Formula Sheet: Vector Analysis and Forces Acting on an Object

Vector Analysis and Forces Acting on an Object

Vector Fundamentals

Vector Notation and Representation

• Magnitude of a vector: \[|\vec{A}| = A = \sqrt{A_x^2 + A_y^2 + A_z^2}\] Where:
  • \(A_x, A_y, A_z\) = components of vector in x, y, z directions
  • Units: same as the vector quantity (m, m/s, N, etc.)
• Unit vector: \[\hat{A} = \frac{\vec{A}}{|\vec{A}|}\] Where:
  • \(\hat{A}\) = unit vector in direction of \(\vec{A}\)
  • Magnitude of unit vector = 1 (dimensionless)

Vector Addition and Subtraction

• Component method: \[\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}\] Where:
  • \(\vec{R}\) = resultant vector
  • \(\hat{i}, \hat{j}, \hat{k}\) = unit vectors in x, y, z directions
• Vector subtraction: \[\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\] Subtract corresponding components

Vector Components and Decomposition

• Component in x-direction: \[A_x = A \cos\theta\] Where:
  • \(\theta\) = angle measured from positive x-axis
• Component in y-direction: \[A_y = A \sin\theta\]
• Angle from components: \[\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)\] Note: Consider quadrant when determining angle
• Magnitude from components: \[A = \sqrt{A_x^2 + A_y^2}\] For 2D vectors

Vector Multiplication

• Dot product (scalar product): \[\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\] Where:
  • \(\theta\) = angle between vectors
  • Result is a scalar
  • If \(\vec{A} \perp \vec{B}\), then \(\vec{A} \cdot \vec{B} = 0\)
• Cross product (vector product): \[|\vec{A} \times \vec{B}| = AB\sin\theta\] Where:
  • Result is a vector perpendicular to both \(\vec{A}\) and \(\vec{B}\)
  • Direction determined by right-hand rule
  • If \(\vec{A} \parallel \vec{B}\), then \(|\vec{A} \times \vec{B}| = 0\)

Newton's Laws of Motion

First Law (Law of Inertia)

• Statement: An object at rest stays at rest, and an object in motion stays in motion with constant velocity unless acted upon by a net external force
  • If \(\vec{F}_{net} = 0\), then \(\vec{v} = \text{constant}\)

Second Law

• Net force equation: \[\vec{F}_{net} = m\vec{a} = \frac{d\vec{p}}{dt}\] Where:
  • \(\vec{F}_{net}\) = net force (N)
  • \(m\) = mass (kg)
  • \(\vec{a}\) = acceleration (m/s²)
  • \(\vec{p}\) = momentum (kg⋅m/s)
• Component form: \[F_{net,x} = ma_x\] \[F_{net,y} = ma_y\] \[F_{net,z} = ma_z\] Each component independent
• Weight: \[W = mg\] Where:
  • \(W\) = weight (N), always directed downward
  • \(g\) = acceleration due to gravity = 9.8 m/s² (or 10 m/s² for MCAT approximation)

Third Law

• Action-reaction pairs: \[\vec{F}_{12} = -\vec{F}_{21}\] Where:
  • Forces are equal in magnitude, opposite in direction
  • Act on different objects

Types of Forces

Gravitational Force

• Universal gravitation: \[F_g = G\frac{m_1m_2}{r^2}\] Where:
  • \(G\) = gravitational constant = 6.67 × 10^-11 N⋅m²/kg²
  • \(m_1, m_2\) = masses (kg)
  • \(r\) = distance between centers of mass (m)
  • Always attractive
• Gravitational field strength: \[g = \frac{GM}{r^2}\] Where:
  • \(g\) = gravitational acceleration (m/s²)
  • \(M\) = mass of planet/body creating field (kg)
  • \(r\) = distance from center (m)

Normal Force

• On horizontal surface: \[N = mg\] When no other vertical forces present
• On horizontal surface with additional vertical force: \[N = mg + F_{\perp}\] Where \(F_{\perp}\) is additional force perpendicular to surface (positive if pushing down)
• On inclined plane: \[N = mg\cos\theta\] Where:
  • \(\theta\) = angle of incline from horizontal
  • Normal force perpendicular to surface
• Note: Normal force is a contact force perpendicular to the surface

Friction Forces

• Static friction (object at rest): \[f_s \leq \mu_s N\] Where:
  • \(f_s\) = static friction force (N)
  • \(\mu_s\) = coefficient of static friction (dimensionless)
  • \(N\) = normal force (N)
  • Maximum value: \(f_{s,max} = \mu_s N\)
  • Direction: opposes potential motion
• Kinetic friction (object in motion): \[f_k = \mu_k N\] Where:
  • \(f_k\) = kinetic friction force (N)
  • \(\mu_k\) = coefficient of kinetic friction (dimensionless)
  • Generally \(\mu_k <>
  • Direction: opposes motion
  • Magnitude constant (independent of velocity and contact area)

Tension Force

• Ideal rope/string: Tension is uniform throughout
  • Massless and inextensible assumption
  • Pulls equally on both ends
  • Always directed along the rope/string
• Rope with mass: Tension varies along length
  • Must account for weight of rope

Spring Force (Hooke's Law)

• Restoring force: \[F_s = -kx\] Where:
  • \(F_s\) = spring force (N)
  • \(k\) = spring constant (N/m)
  • \(x\) = displacement from equilibrium position (m)
  • Negative sign indicates force opposes displacement
• Elastic potential energy: \[U_s = \frac{1}{2}kx^2\] Where \(U_s\) = elastic potential energy (J)

Centripetal Force

• Centripetal force (not a new type of force): \[F_c = \frac{mv^2}{r} = m\omega^2 r = \frac{4\pi^2mr}{T^2}\] Where:
  • \(F_c\) = centripetal force (N), directed toward center
  • \(m\) = mass (kg)
  • \(v\) = tangential velocity (m/s)
  • \(r\) = radius of circular path (m)
  • \(\omega\) = angular velocity (rad/s)
  • \(T\) = period (s)
• Note: Centripetal force is provided by other forces (tension, gravity, friction, normal force, etc.)

Drag Force

• Air resistance (general): \[F_d = \frac{1}{2}C\rho Av^2\] Where:
  • \(F_d\) = drag force (N)
  • \(C\) = drag coefficient (dimensionless)
  • \(\rho\) = fluid density (kg/m³)
  • \(A\) = cross-sectional area (m²)
  • \(v\) = velocity (m/s)
  • Opposes direction of motion
• Terminal velocity: \[v_t = \sqrt{\frac{2mg}{C\rho A}}\] When drag force equals weight (\(F_d = mg\))

Inclined Plane Problems

Force Components on Incline

• Parallel to incline (down the slope): \[F_{\parallel} = mg\sin\theta\] Where \(\theta\) = angle of incline from horizontal
• Perpendicular to incline: \[F_{\perp} = mg\cos\theta\] This component is balanced by normal force
• Normal force: \[N = mg\cos\theta\] When no additional perpendicular forces
• Net force down incline (with friction): \[F_{net} = mg\sin\theta - f\] Where \(f\) = friction force
• Acceleration down frictionless incline: \[a = g\sin\theta\]
• Acceleration down incline with friction: \[a = g(\sin\theta - \mu_k\cos\theta)\] When object is sliding down

Free Body Diagrams (FBD)

FBD Construction Principles

• Isolate the object of interest
• Draw all forces acting on the object (not by the object)
• Represent each force as a vector arrow
• Label each force clearly
• Choose appropriate coordinate system
• Do not include action-reaction pairs on same FBD

Equilibrium Conditions

Translational Equilibrium

• Static equilibrium: \[\sum \vec{F} = 0\] or in components: \[\sum F_x = 0\] \[\sum F_y = 0\] \[\sum F_z = 0\] Where:
  • Net force in all directions is zero
  • Acceleration is zero
  • Velocity is constant (may be zero)

Rotational Equilibrium

• Torque equilibrium: \[\sum \vec{\tau} = 0\] Where:
  • Net torque is zero
  • Angular acceleration is zero
  • Angular velocity is constant

Systems of Objects

Connected Objects

• Two objects connected by rope: Apply Newton's second law to each object separately
  • Same tension throughout ideal rope
  • Same magnitude of acceleration if inextensible rope
• Atwood machine (two masses over pulley): \[a = \frac{(m_2 - m_1)g}{m_1 + m_2}\] \[T = \frac{2m_1m_2g}{m_1 + m_2}\] Where:
  • \(m_2 > m_1\) (assumed)
  • \(a\) = acceleration (m/s²)
  • \(T\) = tension in rope (N)
  • Assumes massless, frictionless pulley and massless rope

Objects in Contact

• Two blocks pushed together:
  • Apply Newton's second law to system: \(F = (m_1 + m_2)a\)
  • Solve for acceleration: \(a = \frac{F}{m_1 + m_2}\)
  • Find contact force by analyzing one block separately

Special Applications

Banked Curves

• Ideal banking angle (no friction needed): \[\tan\theta = \frac{v^2}{rg}\] Where:
  • \(\theta\) = banking angle from horizontal
  • \(v\) = velocity (m/s)
  • \(r\) = radius of curve (m)
  • Normal force provides centripetal force
• Maximum speed on banked curve with friction: \[v_{max} = \sqrt{rg\frac{\sin\theta + \mu_s\cos\theta}{\cos\theta - \mu_s\sin\theta}}\]

Circular Motion in Vertical Plane

• At top of vertical circle: \[T + mg = \frac{mv^2}{r}\] Where:
  • \(T\) = tension in rope (N)
  • Both tension and weight point toward center
• Minimum speed at top (for \(T = 0\)): \[v_{min} = \sqrt{gr}\]
• At bottom of vertical circle: \[T - mg = \frac{mv^2}{r}\] \[T = mg + \frac{mv^2}{r}\] Tension must overcome weight and provide centripetal force

Conical Pendulum

• Period of rotation: \[T = 2\pi\sqrt{\frac{L\cos\theta}{g}}\] Where:
  • \(L\) = length of string (m)
  • \(\theta\) = angle from vertical
• Horizontal component of tension: \[T\sin\theta = \frac{mv^2}{r}\] Provides centripetal force
• Vertical component of tension: \[T\cos\theta = mg\] Balances weight

Problem-Solving Strategy

General Approach for Force Problems

1. Draw a clear free body diagram for each object
2. Choose appropriate coordinate system (align with motion when possible)
3. Resolve all forces into components
4. Apply Newton's second law in each direction: \(\sum F = ma\)
5. Write separate equations for each direction
6. Solve the system of equations
7. Check: units, signs, limiting cases, reasonableness

Sign Conventions

• Choose positive direction consistently
• Forces in positive direction: positive values
• Forces in negative direction: negative values
• If acceleration is unknown, assume a direction; negative answer means opposite direction

Important Relationships and Notes

Common Misconceptions

• Normal force ≠ weight: Normal force equals weight only on horizontal surface with no other vertical forces
• Tension ≠ weight: Tension equals weight only when object hangs at rest
• Friction depends on normal force: Not directly on weight or contact area
• Centripetal force: Not a separate force; name for net force toward center
• Action-reaction pairs: Always act on different objects, never cancel

Key Conditions and Limits

• Object at rest: \(\sum \vec{F} = 0\) and \(\vec{v} = 0\)
• Constant velocity: \(\sum \vec{F} = 0\) and \(\vec{a} = 0\)
• Accelerating: \(\sum \vec{F} \neq 0\)
• Free fall: Only gravity acts; \(a = g\) downward
• Projectile motion: \(a_x = 0\), \(a_y = -g\)

Coefficient Values (Typical Ranges)

• \(\mu_s\) (static friction): 0.1 to 1.5
• \(\mu_k\) (kinetic friction): 0.03 to 1.0
• Always: \(\mu_k < \mu_s\)="" for="" same="" material="">
• Ice: very low coefficients (~0.02-0.1)
• Rubber on dry concrete: high coefficients (~0.7-1.0)