Comprehensive Study Notes on Newton's Laws of Motion

Unit 6: Newton's Third Law of Motion

Definition and Core Principle

• Newton's Third Law of Motion: When object A exerts a force on object B, object B simultaneously exerts an oppositely directed force of equal magnitude on object A.

  • Key Concept: For every action force, there is a reaction force. These forces:

  • Act on different objects

  • Have the same magnitude

  • Are directed in opposite directions

  • Important to remember that action and reaction always act on different objects.

Module 1: Mechanics: Chapter 1: Newton's Laws of Motion

Definitions and Terms

• Frequency: The number of crests of a wave that move past a given point in a given unit of time, measured in hertz (Hz). One hertz corresponds to one crest per second.

• Instantaneous: Referring to a specific moment or spot in time.

• Magnitude: Refers to the 'size' of acceleration, without any direction.

• Acceleration: The rate of change of velocity. The SI unit for acceleration is m-s².

• Net Force: The single force that has the same effect as all the other forces acting together on an object.

  • Defined as the sum of all forces acting on the same object.

Ticker Timer and Velocities

• The ticker timer is a device that prints dots on tape, indicating frequency. For instance:

  • If the frequency of the timer is 20 Hz, it prints 20 dots per second on the tape.

  • The period of the ticker timer is defined as the time interval between two adjacent dots:

  • For 20 Hz frequency, the period = $(0.05 ext{ s})$ or $rac{1}{20} ext{ s}$.

  • Average Velocity (v) can be calculated using the formula:

  • $v = rac{ ext{displacement (m)}}{ ext{time (s)}}$

  • Example: If a trolley covers 40 cm (0.4 m) in 5 seconds:

    • $v = rac{0.4 ext{ m}}{5 ext{ s}} = 0.08 ext{ m/s (forward)}$

Instantaneous Velocity Calculation

• Instantaneous velocity corresponds to the velocity at a specific moment. The average velocity between points A and C is equal to the instantaneous velocity at point B.

  • If displacement between A and C is 30 cm (0.3 m), and time intervals between A and B as well as B and C each are 0.5 seconds:

    • $ext{Average velocity} = rac{ ext{displacement from A to C}}{ ext{time from A to C}} = rac{0.3 ext{ m}}{1 ext{ s}} = 0.3 ext{ m/s (forward)}$

  • Thus, instantaneous velocity at B = 0.3 m/s forward.

Acceleration Calculation

• To calculate the acceleration between points B and C:

  • Use instantaneous velocities at both points:
    $ext{Acceleration} = rac{ ext{instantaneous velocity at C} - ext{instantaneous velocity at B}}{ ext{time between B and C}}$

• Example: If instantaneous velocity at B is $v<em>B$ and at C is $v</em>C$ and time between B and C is 1 second, then:
  $ext{Acceleration} = rac{v<em>C - v</em>B}{1 ext{ s}}$

Worked Examples

Worked Example 9

• A horizontal pulling force of 10 N acts on a block with a weight of 50.96 N on a rough surface. The block accelerates at 1.67 m/s² to the right.

  1. Free Body Diagram: Label all horizontal forces on the block.

  2. Calculate Mass:

     • Use the formula $F = mg$:

     • $50.96 = m imes 9.8$ implies $m = 5.2 ext{ kg}$.

  3. Net Force Calculation:

     • $F_{ ext{net}} = ma$:

     • $F_{ ext{net}} = 5.2 imes 1.67 = 8.684 ext{ N (to the right)}$.

  4. Calculate Frictional Force:

     • Set up: $F<em>{ ext{applied}} - F</em>f = F_{ ext{net}}$

     • $10 - F<em>f = 8.684 o F</em>f = 1.316 ext{ N (to the left)}$.

  5. Normal Force: Equal to weight, $F_N = 50.96 ext{ N (upwards)}$.

  6. Frictional Coefficient: Calculation gives:
     $ext{Frictional coefficient} = rac{F<em>f}{F</em>N} = rac{1.316}{50.96} o ext{approx. } 0.026$.

Worked Example 7

• Force on a Block: Given a horizontal pulling force of 10 N to the left on a 4 kg block on a smooth surface.

  1. Free Body Diagram: Label horizontal forces.

  2. Calculate Net Force:

     • $F = 10 ext{ N (to the left)}$.

  3. Calculate Acceleration using $F = ma$:

     • $10 = 4 imes a o a = 2.5 ext{ m/s² (to the left)}$.

Worked Example 8

• A horizontal pulling force of 24 N to the right on a block with a weight of 39.2 N and a frictional force of 1.4 N.

  1. Free Body Diagram: Label horizontal forces.

  2. Determine Net Force:

     • $F_{net} = 24 - 1.4 = 22.6 ext{ N (to the right)}$.

  3. Calculate Mass:

     • $39.2 = m imes 9.8 o m = 4 ext{ kg}$.

  4. Calculate Acceleration using $F = ma$:

     • $22.6 = 4 imes a o a = 5.65 ext{ m/s² (to the right)}$.

Relationship of Force, Mass, and Acceleration

• When the same force is applied to objects with different masses:

  • Smaller Mass: Larger acceleration.

  • Larger Mass: Smaller acceleration.

  • Relationship Summary:

  • If net force remains constant, acceleration is inversely proportional to mass:
    $a ext{ α } rac{1}{m}$.

  • If mass remains constant, acceleration is directly proportional to (net force):
    $a ext{ α } F_{ ext{net}}$.

Summary of Newton's Laws

Newton's First Law:

• An object remains at rest or in uniform motion unless acted on by an unbalanced force.

• Inertia: The property of a body to resist any change in motion.

• Definitions:

  • Mass: Measure of inertia (SI unit = kg).

  • Scalar: Quantity with magnitude but no direction (e.g., 3 kg).

Newton's Second Law:

• When a net force $F_{ ext{net}}$ is applied to an object of mass $m$, the object accelerates in the direction of the net force:

  • Equation: $F_{ ext{net}} = ma$ (SI unit of force is Newton, N).

Newton's Third Law:

• When object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A.

  • Examples: Action-reaction pairs, demonstrations with spring balances.

Important Concepts

• Force: A push or pull with magnitude and direction (SI unit = N).

• Vector: A quantity with magnitude, unit, and direction (e.g., 5 N, east).

• When a force is applied:

  • Object can change direction, change shape, or accelerate.