle (or phase) of the oscillatory function at time zero. It shifts the sinusoidal function along the time axis, determining the position and velocity of the oscillating particle at the start of the motion, and is crucial for matching the equation of motion to the initial conditions.

Transcript

00:01 A position and a velocity as a function of time in simple harmonic motion.

00:06 X as a function of time equaling a cosine of omega -t, the angular frequency multiplied by time plus the phase constant, phi -not.

00:16 And we have the velocity in the x -direction as a function of time, equaling the negative amplitude multiplied by the angular frequency, multiplied by sine of omega -t, finite this is also equaling the negative maximum velocity multiplied by sine of omega t plus finite and so here we know that from the graph t equals 12 seconds and so the angular frequency would be equaling to 2 pi over the period t this is equaling to 2 pi over 12 seconds and we find that then the angular frequency is equaling to pi over six radians per second.

01:04 For part a, we know the maximum velocity is equaling the product of the amplitude and the angular frequency, and we know that this is equaling then 60 centimeters per second.

01:16 And so the amplitude then would be equaling the maximum velocity divided by the angular frequency.

01:23 This would be equaling than 60 centimeters per second divided by pi over six radians per second.

01:31 And this is giving us approximately 115 centimeters or we can say 1 .15 meters.

01:42 For part b we have t equaling zero seconds and so we can say that the initial velocity in the x direction would be equaling to negative a omega sign of finite this is going to then equal negative 30 centimeters per second and so we can say then that negative 60 centimeters per second, the maximum velocity, multiplied by sine of phi not equals negative 30 centimeters per second.

02:24 And so we can say that then phi not is equaling arc sign of 1⁄2 or 0 .5 radians...

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