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: 14
Grade code: 3.1.2.LI.4
Strand code: 1
Sub-strand code: 2
Content standard code: 3.1.2.CS.1
Indicator code: 3.1.2.LI.4
Theme: MECHANICS AND MATTER
Subtheme: KINEMATICS
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This lesson focuses on the energy transformations that occur in systems undergoing Simple Harmonic Motion (SHM). While we have studied the *kinematics* of SHM (displacement, velocity, acceleration), we will now explore the *dynamics* by analysing the energy involved. Understanding this energy is crucial for explaining many real-world phenomena, from the suspension system in a "tro-tro" navigating a bumpy road in Accra, to the vibration of a guitar string, to a child swinging at the park. We will see that while the energy constantly changes its form, the total energy of an ideal SHM system remains constant.
A. Recap of Key SHM Concepts
Before we discuss energy, let's remember the basics of SHM: Defining Condition: The acceleration (`a`) is directly proportional to the displacement (`x`) from the equilibrium position and is always directed towards it. Mathematically, `a ∝ -x`. Key Terms: Amplitude (A): The maximum displacement from the equilibrium position. Equilibrium Position (x=0): The point where the net force is zero. Extreme Positions (x = ±A): The points of maximum displacement where the object momentarily stops before changing direction. Key Equations (for a mass-spring system): Angular frequency, ω = √(k/m) Velocity at displacement x, v = ±ω√(A² - x²) B. Potential Energy (PE) in a Mass-Spring System
The exemplar from NaCCA guides us to use our knowledge from the "area under the load-extension graph." Let's follow that. Hooke's Law: For a spring, the restoring force (F) needed to cause an extension or compression (x) is given by `F = kx`, where `k` is the spring constant. Work Done: The work done in stretching or compressing the spring is stored as Elastic Potential Energy (EPE). This work is calculated as the area under the Force-extension graph. Derivation: The graph of F vs. x is a straight line through the origin. The area under this graph is a triangle. Area = ½ × base × height Area = ½ × x × F Since F = kx, we substitute it in: Area = ½ × x × (kx) = ½kx² Therefore, the Potential Energy stored in the spring at a displacement `x` is: PE = ½kx² Key Insight: At the equilibrium position (x=0), PE = 0. At the extreme positions (x = ±A), the PE is maximum: PE_max = ½kA². C. Kinetic Energy (KE) in a Mass-Spring System Standard Formula: The kinetic energy of any moving object is given by `KE = ½mv²`. Substituting for Velocity in SHM: We know that for SHM, the velocity `v` at any displacement `x` is given by `v = ±ω√(A² - x²)`. Derivation: KE = ½m[±ω√(A² - x²)]² KE = ½m[ω²(A² - x²)] KE = ½mω²(A² - x²) Now, let's recall the relationship for a spring system: ω² = k/m. This means `mω² = k`. Substituting `k` for `mω²` gives us a very elegant formula for KE in terms of displacement: KE = ½k(A² - x²) Key Insight: At the extreme positions (x = ±A), the object is momentarily at rest (v=0), so KE = ½k(A² - A²) = 0. At the equilibrium position (x=0), the speed is maximum, so KE is maximum: KE_max = ½k(A² - 0²) = ½kA². D. Total Mechanical Energy (TME) and Conservation of Energy
The Total Mechanical Energy (TME) of the system is the sum of its Potential Energy and Kinetic Energy, assuming no energy is lost to friction or air resistance.