Applications of Vectors
| Scalar and Vector Quantities |
| Difference between Scalar and Vector Quantities |
| Distinguish between scalar and vector quantities |
| These are physical quantities which have magnitude only. Examples of scalar quantities include |
| mass, length, time, area, volume, density, distance, speed, electric current and specific heat |
| These are physical quantities which have both magnitude and direction. Examples of vector |
| quantities include displacement, velocity, acceleration, force, pressure, retardation, and |
| Addition of Vectors Using Graphical Method |
| Add vectors using the graphical method |
| Scalar physical quantities have a magnitude only. Thus, they can be added, multiplied, divided, or |
| subtracted from each other. |
| If you add a volume of 40cm |
| of water to a volume of 60cm |
| of water, then you will get |
| Vectors can be added, subtracted or multiplied conveniently with the help of a diagram. |
| A vector quantity can be represented on paper by a direct line segment. |
| The length of the line segment represents the magnitude of a vector. |
| The arrow head at the end represents the direction. |
| Methods of Vector Addition |
| There are two methods that are used to sum up two vectors: |
| A step-by-step method for applying the head-to-tail method to determine the sum of two or more |
| 1. Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will |
| result in a diagram that is as large as possible, yet fits on the sheet of paper. |
| 2. Pick a starting location and draw the first vector |
| in the indicated direction. Label the |
| magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m). |
| 3. Starting from where the head of the first vector ends, draw the second vector |
| indicated direction. Label the magnitude and direction of this vector on the diagram. |
| 4. Draw the resultant from the tail of the first vector to the head of the last vector. Label this |
| 5. Using a ruler, measure the length of the resultant and determine its magnitude by converting to |
| real units using the scale (4.4 cm x 20 m/1 cm = 88 m). |
| 6.Measure the direction of the resultant using the counterclockwise convention. |
| This is the vector drawn from the starting point of the first vector to the end |
| point of the second vector which is the sum of two vectors. |
| Suppose a man walks starting from point A, a distance of 20m due North, and then 15m due |
| East. Find his new position from A. |
| 20m due to North Indicates 4 cm |
| 15m due to East Indicates 3cm. |
| The position of D is represented by Vector AD of magnitude 25M or 5CM at angle of 36 |
| Tan Q = (Opposite /Adjacent) |
| The Resultant displacement is 25m ad direction Q = 36º51” |
| The Triangle and Parallelogram Laws of Forces |
| State the triangle and parallelogram laws of forces |
| Triangle Law of Forces states that “If three forces are in equilibrium and two of the forces are |
| represented in magnitude and direction by two sides of a triangle, then the third side of the |
| triangle represents the third force called resultant force.” |
| A block is pulled by a force of 4 N acting North wards and another force 3N acting North-East. |
| Find resultant of these two forces. |
| Draw a line AB of 4cm to the North. Then, starting from B, the top vectorofAB, draw a line BC |
| Join the line AC and measure the length (AC = 6.5 cm) which represents 6.5N. Hence, AC is the |
| Resultant force of two forces 3N and 4N. |
| In this method, the two Vectors are drawn (usually to scale) with a common starting point , If the |
| lines representing the two vectors are made to be sides of s parallelogram, then the sum of the |
| two vectors will be the diagonal of the parallelogram starting from the common point. |
| The Parallelogram Law states that “If two vectors are represented by the two sides given and the |
| inclined angle between them, then the resultant of the two vectors will be represented by the |
| diagonal from their common point of parallelogram formed by the two vectors”. |
| Two forces AB and AD of magnitude 40N and 60N respectively, are pulling a body on a |
| horizontal table. If the two forces make an angle of 30 |
| between them find the resultant force on |
| between AB and AD. Complete the parallelogram ABCD using the two |
| sides AB and include angle 30 |
| Draw the lineAC with a length of9.7 cm, which is equivalent to 97 N. |
| The lineAC of the parallelogram ABCD represents the resultant force of AB and AD in |
| Two ropes, one 3m long and the other and 6m long, are tied to the ceiling and their free ends are |
| pulled by a force of 100N. Find the tension in each rope if they make an angle of 30 |
| By using parallelogram method |
| Tension, determined by parallelogram method, the length of diagonal using scale is 8.7 cm, |
| which represents 100N force. |
| Tension in 3m rope = 3 X 100 / 8.7 = 34.5N |
| Tension in 6m rope = 6 x 100 / 8.7 =69N |
| Tension force in 3m rope is 34.5N and in 6m rope is 69N. |
| are those that act on a body at rest and counteract the force pushing or |
| pulling the body in the opposite direction. |
| The Concept of Relative Motion |
| Explain the concept of relative motion |
| Relative motion is the motion of the body relative to the moving observer. |
| The Relative Velocity of two Bodies |
| Calculate the relative velocity of two bodies |
| is the velocity relative to the moving observer. |
| CASE 1: If a bus in overtaking another a passenger in the slower bus sees the overtaking bus as |
| moving with a very small velocity. |
| CASE 2: If the passenger was in a stationary bus, then the velocity of the overtaking bus would |
| CASE 3: If the observer is not stationary, then to find the velocity of a body B relative to body A |
| If velocity of body B is VB and that of body A is VA, then the velocity of B with respect to A , |
| the relative velocity VBA is Given by: |
| NOTE:The relative velocity can be obtained Graphically by applying the Triangle or |
| = VB – VA ___________________ (I) |
| VrBA = VB + VA _______________________ (II) |
| A man is swimming at 20 m/s across a river which is flowing at 10 m/s. Find the resultant |
| velocity of the man and his course if the man attempted to swim perpendicular to the water |
| The length of AC is 11.25 cm which is 22.5 m/s making a angle of 65º25 with the water |
| The diagonal AC represent (in magnitude and direction) the resultant velocity of the man. |
| The Concept of Relative Motion in Daily Life |
| Apply the concept of relative motion in daily life |
| Knowledge of relative motion is applied in many areas. In the Doppler effect, the received |
| frequency depends on the relative velocity between the source and receiver. Friction force is |
| determined by the relative motion between the surfaces in contact. Relative motions of the |
| planets around the Sun cause the outer planets to appear as if they are moving backwards relative |
| The Concept of Components of a Vector |
| Explain the concept of components of a vector |
| Is the Splits or separates single vector into two vectors (component vectors) which when |
| compounded, provides the resolved vector. |
| is asingle vector which can be split up into component vectors. |
| are vectors obtained after spliting up or dividing a single vector. |
| Resolution of a Vector into two Perpendicular Components |
| Resolve a vector into two perpendicular components |
| Components of a vector are divided into two parts: |
| Horizontal component, FX = FSinQ |
| Vertical component: Fy = FCosQ |
| Resolution of Vectors in Solving Problems |
| Apply resolution of vectors in solving problems |
| Find the horizontal and vertical components of a force of 10N acting at 30 |
| FX = F CoS60º _________________(1) |
| Fy =F SinQ __________________________ (ii) |
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