Pure: Straight Line

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Grade 1 · Term 3

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About A/L Combined Mathematics: Combined Mathematics consists of Pure Mathematics and Applied Mathematics across two years.

• Draw a Free Body Diagram: Sketch the particle, string, and the forces acting on it: Tension (T) acting along the string, and Weight (mg) acting vertically downwards. Indicate the angle θ (30°) with the vertical. 2. Resolve Forces: Resolve the tension T into horizontal and vertical components. Vertical component: T cosθ (upwards)
• Analyze Condition for Completing a Full Circle: For the particle to complete a full circle, the string must remain taut even at the highest point. This means the tension (T) must be greater than or equal to zero at the highest point. At the highest point, the velocity (vtop) must be non-zero. 2. Apply Conservation of Energy (Lowest to Highest Point): Let the lowest point be the reference for potential energy (PE=0). The height of the highest point is 2a. KEbottom + PEbottom = KEtop + PEtop
• Analyze Slack String Condition (Part ii): If u = √(3ag), this is less than √(5ag), so the string will become slack before reaching the top. The string becomes slack when T = 0. 6. Apply Conservation of Energy (Lowest Point to Angle θ): Let v be the velocity at angle θ from the lowest point. The height gained is h = a - a cosθ = a(1 - cosθ). (1/2)mu² = (1/2)mv² + mga(1 - cosθ)
• Apply Newton's Second Law (at Angle θ): Resolve forces radially. Tension (T) acts inwards, and the component of weight (mg cosθ) acts outwards (away from the center). T - mg cosθ = mv²/a
• Solve for θ: Equate Equation 2 and Equation 3. ag + 2ag cosθ = -ag cosθ

Source: Idasara knowledge pack — english/AL_SCIENCE/combined_mathematics v13, short_notes: Lesson 05 - Derives the perpendicular distance from a given point to a given straight line.

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