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: 12
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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Welcome, my dear students. In our previous lessons, we learned to describe the motion of objects that oscillate back and forth, a special type of motion we call Simple Harmonic Motion (SHM). We see this everywhere in Ghana – from a child on a swing during playtime (ampe), to the suspension of a tro-tro navigating a bumpy road, to the pendulum of a grandfather clock. Today, we will go deeper. Motion involves energy. So, how does energy behave in a system performing SHM? We will investigate the transformation between different types of energy and learn how to calculate the total energy stored in an oscillating system.
Recap: What is SHM? Simple Harmonic Motion (SHM) is a periodic motion where the restoring force (and hence acceleration) is directly proportional to the displacement from the equilibrium position and is always directed towards that equilibrium position. Mathematically: `F = -kx` (Hooke's Law) Equilibrium Position (x=0): The point of zero net force. Extreme Positions (x = ±A): The points of maximum displacement, where A is the amplitude. Energy in an Oscillating System Any object in motion has Kinetic Energy (KE), and due to its position or state, it can have Potential Energy (PE). In SHM, the system possesses both, and they constantly transform into one another. The sum of KE and PE is the Total Mechanical Energy (E).
`E = KE + PE`
For an ideal system with no friction or air resistance, this total energy is conserved (remains constant). Case Study 1: The Mass-Spring System
Let's consider a mass `m` attached to a horizontal spring with spring constant `k`, oscillating on a frictionless surface. Potential Energy (PE) in the Spring Source: The PE is stored in the spring when it is stretched or compressed. This is called Elastic Potential Energy. Derivation (from Area under the Load-Extension Graph): Recall from our study of elasticity that the work done in stretching a spring by a displacement `x` is stored as elastic potential energy. The force required is not constant; it increases from 0 to `F = kx`. The work done is the area under the force-displacement graph, which is a triangle. Work Done (W) = Area of triangle = ½ × base × height W = ½ × displacement × maximum force W = ½ × `x` × (`kx`) Therefore, the Potential Energy stored is: PE = ½ kx² At different positions: At the equilibrium position (x=0), the spring is not stretched or compressed. So, PE = 0. At the extreme positions (x = ±A), the displacement is maximum. So, the PE is maximum: PE_max = ½ kA². Kinetic Energy (KE) of the Mass Source: The KE comes from the motion of the mass `m`. The formula is KE = ½ mv². At different positions: At the extreme positions (x = ±A), the mass momentarily stops before changing direction. Its velocity `v` is zero. Therefore, KE = 0. As the mass moves towards the equilibrium position, the restoring force accelerates it, and its speed increases. At the equilibrium position (x=0), the mass is moving at its fastest. Its velocity is maximum (`v_max`). Therefore, the KE is maximum: KE_max = ½ mv_max². Total Mechanical Energy (E)