Scalar and Vector Quantities Intermediate Notes PDF

INTERMEDIATE LEVEL PHYSICS 1st Years Scalar and Vector Quantities 0  MATSEC IM Syllabus Section 1.3: Scalar and Vector Quantities Candidates should be able to: (a) Define vector and scalar quantities, giving examples of each. (b) State...

INTERMEDIATE LEVEL PHYSICS 1st Years Scalar and Vector Quantities 0  MATSEC IM Syllabus Section 1.3: Scalar and Vector Quantities Candidates should be able to: (a) Define vector and scalar quantities, giving examples of each. (b) State whether a given quantity is a vector or a scalar. (c) Define the resultant of a set of vectors. (d) Define a sign convention or use a given sign convention to add (subtract) vectors. (e) Resolve a vector into two perpendicular components. (f) State and use the fact that mutually perpendicular vectors can be treated separately. (g) Determine the resultant (giving both the magnitude and direction) of any set of coplanar vectors. 1 Ø Scalars and Vectors A scalar quantity is a physical quantity which has only magnitude (size). Examples: temperature, distance, mass, length, time, volume, density, speed, work, energy, power, charge, electric potential … … A vector quantity is a physical quantity which has both magnitude and direction. Examples: displacement, force, weight, tension, thrust, drag, friction, velocity, acceleration, momentum, torque, magnetic field strength, electric field strength … … Note: A physical quantity is a vector quantity if It has magnitude and direction. It obeys the laws of vector addition – the triangle law of vector addition and the parallelogram law of vector addition. Current is a scalar quantity even though it has both magnitude and direction. This is because current does not obey the laws of vector addition. Ø Representing vectors Vectors can be represented in diagrams by arrows O T The length of the arrow OT denotes the size of the vector The direction of the arrow denotes the direction of the vector Note: The point O is the origin of the vector The tip of the arrow T is the terminal point (the end) of the vector. 2 Examples: a) Some force vectors: (i) A horizontal force of 15 N: (ii) A vertical force of 5 N: 15N 5N (iii) A force of 20 N at 30° to the horizontal: 20N 30° b) Some velocity vectors: (i) A velocity of 4 m s-1 at a direction of E 20° S (20o South of East) 20° 4 m s-1 (ii) A velocity of 2 m s-1 due North 2 m s-1 (iii) A velocity of 5 m s-1 at a direction of 40o West of South (S 40o W) 40o 5 m s-1 3  Ø Adding Vectors Scalars are added using simple arithmetic. Adding two masses of 10 kg always gives the same answer: 20kg. Mass is a scalar quantity. Adding two forces of 10N can give any answer between 0N and 20N. Force is a vector quantity. Direction needs to be taken into account when adding vectors. § 2 forces of 10 N acting in opposite direction: § Resultant is a force of 0 N 10 N 10 N § 2 forces of 10 N acting at 120o to each other: 10 N Resultant is a force of 10 N acting at an angle of 60o to the horizontal. 10 N § 2 forces of 10 N acting at 90o to each other: 10 N Resultant is a force of 14.1 N acting at an angle of 45o to the horizontal. 10 N § 2 forces of 10 N acting at 30o to each other: 10 N Resultant is a force of 19.3 N acting at an angle of 15o to the horizontal. 10 N § 2 forces of 10 N acting in the same direction: 10 N Resultant is a force of 20 N acting to the right 10 N 4 The result of combining two or more vectors together is known as the resultant vector. Definition: The resultant of two or more vectors is the single vector which produces the same affect in both magnitude and direction as the original set of vectors. (a) Adding vectors acting along the same straight line Vectors acting along the same straight line are added algebraically. (i) Vectors acting in the same direction Vectors acting in the same direction can simply be added together: F1 F2 Resultant F1 + F2 Resultant = F1 + F2 (ii) Vectors acting in opposite direction Vectors in opposite directions are given opposite signs before adding vectors in opposite directions. F1 F2 Resultant F1 – F2 Resultant = F1 + (- F2) = F1 – F2 5  Sign Convention: Vectors in the same direction are given the same sign. Vectors in opposite directions are given opposite signs Vectors in the horizontal direction: vectors pointing to the right are given a positive sign. ® + vectors pointing to the left are given a negative sign. -¬ Vectors in the vertical direction: vectors pointing upwards are given a positive sign ­ + vectors pointing downwards are given a negative sign. ¯- For vectors acting along the same straight line, one direction is taken as positive and the other as negative. (b) Adding vectors acting at an angle To add two vectors which are acting at an angle, the Parallelogram law or the Triangle law of vector addition has to be applied. § If the two vectors are acting at a point, the Parallelogram law is used. Parallelogram law: If two vectors acting at a point are represented in magnitude and direction by the sides of a parallelogram drawn from the point, their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the point. Q nd X Pa Y and ors ors v ect Y v ect P of t of t ant tan su l sul Re Re X Q 6  § If the two vectors are trailing each other, the Triangle law of vector addition is used. Triangle law of vector addition: If two vectors are drawn such that the terminal point of one joins the origin of the other, the resultant vector is the line which joins the origin of the first vector to the terminal point of the second vector. Q nd Y Pa and r s sX cto ve ctor to f ve Y t an P t of su l l tan Re su Re X Q (i) Vectors acting perpendicular to each other Pythagoras theorem is used to find the magnitude of the resultant R Y R Y tR t ant ultan sul Re s Re a a X X By Pythagoras: 𝑅 = $𝑌 ! + 𝑋 ! Simple trigonometry is used to find the angle which gives the direction of the resultant. 𝑜𝑝𝑝 𝑌 𝑇𝑎𝑛 𝑎 = = 𝑎𝑑𝑗 𝑋 $ 𝑎 = 𝑡𝑎𝑛"# 0%1 For vectors acting at 90o to each other, Pythagoras Theorem can be used to find the magnitude and trigonometry is used to find the angle which gives the direction of the vector. 7 (ii) Vectors acting at an angle to each other The Cosine rule is used to find the magnitude of the resultant R P R R b c b c Q The Sine rule is used to find the angle which gives the direction of the resultant. A mathematical note The Cosine and Sine Rules: For any triangle: a, b and c are sides. A, B and C are angles A is the angle opposite side a B is the angle opposite side b C is the angle opposite side c. Cosine rule: a2 = b2 + c2 – [2 b c cos(A)] Sine rule: 𝒂 𝒃 𝒄 𝑺𝒊𝒏 𝑨 = 𝑺𝒊𝒏 𝑩 = 𝑺𝒊𝒏 𝑪 For vectors acting at any other angle to each other, the cosine rule is used to find the magnitude of the resultant whereas to find the direction (an angle) the sine rule is used 8 Ø Resolving Vectors Vector resolution is a process in which a single vector is split into two smaller parts. These parts are called the components of the vector. Components are usually taken perpendicular to each other. A vector’s components describe the effect of the vector in a given direction. Note: Vector resolution is the reverse process to finding the resultant of two vectors. To determine the components of a vector the following procedure is adopted: § Draw a sketch of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal. q § Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. q 9  § Draw the components of the vector. The origin of each component vector is the same as the origin of the vector. Place arrowheads on these components to indicate their direction (up, down, left, right). Y F q X § To determine the size of the component of the vector opposite the given angle, use the sine function. § To determine the size of the component of the vector adjacent to the given angle, use the cosine function. (i) Vertical and horizontal components Vector F which acts at an angle q to the horizontal is resolved into two components, Y and X at right angles to each other. Y Y is the vertical component X is the horizontal component q X To find the size of Y – The vertical component: &'' $ Sin q = ()' = * Rearranging: Y = F Sin q To find the size of X – The horizontal component: +,- % Cos q = ()' = * Rearranging: X = F Cos q 10 (ii) Components perpendicular and parallel to a plane In the case when an object is situated on an inclined plane, it is convenient to resolve the vectors into two components which are perpendicular and parallel to the plane. In the following example, the weight W of the object is resolved into two components perpendicular to each other. P is the component of the weight parallel to (along) the plane; Q is the component of the weight perpendicular to the plane P q q W Q To find the size of P – the component of the weight parallel to the plane: &''. Sin q = ()' = / Rearranging: P = W Sin q To find the size of Q – the component of the weight perpendicular to the plane: +,- 0 Cos q = ()' = / Rearranging: Q = W Cos q Ø Combining non-perpendicular coplanar vectors by calculation: Coplanar vectors are vectors which are parallel to the same plane. To find the resultant of two or more coplanar vectors acting at an angle to each other the following steps are followed: 1. resolve all vectors in vertical and horizontal components (note: any convenient mutually perpendicular directions can be used) 11  2. find the resultant of the horizontal vectors and find the resultant of the vertical vectors 3. combine the resultant horizontal and vertical vectors to obtain the resultant of the coplanar vectors. Pythagoras is used to find the magnitude of the resultant. Trigonometry is used to find the direction of the resultant. Example: Find the resultant of a force of 6N acting at an angle of 27o to the horizontal and a vertical force of 10N. Solution: Step 1: Draw a vector diagram 10N 6N 27o Step 2: Resolve the 6N force into its horizontal and vertical components: Y 6N 27o X To find the vertical component, Y: &'' $ Sin 27 = ()' = 1 Rearranging: Y = 6 Sin 27 = 2.72 N To find the horizontal component, X: +,- % Cos 27 = ()' = 1 Rearranging: X = 6 Cos 27 = 5.35 N 12 Step 3: Replace the 6N force with its components in the diagram. These are shown in red in the diagram. 10N 2.72N 5.35N Step 4: Find the total force in the horizontal and vertical directions: Total vertical force: 10 + 2.72 = 12.72 N (vertically up) Total horizontal force: 5.35 N (to the right) Step 5: Combine the total horizontal and vertical forces into a single resultant force: 12.72 N Use Pythagoras theorem to find the magnitude: R = √12.72! + 5.35! = √190.42 = 13.8 𝑁 Use trigonometry to find the direction: &'' #!.3! R tan 𝜃 = +,- = 4.54 = 2.38 q = tan-1 (2.38) = 67.2o q 5.35N Ø Combining vectors by using the polygon law of vector addition: The resultant of a number of vectors can be found by applying the polygon law of vector addition: the vectors are drawn such that the terminal point of one joins the origin of the other; the resultant vector would be the line which joins the origin of the first vector to the terminal point of the last vector. 13 Example: A girl delivering door-to-door advertising magazines covers her route by travelling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (i) Sketch the path taken by the girl. (ii) What is her resultant displacement? (iii) What is the total distance she travels? Advanced MATSEC Paper: May 2016 Paper 1 No.1b Solution: (i) 6.00 blocks t R en 4.00 blocks m ce la sp di nt lta su Re 3.00 blocks (ii) 6.00 blocks 3.00 blocks 3.00 blocks 4.00 blocks 4.00 blocks R q 3.00 blocks To find the magnitude of the resultant displacement R use Pythagoras: R2 = 4.002 + 3.002 R2 = 16.00 + 9.00 R2 = 25.00 R = √25.00 = 5.00 blocks 14  To find the direction use trigonometry: &'' 5 𝑇𝑎𝑛 𝜃 = +,- = 6 = 0.75 q = tan-1(0.75) = 36.9o The resultant displacement is 5.00 blocks on a bearing of N 36.9o E (iii) Total distance travelled = 3.00 + 4.00 + 6.00 = 13.00 blocks Ø Subtraction of Vectors Vector subtraction is the same as the addition of a vector of the same size acting in the opposite direction. The negative of a vector has the same magnitude as the vector but is in the opposite direction. Vector F Vector –F Example: John kicks a ball against a wall with a horizontal velocity of 3 ms-1. The ball bounces back horizontally with the same speed. Find the ball’s change in velocity. Solution: Motion towards wall Motion away from wall - 3 m s-1 + 3 m s-1 Change in velocity = final velocity – initial velocity = 3 – (-3) = 6 m s-1 Change in velocity is 6 m s-1 to the right. 15 When subtracting vectors at an angle: Draw the final vector as it is Draw the initial vector in the opposite direction Add the two vectors using either the parallelogram or the triangle law of vector addition. Example: A car changes speed from 30 m s-1 to 20 m s-1 while turning a corner and changing direction by 90o. What is the change in the velocity of the car? Solution: vi = 30 m s-1 vf = 20 m s-1 The vector vi : 30 m s-1 The vector - vi : 30 m s-1 (same magnitude but opposite direction) Change in velocity = final velocity – initial velocity = vf - vi Drawing the vector diagram and applying the triangle law of vector addition: -v q 20 m s-1 i v f 30 m s-1 To find the magnitude use Pythagoras Theorem: vf - vi = √20! + 30! = √1300 = 36 𝑚 𝑠 "# To find the direction use trigonometry: &'' 57 𝑇𝑎𝑛 𝜃 = +,- = !7 = 1.5 q = tan-1 1.5 = 56.3o The change in velocity is 36 m s-1 at an angle of 56.3o to the direction of the final velocity. 16 References: Ordinary Level Physics - Abbott, Heinemann Educational Books, London The World of Physics - John Avison, Thomas Nelson and Sons, Surrey GCSE Physics (4th Edition)- Tom Duncan, Heather Kenneth, John Murray. Advanced Physics - Muncaster, R., Stanley Thomas (Publishers) Ltd. Understanding Physics for Advanced Level- Breithaupt, J., Nelson Thornes. Advanced Physics - Adams, S. & Allday, J., Oxford Press. Advanced Physics for You – Keith Johnson, Simmone Hewett, Sue Holt, John Miller, Oxford University Press. Intermediate Physics 16 – 18 (5th Edition) - Martin Peter Farrell, Agenda Publishers. http://www.s-cool.co.uk/alevel/physics.html https://stepbystepscience.com https://www.physicshigh.com https://www.fizzics.org 17