Kinematics and Dynamics

Question 1

Dot Product and Cross Product

Answer 1

• Dot Product: A • B = |A| × |B| × cos(θ)
• Work: W = Δx × F × cos(θ)

• Cross Product: A • B = |A| × |B| × sin(θ)
• Torque: τ = r × F × sin(θ)
• Use Right Hand Rule

Question 2

Friction

Answer 2

• Static Friction: 0 < fₛ ≤ µₛN
• Equal but opposite static force exists to applied force until µₛN threshold is reached.

• Kinetic Friction: fₖ = µₖN
• Constant value for kinetic friction regardless of surface area contact or velocity.

• µₛ > µₖ
• Objects tend to stick until they start moving but will slide more easily over one another once they do start moving.

Question 3

Newton’s Laws

Answer 3

• First Law (Law of Inertia): An object at rest or in motion with constant velocity will remain as such, unless acted upon by an external unbalanced force; F = ma = 0.

• Second Law (Corollary of First Law): An object will accelerate when the vector sum of the forces results in some nonzero resultant force vector: F = ma.

• Third Law (Law of Action and Reaction): For every action, there is an equal but opposite reaction; F(AB) = –F(BA).

Question 4

Motion

Answer 4

• Linear Motion: One-Dimensional Equations
• v = vₒ + at
• v² = vₒ² + 2ax
• x = vₒt + ½at²
• x = ṽt

• Projectile Motion: Vertical velocity changes, but horizontal velocity remains constant; Height of 0 gives you two time values, and time value in between gives you maximum height of parabola.

• Inclined Planes: Force vectors must be divided into components that are parallel and perpendicular to the inclined plane; Normal force is perpendicular to the plane.

• Uniform Circular Motion: Tangential force is zero as instantaneous velocity is constant and tangent to the circular path; Centripetal Force points radially inwards and causes object to experience Centripetal Acceleration.
• Fc = mv²/r.

Question 5

Mechanical Equilibrium

Answer 5

• Translational Equilibrium: Exists when the vector sum of all the forces acting on an object is zero; object is either stationary or moving with constant nonzero velocity.
• Think of two blocks connected via pulley in static equilibrium.

• Rotational Equilibrium: Exists when the vector sum of all the torques acting on an object is zero; torques that generate counterclockwise rotation are positive.
• Forces are applied to rotate an object around a fulcrum; Torque (τ) generates rotational motion due to Applied Force (F) acting at some distance away from fulcrum, known as the Lever Arm (r).
• Torque is greatest when applied force is directly perpendicular to the lever arm.
• τ = r × F × sin(θ).

Question 6

Gravitational Force

Answer 6

• Quantifies the attractive force of gravity between two objects of masses m₁ and m₂ using the distance (r) between them and the universal gravitational constant (G).
• F = (Gm₁m₂)/r²