Grade 11 Physical Sciences: Vectors

• This document contains learner support material for Physical Sciences, specifically Mechanics, for Grade 11 students in KwaZulu-Natal in 2020.

Vectors in One Dimension

• A vector is a physical quantity that has both magnitude and direction
• A scalar is a physical quantity that has magnitude only
• Examples of vectors include force, weight, velocity, displacement, and acceleration.
• Examples of scalars include time, energy, mass, speed, and distance.
• Vectors are graphically represented by arrows, where the length indicates magnitude and the arrowhead indicates direction
• A positive or negative sign indicates the direction of a horizontal or vertical vector

Describing Vector Direction

• Direction can be described using three methods for vectors that are not horizontal or vertical.
• Vectors can be described on a graph using angles relative to the positive or negative x- or y-axis
• Bearing is only useable for vectors in the horizontal plane, parallel to the Earth's surface, using North as 0° and measuring clockwise.
• Compass can be used to describe using cardinal points such as North, South, East, and West

Resultant Vectors

• A resultant is defined as the vector sum of two or more vectors, i.e., a single vector with the same effect as the original vectors combined.
• The resultant vector is greatest when vectors are in the same direction and smallest when vectors are in opposite directions

Combining Vectors in One Dimension

• When two vectors act in the same direction, their magnitudes are added, and the resultant has the same direction.
• When two vectors act in opposite directions, the smaller magnitude is subtracted from the larger magnitude, and the resultant has the direction of the larger vector.
• For multiple vectors acting in different directions, assign a positive sign to one direction then sum all vectors, using positive signs for vectors in that direction and negative signs for vectors in the opposite direction

Vectors in Two Dimensions

• Perpendicular vectors are at right angles to each other.
• Co-linear vectors act in one dimension along the same line
• The net x-component (Rx) has the sum of the vector's parallel with the x-direction, Rx = Rx1 + Rx2
• The net y-component (Ry) has the sum of the vector's perpendicular to the x-direction, Ry = Ry1 + Ry2

Finding Resultant Vectors

• To find the resultant of vectors using tail-to-tail, a parallelogram must be completed, and the diagonal represents the resultant.
• Pythagoras' theorem is used to calculate the magnitude of the resultant R² = Rx² + Ry²
• Trigonometry is useable to find the direction of the resultant using tanθ = Ry / Rx

Graphical Determination of Resultant Vector

• The tail-to-head method finds the resultant of two or more consecutive vectors.
• The steps involved are:
  • Choose a suitable scale.
  • Accurately draw the first vector according to the chosen scale and direction.
  • Draw the second vector by placing its tail at the tip of the first vector.
  • Complete the diagram by drawing the resultant from the tail of the first vector to the head of the last vector.
  • Measure the angles correctly with a protractor
  • Always add arrow heads to vectors
  • Use the scale to determine the real magnitude of the resultant.

Triangle Rule for Forces in Equilibrium

• When drawing force vectors at equilibrium, a closed quadrilateral, such as a triangle, is formed, the resultant is zero, and all vectors are drawn head-to-tail.
• Forces acting on the same object in equilibrium maintain constant velocity, with no motion change.
• These forces form a closed triangle where the sides represent magnitude and direction.
• If three forces at a point are in equilibrium, they form a triangle in both magnitude and direction, taken in order F1+F2+F3=0

Resolution of a Vector

• Resolving a vector in its parallel and perpendicular components involves breaking it down into components at right angles to each other
• For a force at an angle 0 to the horizontal, the horizontal component uses cos 0 = adjacent / hypotenuse
• The vertical component uses sin 0 = opposite / hypotenuse
• The horizontal and vertical components of each force are needed, after which they can be added

Different kinds of Forces

• Normal force (N) is the force exerted by a surface on an object in contact with it, always perpendicular to the surface
• Frictional force opposes the motion of an object, acting parallel to the surface in contact
• Frictional force is proportional to the normal force and independent of the contact area
• Static friction opposes the tendency of motion of a stationary object
• Maximum static friction can overcome static friction and cause an object to slide
• Kinetic friction opposes the motion of a moving object relative to the surface
• Gravitational force or weight ("g or w) attracts an object towards itself.
• Tension is the force transmitted through a rope, string, or wire when pulled by forces from opposite sides
• Forces can be either contact (touching) or non-contact (separated)
• fsmax = µsN, fk = μκΝ (µs is coefficient of static friction and µk is coefficient of kinetic friction)

Effect of a Changing Angle

• For a pulling force: When the angle is increased, the normal will decrease, hence the frictional force will also decrease, when it is decreased, the normal will increase, hence the frictional force will also increase.
• For a pushing force: When the angle is increased, the normal will increase, hence the frictional force will also increase. When the angle is decreased, the normal force will decrease, hence the frictional force will also decrease.
• A change in angle will affect the coefficient of kinetic friction (μκ) used to determine the force of the object

Diagrams

• A force diagram is a picture of the object(s) of interest with all the forces acting drawn in arrows - Picture includes the object and all arrows going outwards
• A free body diagram is a picture of an object of interest drawn as a dot and all the forces acting on it are drawn as arrows pointing away from the do