GRADE 11 PHYSICAL SCIENCES: MECHANICS

GRADE 11 PHYSICAL SCIENCES: MECHANICS STUDY NOTES

TABLE OF CONTENTS

1. Vectors in One Dimension – Revision
2. Vectors in Two Dimension
3. Solutions to Vectors Activities
4. Newton’s Laws and Application of Newton’s Laws
5. Newton’s Laws Activities
6. Solutions to Newton’s Laws Activities

1. Vectors in One Dimension – Revision

• Vector: A physical quantity having magnitude and direction.
• Scalar: A physical quantity having magnitude only.

Examples of Vector and Scalar Quantities

• Vector:
  • Force
  • Velocity
  • Displacement
  • Acceleration
• Scalar:
  • Mass
  • Time
  • Energy
  • Weight
  • Distance
  • Speed

Graphical Representation of a Vector

• A vector is represented by an arrow:
  • The length of the arrow represents the magnitude of the vector.
  • The arrow-head indicates the direction of the vector.

Direction of a Vector

• Horizontal Vector:
  • Positive sign (+) indicates right direction.
  • Negative sign (−) indicates left direction.
• Vertical Vector:
  • Positive sign (+) indicates upward direction.
  • Negative sign (−) indicates downward direction.

Examples of Direction

1. Right is +:
   • Example: + 5 N
2. Left is −:
   • Example: - 3 N

Three Methods to Describe Non-Horizontal/Vertical Vectors

1. Polar Coordinates:
   • FA: 10 N at 30° above positive x-axis
   • FB: 8 N at 12° left of negative y-axis
   • FC: 5 N at 65° above negative x-axis
2. Bearing:
   • Use North as 0° and measure clockwise:
   • FA: 10 N on a bearing of 60°
   • FB: 8 N on a bearing of 192°
   • FC: 5 N on a bearing of 335°
3. Compass Directions:
   • FA: 10 N at 30° North of East
   • FB: 8 N at 12° West of South
   • FC: 5 N at 65° North of West

Resultant of Vectors

• Resultant Vector: The vector sum of two or more vectors, producing a single vector with the same effect as the original vectors combined.

Key Properties

1. Greatest when vectors act in the same direction.
2. Smallest when vectors act in opposite directions.

Worked Examples

1. Vectors Acting in the Same Direction:     A girl walks 120 m due East and then 230 m in the same direction.
   • Resultant displacement:
   • 
     $R = 120 m + 230 m = 350 m ext{ East}$
2. Vectors Acting in Opposite Directions:     A boy walks 210 m due East and then walks back 60 m due West.
   • Resultant displacement:
   • 
     $R = 210m + (-60m) = 150 m ext{ East}$
3. Multiple Vectors Acting in Different Directions:     Given forces: 8 N right, 10 N right, 25 N left, 12 N left.
   • Resultant force calculation:
   • $F<em>{net} = F</em>1 + F<em>2 + F</em>3 + F_4$
   • 
     $F_{net} = 8 + 10 - 25 - 12 = -19 N$
   • Thus,
     $F_{net} = 19 N ext{ left}$

2. Vectors in Two Dimensions

• Resultant of Perpendicular Vectors:
  • Example: A horizontal force of 30 N and a vertical force of 40 N.

Adding Co-linear Vectors

• Vectors that act in one dimension come under this category.
• Net x-component ($ Rx $): $R</em>x = R<em>{x1} + R</em>{x2}$
• Net y-component ($ Ry $): $R</em>y = R<em>{y1} + R</em>{y2}$

Worked Example

1. Two forces of 3 N and 2 N apply an upward force to an object; 2 forces each of 2 N act horizontally to the right.
   • Calculating Net Forces:
   • $R<em>y = R</em>{y1} + R_{y2} = 2 + 3 = 5 N ext{ upwards}$
   • $R<em>x = R</em>{x1} + R_{x2} = 2 + 2 = 4 N ext{ right}$

Graphical Representation of Vectors

1. Tail-to-Tail Method/Parallelogram:

   • Used for constructing resultant of multiple vectors.

   • Pythagorean theorem applied to find magnitude.

     $R = ext{hypotenuse} = ext{using } R^2 = R<em>x^2 + R</em>y^2$

2. Direction Calculation:

   • an heta = rac{Ry}{Rx}

Example

• A force of F1 = 5 N is applied at an angle of 30° above the horizontal.
  • Vector diagrams drawn, then using a protractor the direction and magnitude measured and calculated.
  • Resultant found to be 8.7 N at 13.3° above horizontal.

3. Solutions to Vectors Activities

4. Newton’s Laws and Application of Newton’s Laws

• Normal Force (N): Contact force exerted perpendicularly to the surface.
• Frictional Force: Opposes motion; proportional to normal force, independent of the area of contact.
• Maximum Static Friction ($ f{s max} $):
  $f</em>{s max} = \mu_s N$
• Kinetic Friction ($ fk $):
  $f</em>k = \mu_k N$
• Applied Forces:
  • Tension (F_T) in ropes, strings, or wires.

Applications of Newton’s Laws

1. First Law: An object remains at rest or in uniform motion unless acted on by a resultant force.
2. Second Law:
   • The acceleration of an object is proportional to the net force acting on it, and inversely proportional to its mass.
   • 
     $F_{net} = ma$

Examples of Newton’s Laws

• Example calculations using net forces and tensions in different arrangements like pulleys, inclined planes, etc.

5. Newton’s Laws Activities

• Exercises exploring scenarios hypothesizing potential changes in frictional force, analyzing graphs of steady-state values.

6. Solutions to Newton’s Laws Activities

• Detailed solutions provided for multiple example problems applying theories regarding forces, systems in equilibrium, vector resolution, and equilibrium conditions.