Lesson Notes By Weeks and Term v4 - SHS 3
KINEMATICS
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KINEMATICS
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Subject: Physics
Class: SHS 3
Term: 1st Term
Week: 11
Grade code: 3.1.2.LI.2
Strand code: 1
Sub-strand code: 2
Content standard code: 3.1.2.CS.1
Indicator code: 3.1.2.LI.2
Theme: MECHANICS AND MATTER
Subtheme: KINEMATICS
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This lesson explores the quantitative aspects of Simple Harmonic Motion (SHM). We have previously described SHM as a special type of to-and-fro motion. Today, we will use mathematics to precisely describe the position, speed, and acceleration of an object undergoing SHM at any given moment. This is crucial for understanding many real-world phenomena, from the swaying of a tall building like Villaggio Vista in the wind, to the operation of a car's suspension system on a bumpy road in our communities, or even the rhythmic swing of a child's `nkɔn nkɔn` (swing) at the park. By mastering these equations, we can predict and engineer systems that rely on oscillatory motion.
Recap: What is SHM? Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force (and therefore acceleration) is directly proportional to the displacement from the equilibrium position and is always directed towards that equilibrium position. Mathematically, this defining condition is: `a ∝ -x` Deriving the Equations of SHM using a Reference Circle The easiest way to understand the equations for SHM is to visualize it as a projection of uniform circular motion.
Imagine a particle 'P' moving in a circle of radius `A` with a constant angular velocity `ω`. Now, imagine a lamp casting a shadow of this particle onto a vertical screen (the diameter). The shadow, let's call it 'N', will move up and down along the diameter. When P is at the top or bottom, N is at the ends of its path. When P is at the sides, N is at the center of its path.
The motion of this shadow 'N' is a perfect example of Simple Harmonic Motion. We can use trigonometry on this reference circle to find the equations for displacement, velocity, and acceleration. A. Displacement (x) Displacement is the position of the oscillating object from its equilibrium (center) position at any time `t`. From the diagram, `θ` is the angle swept by the particle P. Since `ω = θ/t`, we have `θ = ωt`. Using trigonometry in the triangle OPN, `cos(θ) = Adjacent / Hypotenuse = x / A`. Rearranging this gives us the displacement `x`:
> Equation for Displacement: > `x = A cos(ωt)`