A 1.0 kg block oscillates on a spring with spring constant...

A $1.0 \mathrm{kg}$ block oscillates on a spring with spring constant 20 N/m. At $t=0$ s the block is $20 \mathrm{cm}$ to the right of the equilibrium position and moving to the left at a speed of $100 \mathrm{cm} / \mathrm{s}$ Determine the period of oscillation and draw a graph of position versus time.

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic oscillatory behavior where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This concept is a fundamental topic in classical mechanics and is characterized mathematically by sinusoidal functions, which describe the system's displacement, velocity, and acceleration over time.

Angular Frequency and Period

Angular frequency, commonly denoted by ?, quantifies how quickly the system oscillates in radians per second and is derived from the physical parameters of the system, typically using the formula ? = ?(k/m) for a mass-spring system. The period of oscillation, T, is the time it takes for one complete cycle of motion and is related to angular frequency by T = 2?/?. These relationships are pivotal in analyzing and predicting the behavior of oscillatory systems.

Position vs. Time Graph

A position versus time graph for a system undergoing simple harmonic motion is a sinusoidal curve that illustrates the periodic oscillation between the maximum and minimum displacements. The graph provides visual insights into key aspects of the motion, such as amplitude, period, phase, and equilibrium position, enabling a comprehensive understanding of the oscillator's dynamics.

Mass-Spring System

A mass-spring system is a physical setup that exemplifies simple harmonic motion, consisting of a mass attached to a spring that obeys Hooke’s law. The spring provides a linear restoring force proportional to the displacement from equilibrium, and the mass's inertia determines how quickly it responds to the force. This system is used extensively to model oscillatory phenomena in both mechanical and other contexts.

Transcript

00:01 So here we have simple harmonic motion and this would be modeled by x position as a function of time equaling the amplitude times cosine of the angular frequency multiplied by time plus the phase constant.

00:15 The velocity in the x -direction as a function of time would be equaling to the negative amplitude multiplied by the angular frequency multiplied by sine of the angular frequency multiplied by time plus again the phase constant.

00:29 So we need to then determine the angular frequency, the phase constant, and the amplitude.

00:37 So we can first start off by saying the period equaling 2 pi multiplied by the mass over the spring constant k.

00:46 This would be equaling to 2 pi multiplied by 1 .0 kilograms.

00:52 This would be divided by 20 neutins per meter.

00:56 And this is giving us 1 .405 seconds.

01:00 So the angular frequency would then be equaling to 2 pi divided by t, the time period, 2 pi divided by 1 .405 seconds.

01:12 And this is giving us then 4 .472 radians per second.

01:20 We know that initially we can say at t equaling 0 seconds, the initial position equals the amplitude multiplied by cosine of phi not and we know that the initial velocity in the x direction would be equaling to negative a omega sine of phi not dividing these two equations we can say that then tangent of phi not would be equaling the negative initial velocity in the x direction divided by the angular frequency multiplied by the initial exposition this is equaling to then negative negative 1 .0 meters per second, this would be divided by 4 .472 radians per second, multiplied by 0 .20 meters.

02:19 And we find that then this is equaling 1 .1181...

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