Exam Study Notes

Free Body Diagrams

  • When drawing free body diagrams:
    • Only draw forces.
    • Do not draw velocity or acceleration.

Force Estimation

  • Try to accurately represent the size of force arrows to reflect ratios.
  • Estimation is important when the exact force value is unknown, which is often the case.
  • Forces are always drawn from the body outwards; drawing them inwards can cause confusion.

Multiple Objects

  • When dealing with multiple objects, draw a free body diagram for each.
  • Represent each object as a point mass (circle), regardless of size differences.

Example: Two Objects

  • Consider two objects, m1 and m2, with m_1 being pushed.
    • m_1 experiences:
      • Force of push.
      • Normal force.
      • Gravitational force downwards (F_g).
      • Force from the second object (F_{21}).
    • m_2 experiences:
      • Normal force.
      • Gravitational force downwards.
      • Force applied from the first object (F_{12}).

Newton's Third Law

  • Action-reaction pairs exist, meaning F{21} = -F{12} (equal magnitude, opposite direction).

Common Forces

  • Contact Forces:
    • Normal force.
    • Forces between objects in contact (F{12}, F{21}).
  • Gravity:
    • Always present unless specified otherwise.
  • Tension:
    • Force within ropes or cables.
  • Friction:
    • Arises from the motion of surfaces in contact.

Normal Force and Weight

  • Normal force is not always equal to weight.
  • Weight (F_g) is the force due to gravity.
  • On a flat surface with no external vertical forces, normal force cancels weight.

Example

  • If an external force is applied upwards at an angle, the normal force will be reduced.
  • The sum of normal force and the y-component of the applied force equals the gravitational force: Fn + F{py} = F_g
  • Scales read normal force, not weight. They read weight accurately only on flat surfaces with no external vertical forces.

Vector Equations and Components

  • Newton's laws are vector equations and can be split into components (x, y).
  • Solve for each component separately, treating them as independent motions.
  • Information about acceleration in each direction may not be explicitly given; it must be inferred from the problem.

Example: Object Pulled Horizontally

  • If an object is pulled horizontally with acceleration a = 2 m/s^2, then ax = 2 m/s^2 and ay = 0.

Equilibrium

  • Equilibrium: Net force equals zero, implying no acceleration or change in velocity.
  • Equilibrium can exist in one direction but not another.
    • Example: An object may be in equilibrium vertically but accelerating horizontally.

Friction

  • Friction opposes motion and arises from contact between surfaces.
  • Static friction cancels the tendency to slide.
  • Friction coefficient "mu is material dependent:
    • Ff = "mu vert Fn (Frictional force equals friction coefficient times normal force).
    • The friction coefficient "mu will be provided or asked for in the problem.
    • On an inclined surface, normal force is not simply mg; it's mg times a trigonometric term.

Static vs. Kinetic Friction

  • Static friction (F{fs}) is variable up to a threshold; kinetic friction (F{fk}) is constant.
  • It requires more force to start motion than to keep it moving.
  • Static friction is not always "mus N; it is whatever force is required to prevent motion, up to "mus N.
  • Only when an object is at the verge of slipping is F{fs} = "mus N.
  • After exceeding the static friction threshold, kinetic friction takes over: F{fk} = "muk N.
  • Microscopic gaps and interactions between surfaces cause friction.
  • Typically, "mus > "muk.

Inclined Surfaces

  • Choose a coordinate system to simplify equations.
  • For inclined surfaces, rotate the coordinate system so that the x-axis is parallel to the surface.
  • Friction force is always parallel to the surface.
  • Normal force is always perpendicular to the surface.

Coordinate System Choice

  • Rotating the coordinate system aligns normal force, friction, and motion along axes, simplifying calculations.
  • Gravity will have components in both x and y directions.

Angle Considerations

  • Pay close attention to angles; avoid memorizing formulas without understanding.
  • Use limiting cases (0 or 90 degrees) to check whether sine or cosine should be used.
  • If "theta = 0 (flat surface), gravitational force should have zero x-component and a full y-component.

Pulleys and Connected Objects

  • Pulleys redirect force without changing magnitude.
  • Tension is the same throughout a rope.
  • Objects connected by a rope have the same acceleration magnitude.
  • Consider direction of acceleration for each object separately.

Problem-Solving Strategy

  • Start with free body diagrams.
  • Recognize connecting constraints (e.g., same acceleration).
  • This reduces the number of unknowns and allows solving equations.

Example Problem: Masses Connected by a Rope over a Pulley

  • Two masses, m1 = 2 kg and m2 = 8 kg, are connected by a rope over a pulley.
  • Find tension in the rope and acceleration of the masses.

Free Body Diagrams

  • m1 experiences tension upwards and gravity downwards (m1g).
  • m2 experiences tension upwards and gravity downwards (m2g).
  • Since m2 is heavier, the system will accelerate towards m2.

Coordinate Systems

  • Establish a positive y-direction for each mass (upwards for m1, downwards for m2).

Newton's Second Law Equations

  • For m1: T - m1g = m_1a.
  • For m2: T - m2g = -m2a (or m2g - T = m_2a if downwards is positive).

Solving for Acceleration and Tension

  • Add the equations to eliminate T and solve for a.
  • Substitute a back into one of the equations to solve for T.
  • a = g\frac{m2 - m1}{m1 + m2}.

Result Analysis

  • Consider limiting cases to check the result.
    • If m2 is much larger than m1, a should approach g.
    • If m1 = m2, a should be zero.

Elevator Example

  • The perceived weight in an elevator changes due to acceleration.
  • Scales read normal force, not true weight.

Elevator at Rest

  • Scale reads true weight.

Elevator Accelerating Upwards

  • You feel heavier.
  • The equation is: Fn - mg = ma, so Fn = mg + ma.

Elevator Accelerating Downwards

  • You feel lighter.
  • The equation is: mg - Fn = ma, so Fn = mg - ma.

Limiting Cases

  • If the elevator accelerates downwards at g, you feel weightless (F_n = 0).

Quiz Problems

  • Sample quiz problems and their solutions were reviewed, covering units, dimensions, and conceptual understanding.

Units and Dimensions

  • Dimensional analysis is important.
    • Example: Kinetic energy K = (1/2)mv^2. Units of K in kg, m, s

[K] = kg \cdot (m/s)^2 = kg \cdot m^2 / s^2

Ratio Problems

  • Algebra and ratios must be handled.
  • Ratios of a given variable is derived

Free Fall

  • Two objects are dropped from a bridge with an interval of one second then they will always have a constant speed. V(t) = gt +Vo

Average Speed, Velocity from displacement graph

  • The figure shows the position of the object as a function of time.

  • Average speed is distance travelled over a given time,

\text{average speed} = \frac{\text{distance}}{\text{time}}

  • Whereas Average velocity is displacement in a given time

\text{velocity} = \frac{\triangle x}{\triangle t}

  • In addition to Instantaneous speed is calculated by measuring slopes in different areas in a graph, for example a straight line.