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Lecture 5

1. The Universal Problem-Solving Strategy

  • Step 1: Understand & Sketch: What is the system doing? (At rest? Constant velocity? Accelerating?)
  • Step 2: Free-Body Diagram (FBD): Isolate one object. Draw all forces acting ON it. Draw the acceleration vector a nearby.
  • Step 3: Coordinate System: Choose an x-y axis. Tip: Align one axis with the direction of acceleration a. This makes the other component a_other = 0.
  • Step 4: Component Equations: Write ΣF_x = ma_x and ΣF_y = ma_y. Use trigonometry to find force components.
  • Step 5: Solve: You should have as many equations as you have unknown variables.

2. Key Forces & Concepts

Force Name Symbol Description & Key Formula
Weight W or mg Force due to gravity. Always acts vertically down.
Normal Force N Contact force from a surface. Always acts perpendicular to the surface. WARNING: N is NOT always equal to mg. You must solve for it using ΣF_y = ma_y.
Tension T Pulling force from a rope or string. Acts along the rope.
Friction F_f Opposes motion or attempted motion. Acts parallel to the surface.

3. Friction Formulas (Essential)

  • Static Friction (No motion): F_s ≤ μ_s N
  • It only provides the force needed, up to a maximum.
  • Kinetic Friction (Sliding): F_k = μ_k N
  • Usually, μ_k < μ_s.
  • Coefficients: μ_s (static) and μ_k (kinetic) are properties of the surfaces. They have no units.

4. Circular Motion (Uniform)

  • Centripetal Acceleration: a_c = v²/R. It is always directed toward the center of the circle.
  • Centripetal Force: F_c = ma_c = m(v²/R). This is the NET FORCE pointing toward the center.
  • It is NOT a new force. Find it by summing the relevant forces (e.g., tension, friction, components of N or mg).
  • NEVER draw "centrifugal force" on an FBD in an inertial frame.

5. Classic Exam Scenarios & Formulas

  • Block on an Inclined Plane (angle α):
  • Choose tilted coordinates (x-axis parallel to the ramp).
  • Component of weight down the ramp = mg sin(α)
  • Component of weight perpendicular to ramp = mg cos(α)
  • Normal Force (if no other vertical forces/acceleration): N = mg cos(α)
  • Friction Force: F_f = μN = μmg cos(α)
  • Car on a Flat Curve (radius R):
  • The centripetal force is provided by static friction.
  • Maximum safe speed: μ_s mg = m(v_max²/R) => v_max = sqrt(μ_s g R)
  • Conical Pendulum (angle β from vertical):
  • Vertical forces: T cos(β) = mg (y-direction is in equilibrium).
  • Horizontal forces: T sin(β) = m(v²/R) (x-direction provides the centripetal force).