Mechanics Notes for Cambridge International AS & A Level
1. Forces and Resultants
Key Concepts:
- Force: A vector quantity that can change an object’s motion (measured in Newtons, N).
- Resultant Force: The single force that has the same effect as all the forces acting on a body
combined.
- Vector Addition: Forces can be combined using the parallelogram or triangle method.
- Parallelogram Rule: Draw two forces as adjacent sides of a parallelogram; the diagonal
represents the resultant.
- Triangle Method: Arrange forces tip-to-tail; the resultant is the vector joining the start of
the first force to the end of the last.
- Components of Forces:
- Horizontal Component: F_x = F cos θ
- Vertical Component: F_y = F sin θ
- Equilibrium:
- If the resultant force is zero, the object is in equilibrium.
- Conditions for equilibrium:
1. The vector sum of all forces is zero.
2. The sum of moments about any point is zero.
2. Newton's Laws of Motion
First Law (Law of Inertia):
- An object remains at rest or moves with constant velocity unless acted upon by an external
force.
- Inertia: Resistance to changes in motion; proportional to mass.
Second Law:
- The rate of change of momentum of an object is directly proportional to the resultant force
acting on it.
- Formula: F = ma, where:
- F: Force (N)
- m: Mass (kg)
- a: Acceleration (m/s²)
- For systems with multiple forces: F_net = ΣF = ma.
Third Law:
- For every action, there is an equal and opposite reaction.
- Forces always occur in pairs (action and reaction).
3. Friction
Key Concepts:
- Definition: A force that opposes motion or attempted motion between two surfaces in
contact.
- Types of Friction:
- Static Friction (F_s): Prevents motion; F_s ≤ μ_s R
- Kinetic Friction (F_k): Opposes motion when an object is sliding; F_k = μ_k R
- μ_s: Coefficient of static friction
- μ_k: Coefficient of kinetic friction
- R: Normal contact force
- Factors Affecting Friction:
- Nature of the surfaces in contact (rough or smooth).
- Normal force (greater normal force increases friction).
- Frictional Force on an Inclined Plane:
- F_friction = μR, where R = mg cos θ.
Applications:
- Limiting Friction: Maximum force before sliding starts.
- Angle of Repose: The maximum angle at which an object remains stationary on an inclined
plane:
- tan θ = μ.
4. Types of Forces
1. Gravitational Force:
- Definition: The force of attraction between two masses due to gravity.
- Formula: F = Gm₁m₂ / r², where:
- G: Gravitational constant (6.674 × 10⁻¹¹ Nm²/kg²)
- m₁, m₂: Masses of the objects (kg)
- r: Distance between the centers of the masses (m)
- Characteristics:
- Always attractive.
- Acts over long distances.
- Example: Weight (W = mg), where g = 9.8 m/s² near Earth.
2. Normal Force:
- Definition: A contact force exerted by a surface perpendicular to the surface to support the
weight of an object.
- Characteristics:
- Acts perpendicular to the surface.
- Example: A book resting on a table experiences a normal force equal to its weight.
3. Frictional Force:
- Definition: A contact force that opposes the motion (or attempted motion) of an object.
- Types:
- Static Friction: Prevents an object from starting to move.
- Kinetic Friction: Opposes motion when the object is sliding.
- Formula: F_friction = μR.
- Example: Sliding a box across a rough floor.
4. Tension Force:
- Definition: A pulling force transmitted through a string, rope, or cable when it is stretched
by forces acting from opposite ends.
- Characteristics:
- Acts along the length of the string/rope.
- Example: Tension in a rope holding a hanging mass.
5. Air Resistance (Drag):
- Definition: A force exerted by air that opposes the motion of an object through it.
- Characteristics:
- Increases with speed and surface area.
- Example: A parachute experiences air resistance to slow its descent.
6. Magnetic Force:
- Definition: The force between magnetic poles or a magnetic field and a moving charged
particle.
- Formula: F = qvB sin θ.
- Example: Force on a current-carrying conductor in a magnetic field.
7. Electrostatic Force:
- Definition: The force between two charged objects.
- Formula: F = kq₁q₂ / r².
- Characteristics:
- Can be attractive or repulsive.
- Example: Force between two charged particles.
8. Buoyant Force:
- Definition: The upward force exerted by a fluid that opposes the weight of an object
submerged in it.
- Formula: F_buoyant = ρVg.
- Example: A boat floating on water experiences buoyant force equal to its weight.
9. Spring Force:
- Definition: The restoring force exerted by a stretched or compressed spring.
- Formula: F = -kx.
- Characteristics:
- Acts in the opposite direction of displacement.
- Example: Force in a spring compressed by 0.1 m with k = 100 N/m is F = -10 N.
10. Centripetal Force:
- Definition: The force directed towards the center of a circular path that keeps an object in
circular motion.
- Formula: F = mv² / r.
- Example: Tension in a string when swinging a mass in a circular path.