1.1 Resultant Force
Resultant Force ($F$) is the single force that represents the combined vector effect of two or more forces acting on an object.
1. Resolving Vectors and Determining Resultant Force
- Parallel / Linear Forces: Forces acting in the same direction add up directly ($F_R = F_1 + F_2$). Opposite forces subtract ($F_R = F_1 - F_2$).
- Perpendicular Forces: When two forces act at right angles ($90^\circ$):
- Magnitude: $$F = \sqrt{F_x^2 + F_y^2}$$
- Direction ($\theta$ with respect to the horizontal): $$\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)$$
- Non-Perpendicular Forces: Can be determined using scale drawings (Triangle Method or Parallelogram Method) or resolved into orthogonal $x$ and $y$ components.
2. Newton's Second Law of Motion
The relationship between resultant force, mass, and acceleration is given by:
$$F = ma$$
- If $F > 0$, the object accelerates in the direction of the resultant force.
- If $F = 0$, the object is either stationary or moving at a constant velocity (zero acceleration).
3. Real-World Applications of Resultant Force
- Moving Elevator (Lift):
- Stationary or moving at constant velocity: Apparent weight $R = mg$.
- Accelerating upwards at $a$: $R - mg = ma \implies R = m(g + a)$ (Feels heavier).
- Accelerating downwards at $a$: $mg - R = ma \implies R = m(g - a)$ (Feels lighter).
- Incline Plane:
- Component of weight parallel to the slope (pulling down): $W_{\parallel} = mg \sin\theta$.
- Component of weight perpendicular to the slope: $W_{\perp} = mg \cos\theta$.
- Normal reaction force: $R = mg \cos\theta$.
- Acceleration down a frictionless slope: $a = g \sin\theta$.
1.2 Resolution of Forces
Resolution of Forces is the process of splitting a single force into two mutually perpendicular components ($F_x$ and $F_y$).
Formulae for Resolution
Given a force $F$ acting at an angle $\theta$ relative to the horizontal plane:
- Horizontal Component: $$F_x = F \cos\theta$$
- Vertical Component: $$F_y = F \sin\theta$$
Inclined Planes and Pulley Systems
- Inclined Plane: Weight $W$ acts vertically downwards. Resolving along the incline gives $F_{\text{parallel}} = W \sin\theta$ and $F_{\text{perpendicular}} = W \cos\theta$.
- Lawn Mower / Suitcase Problem:
- Pushing: Downward component adds to gravity. Total vertical force on ground = $W + F \sin\theta$. Normal force increases.
- Pulling: Upward component acts against gravity. Total vertical force on ground = $W - F \sin\theta$. Normal force decreases (easier to pull over rough terrain).
1.3 Forces in Equilibrium
An object is in forces equilibrium when the forces acting on it produce a resultant force of zero ($F_{\text{net}} = 0$).
Key Characteristics
- Vector sum of forces is zero: $\sum F_x = 0$ and $\sum F_y = 0$.
- The object remains stationary or moves at a constant velocity ($a = 0$).
- When three forces act on an object in equilibrium, their vectors form a closed triangle of forces.
Triangle of Forces & Lami's Theorem
- Triangle of Forces: Three forces in equilibrium can be arranged head-to-tail to form a closed triangle.
- Sine Rule for Forces: $$\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}$$
1.4 Elasticity
Elasticity is the property of a material to return to its original shape and size after the deforming force applied to it is removed.
1. Hooke's Law
Hooke's Law states that the extension of a spring, $x$, is directly proportional to the applied stretching force, $F$, provided the elastic limit is not exceeded.
$$F = kx$$
- $F$ = Applied force ($\text{N}$)
- $x$ = Extension or compression ($\text{m}$)
- $k$ = Spring constant ($\text{N m}^{-1}$), representing stiffness. A larger $k$ indicates a stiffer spring.
2. Elastic Potential Energy ($E_p$)
The work done to stretch or compress a spring equals the elastic potential energy stored in it (area under the $F-x$ graph):
$$E_p = \frac{1}{2} F x = \frac{1}{2} k x^2$$
3. Factors Affecting the Spring Constant ($k$)
- Length of Spring: Shorter springs have higher $k$ (stiffer).
- Diameter of Spring Wire: Thicker wire yields higher $k$ (stiffer).
- Diameter of Spring Coil: Smaller coil diameter yields higher $k$ (stiffer).
- Material of Spring: Steel has a higher $k$ than copper.
4. Arrangement of Springs
- Series Arrangement:
- Each spring experiences the full load $F$.
- Total Extension: $x_{\text{total}} = x_1 + x_2 + \dots$
- Effective Spring Constant: $\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2}$
- Parallel Arrangement:
- Load is divided equally among springs: $F_i = \frac{F}{n}$.
- Total Extension: $x_{\text{total}} = \frac{x}{n}$
- Effective Spring Constant: $k_{\text{eff}} = k_1 + k_2 + \dots = n \cdot k$ (Stiffer system).