8. A Small Body A small body of mass 0.12 kg is undergoing simple harmonic motion of amplitude

8. A Small Body A small body of mass 0.12 kg is undergoing...

8. A Small Body A small body of mass $0.12 \mathrm{~kg}$ is undergoing simple harmonic motion of amplitude $8.5 \mathrm{~cm}$ and period $0.20 \mathrm{~s}$ (a) What is the magnitude of the maximum force acting on it? (b) If the oscillations are produced by a spring, what is the spring constant?

8. A Small Body A small body of mass $0.12 \mathrm{~kg}$ is undergoing simple harmonic motion of amplitude $8.5 \mathrm{~cm}$ and period $0.20 \mathrm{~s}$
(a) What is the magnitude of the maximum force acting on it? (b) If the oscillations are produced by a spring, what is the spring constant?

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion well known in physics where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. This concept is central to understanding oscillatory systems, such as springs and pendulums, and it underpins the mathematical descriptions involving sinusoidal functions.

Amplitude

Amplitude in SHM represents the maximum displacement of the oscillating object from its equilibrium position. It is a key concept because it directly affects other properties of the motion, such as the maximum acceleration and the maximum force experienced by the object.

Period and Angular Frequency

The period is the time required for one complete cycle of motion in SHM. Angular frequency, derived from the period as ? = 2?/T, is crucial because it links the time domain properties of the oscillation with dynamic quantities such as acceleration and force, thereby providing a bridge between temporal and mechanical analyses in SHM.

Maximum Force in SHM

The maximum force in SHM occurs when the displacement is at its maximum (i.e., at the amplitude) and is calculated using the relation F_max = m?²A, where m is the mass, ? is the angular frequency, and A is the amplitude. This concept helps in determining the extreme dynamic loads in oscillatory systems.

Hooke's Law and Spring Constant

Hooke's Law states that the restoring force in a spring is proportional to the displacement from its equilibrium position, given by F = -kx, where k is the spring constant. In the context of SHM, the spring constant is related to the mass and the angular frequency by the equation k = m?². This relationship is fundamental for characterizing the stiffness of a spring in oscillatory systems.

Transcript

00:01 So in this question we're asked to determine first the maximum force, the magnitude of the maximum force experienced by a small body oscillating with simple harmonic motion and then the spring constant if this oscillation was provided by a spring.

00:18 So we're given that the mass is of 0 .2.

00:22 We're given that the mass, so we're given m is equal to 0 .12 kilograms.

00:31 The maximum or the amplitude of the oscillations equal to 8 .5 centimeters which is equal 0 .085 meters and we're also given that the that the period of oscillation is equal to zero point it's equal to 0 .2 seconds so then using all this we can answer the question.

01:06 So for part a, we know that the relationship between omega and the normal frequency is equal to 2 pi f.

01:15 But we also know that f is equal to 1 over the period.

01:19 So if we substitute both of these in, we get that omega is equal to 2 pi over t, which is equal to in this case 2 pi over t, which is 0 .2.

01:32 And this is equal to, in this case, 31 .3.

01:36 0 .42 rads per second.

01:41 Then using this, we can calculate the maximum force.

01:45 So we know that the force is equal to mass times acceleration, which is equals the mass times the acceleration for an oscillator is equal to omega squared amplitude times cost of omega t.

02:05 And since we're looking for the maximum force at max, we're going to need the maximum acceleration.

02:12 This is equal to m by a max...

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