Chapter 1: Force and Motion II

1.1 Resultant Force

• Resultant force is the single force that represents the vector sum of two or more forces acting on an object. (p.3)
• Two equal forces in opposite directions: resultant force is zero and a stationary object stays at rest. (p.3)
• Forces in the same direction: magnitude = sum of the forces, direction = same as the forces. (p.4)
• Forces in opposite directions: magnitude = difference of the forces, direction = that of the larger force. (p.4)
• Perpendicular forces: F = √(F₁² + F₂²) (Pythagoras); angle θ = tan⁻¹(opposite/adjacent). (p.5)
• Forces not perpendicular: use the triangle of forces method or the parallelogram of forces method (scaled drawing). (p.6)
• A Vector Force Table Kit can be used to find the resultant of two forces at an angle; the balancing force through pulley C equals the resultant. (p.7)
• A free body diagram shows all the forces acting on that object only. (p.8)
• Newton's Second Law: F = ma, where F is the resultant force. (p.9)
• Stationary or uniform velocity: a = 0, resultant force = 0. Accelerating: a ≠ 0, resultant force ≠ 0. (p.9)
• Lift problem: stationary R = W = mg; accelerating upwards R − W = ma. (p.11)
• Trolley and load connected by a string move with the same acceleration; tension T acts on both. (p.12)

1.2 Resolution of Forces

• Resolution of forces is the process of resolving a force into two components. (p.13)
• Horizontal component F_x = F cos θ; vertical component F_y = F sin θ. (p.14)
• Resolving a force allows each perpendicular direction to be analysed separately for the resultant. (p.15)
• On an inclined plane at angle θ: component of weight parallel to the plane W_x = W sin θ; perpendicular W_y = W cos θ. (p.17)

1.3 Forces in Equilibrium

• An object is in equilibrium of forces when the forces acting on it produce a zero resultant force (a = 0). (p.18)
• A triangle of forces shows the equilibrium of three forces; the forces are drawn in sequence to form a closed triangle. (p.18)
• Three forces in equilibrium can be solved by resolution of forces or by a scaled triangle of forces. (p.21)
• Box on an incline in equilibrium: F_R = W sin θ; R = W cos θ. (p.23)
• The sine rule and cosine rule can be used on the triangle of forces for accurate answers. (p.23)

1.4 Elasticity

• Elasticity is the property of a material that enables an object to return to its original shape and size after the applied force is removed. (p.25)
• Extension of a spring, x = l − l₀; do not exceed half the original length to avoid overstretching. (p.26)
• Hooke's law: extension of a spring is directly proportional to the applied force provided the elastic limit is not exceeded; F = kx. (p.27)
• Spring constant, k = F/x = gradient of the graph of F against x. (p.27)
• Elastic potential energy = area under the F–x graph: E_P = ½Fx = ½kx². (p.28)
• Factors affecting k: material; shorter spring → higher k; smaller coil diameter → higher k; thicker wire → higher k. (p.30)
• Springs in series: each carries tension F and extension x; total extension = 2x. Springs in parallel: force is divided equally; extension of system = x/2. (p.30)