3. Oscillator An oscillator consists of a block of mass 0.500 kg connected to a spring. When se

3. Oscillator An oscillator consists of a block of mass...

Key Concepts

Maximum Force

The maximum force in a mass-spring system occurs at the points of maximum displacement, where the spring is either maximally extended or compressed. By Hooke’s law, this force is proportional to the amplitude and the spring constant. This concept is essential in understanding the limits of the system's response to displacement.

Maximum Speed

In SHM, the maximum speed is reached as the mass passes through the equilibrium position. It is directly proportional to both the amplitude of the motion and the angular frequency. This concept helps in quantifying the dynamic range of the oscillatory motion.

Spring Constant

The spring constant is a parameter that quantifies the stiffness of the spring in a mass-spring system. According to Hooke’s law, the restoring force exerted by a spring is proportional to the displacement, with the constant of proportionality being the spring constant. It plays a critical role in determining the period and overall dynamics of the oscillator.

Angular Frequency

Angular frequency is a measure of how rapidly the oscillator cycles through its motion in terms of radians per second. It is related to the frequency by the relation ? = 2?f, and it provides a convenient measure in the analysis of SHM and oscillatory systems.

Mass-Spring Oscillator

A mass-spring system is a classical example of SHM where a mass attached to a spring moves back and forth when displaced from its equilibrium position. The system’s properties, such as the mass and the spring constant, determine the characteristics of the oscillation, including the period and frequency.

Period

The period is the time interval required for the oscillator to complete one full cycle of motion. In SHM, the period depends on system parameters; for a mass-spring system, it is influenced by both the mass and the spring constant.

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion in which the restoring force is directly proportional to the displacement and acts in the opposite direction. This behavior is typical in systems like mass-spring oscillators, where the motion is sinusoidal in time and can be characterized by specific parameters such as amplitude, period, and frequency.

Frequency

Frequency refers to the number of oscillations or cycles completed per unit time. It is the reciprocal of the period and is measured in hertz (Hz), representing how many cycles occur in one second.

Transcript

00:01 So in this question, we're given that an oscillator consisting of a block spring system with a mass of 0 .5 kilograms is set into oscillations with amplitude of 3 .5 centimeters, which is equal to 0 .35 meters, and repeats this motion every 0 .5 seconds.

00:21 Then we're asked to find the period, the frequency, the angular frequency, the spring constant, the maximum speed, and the magnitude of the maximum force on the.

00:30 Block due to the spring of given these initial conditions so for a for the period for the period what so some background so like the bit of background that we need for this question so since we're gonna assume simple harmonic motion so then we can assume that it oscillates it the displacement that is oscillates where the distance that oscillates by is given by x or the distance that oscillates by is given by x equal the a which is amplitude times cost of omega, the angular frequency by t at the time.

01:05 And then the velocity is given by, of course, then the derivative of x, which is the minus omega a sign of omega t.

01:17 Then for part eight, so we know that, so priority we're asked for the period.

01:24 So the period by definition is the time for repetition.

01:26 So that means that the period is just equals of what we're given.

01:30 So that's equal to 0 .5 seconds.

01:34 Then for part b, we're asked what the frequency is, so we know that the frequency is equal to 1 over the period, which is equal to 1 over 0 .5 seconds, which is equal to which is equal to 2 hertz, if we work this out, since the unit of inverse seconds is hertz.

01:59 Then for part c, we're asked what the angular frequency is, so angular frequency, omega is equal to 2.

02:06 2 pi times the frequency f.

02:10 So this is equal to if we work it out 2 pi times 2 hertz, which is equal to 4 pi, and this is in units of rads per second, which is approximately equal to 12 .57 rads per second.

02:32 Then for part d, we're asked, what is the spring constant? so we know what the relationship between angular frequency and the spring constant is.

02:40 So omega is equal to the square root of the spring constant k over the mass of the particle.

02:46 And this is true for a spring mass system.

02:50 So then rearranging this, we get that k is equal to omega squared times the mass.

03:02 And if we work this out, then this is equal to, so if we square it, so this is equal to 12 .9, 12 .57.

03:14 Squared, 12 .57 squared multiplied by mass, which is 0 .5, which is equal to 0 .5 kilogram, so 0 .5.

03:25 And if we work this out, this is equal to, so this is equal to 7 .81, and the units are newtons per meter.

03:43 Sorry, there's a mistake here.

03:45 So this, i missed a decimal place, so this is actually equal to 79 newtons per meter...

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