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: 13
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 focuses on the kinematics of Simple Harmonic Motion (SHM), which is the mathematical description of a special type of oscillatory or vibratory motion. We see this motion all around us in Ghana: from the gentle swing of a child's 'aboya' (swing), to the vibration of a guitar string, and even the up-and-down motion of a piston in a car or a fufu pounding machine. By understanding the equations that describe the displacement, velocity, and acceleration of an object in SHM, we can predict its motion and analyse the energy involved. This knowledge is fundamental to engineering, music, and many other fields.
A. Recap: What is SHM? Remember that an object is performing Simple Harmonic Motion (SHM) if: Its acceleration is directly proportional to its displacement from a fixed equilibrium position. The acceleration is always directed towards that equilibrium position. Mathematically, this is expressed as: `a ∝ -x` Which leads to the defining equation of SHM: `a = -ω²x` Where: `a` is acceleration. `x` is displacement from equilibrium. `ω` is the angular frequency (a constant for the system, measured in radians per second, rad/s). B. Deriving the Kinematic Equations using Circular Motion Analogy
The easiest way to understand the equations of SHM is to see it as the *projection* of an object moving in a uniform circle. Imagine a point `P` moving at a constant angular velocity `ω` around a circle of radius `A`. Now, imagine a light shining from above, casting a shadow of `P` onto the horizontal diameter (the x-axis). We will call this shadow point `N`.
As `P` goes around the circle, its shadow `N` moves back and forth along the diameter. This back-and-forth motion of `N` is perfect Simple Harmonic Motion. Displacement (x) Let's assume the particle `P` starts at the extreme right (at time `t=0`). After time `t`, it has moved through an angle `θ = ωt`. From the right-angled triangle OPN, `cos(θ) = Adjacent / Hypotenuse = ON / OP`. The displacement of the shadow `N` from the centre `O` is `x = ON`. The radius of the circle is the maximum displacement, which we call the Amplitude (A). So, `OP = A`. Substituting these in, we get `cos(ωt) = x / A`.
Rearranging for `x`, we get the equation for displacement: `x = A cos(ωt)` `x` = displacement at time `t` (m) `A` = Amplitude (maximum displacement) (m) `ω` = Angular frequency (rad/s). Remember, `ω = 2πf = 2π/T`. `t` = time (s)