Lamlad Physics - Measurement, Scalars, and Vectors PDF

## Measurement: Scalars and Vectors ### Introduction A list of experiments which candidates are expected to learn for the WASSCE examination is provided under the check-list section in each chapter of this book. Make sure you can repeat the above procedure for each of the listed experiments. ### Nu...

## Measurement: Scalars and Vectors ### Introduction A list of experiments which candidates are expected to learn for the WASSCE examination is provided under the check-list section in each chapter of this book. Make sure you can repeat the above procedure for each of the listed experiments. ### Numerical Problems In approaching a numerical problem in the examination, it is often helpful to: 1. read the question carefully to ascertain what exactly is required. Quite often, many candidates provide answers which bear little relevance to the question being asked; 2. list the given data using appropriate symbols; 3. write down the relevant equation(s) and define all symbols; 4. sketch a diagram where necessary; 5. ensure that all the data to be substituted into the equation are in the correct system of units (SI) 6. rearrange the equation so that the required parameter becomes the subject; and 7. write carefully to avoid computational errors. ### Suggested Study Procedure To get the best out of this book, you are advised to adopt the following study procedure: 1. Study the revision notes thoroughly. Where necessary, refer to your textbooks for further clarification. 2. Attempt all the true or false - type questions before looking up the answers. 3. Attempt all the SSCE/JME - type objective questions. Allow yourself approximately one minute to each question, on average. Use the answer key provided at the end of the problem set to score yourself. 4. If you are unable to complete the whole problem set within the appropriate time, continue all the way to the end and note how much extra time you have consumed. This is an indication of how much faster you should try to work under real examination conditions. 5. Go over the solutions carefully and note your areas of weakness. If the explanation is not clear, you may need to revisit the revision note, and even your textbook in order to gain a better understanding of the subject. Consultation with your teacher will also be helpful. 6. You should take the model objective tests after completing the whole book, and as close to the examination as possible. Your average score on these should give you an indication of your expected performance in the actual examination. **GOOD LUCK!** ### 1.1 Fundamental Concepts - The quantities length, mass and time which occur frequently in mechanics are referred to as fundamental quantities. - Other fundamental quantities are electric current and temperature. - The units in which these quantities are measured are called fundamental units. - In the international system (SI), the units are: | Quantity | Unit | Symbol | |---|---|---| | Length | metre | m | | Mass | kilogram | kg | | Time | second | s | | Temperature | kelvin | K | | Electric current | ampere | A | - The prefixes mega, kilo, centi, milli, etc. are used to indicate multiples and sub-multiples of these units. e.g. the kilometre is 1,000 metre and 1 centimetre is 10⁻² metre. | Prefix | Factor | Example | |---|---|---| | mega (M) | 10⁶ | 1Ms = 10⁶ s| | kilo (k) | 10³ | 1km = 10³ m| | deca (da) | 10¹ | 1 dam = 10¹m | | deci (d) | 10⁻¹ | 1dm = 10⁻¹ m | | centi (c) | 10⁻² | 1cm = 10⁻² m | | milli (m) | 10⁻³ | 1mg = 10⁻³ g | | micro (μ) | 10⁻⁶ | 1μs = 10⁻⁶ s | | nano (n) | 10⁻⁹ | 1ns = 10⁻⁹ s | | pico (p) | 10⁻¹² | 1ps = 10⁻¹² s | ### 1.2 Derived Concepts - Quantities such as area, volume, speed, density, etc. which can be derived from the fundamental quantities are called derived quantities. - Their units are referred to as derived units. Examples are provided below. | Quantity | Derived Unit | Symbol | |---|---|---| | Area | square metre | m²| | Density | kilogram per cubic metre | kgm⁻³ | | Speed | metre per second | ms⁻¹ | | Force | newton | N= kg.m.s² | | Energy | joule | J = N.m = kg m²s⁻² | ### 1.3 Measurement of Length - Lengths are measured using a graduated scale e.g a meter rule in measuring the length of a table. - The precision of a measurement depends on the graduation of the instrument being used. One can estimate the length to a fraction of the least graduation on the scale. ### 1.4 Position and Direction - The position of an object is usually specified relative to some fixed reference position, e.g. the origin (0) of the x-y coordinate axes (for a plane which is two dimensional) or x-y-z coordinate axes (for three-dimensional space). - In Fig. 1.4 the position of an object P is written as (x,y) where x is the perpendicular distance from P to OY and y is the perpendicular distance from P to OX. - The direction of P relative to O is described by the angle which OP makes with a reference line such as OX. In terms of the four cardinal points (north, south, east and west), the direction is specified relative to the cardinal points, e.g. direction OA (Fig. 1.5) is 30°N of E or N60° E while direction OB is 15° Wof S or S 15° W. ### 1.5 Scalars and Vectors - Scalars are physical quantities which have magnitude but no direction associated with them. Examples include length, mass, time and speed. - Vectors are physical quantities which have both magnitude and direction. Examples include displacement, velocity, acceleration and force. - A vector quantity is represented by a line with an arrow on it. The length of the line represents the magnitude while the arrow indicates the direction of the vector. ### 1.6 Addition of Vectors - The addition of two vectors A and B gives a third vector R which is also known as the resultant of the two vectors. - Parallelogram of Vectors: If two vectors are represented by two adjacent sides of a parallelogram the resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the point of intersection of the two sides. In Fig. 1.6 the resultant of A and B is written as $R=A+B$. - The vector -B is equal in magnitude to B but in opposite direction. Thus A-B = A+ (-B) = resultant of A and -B (Fig 1.7). - In solving problems on parallelogram of vectors, the following trigonometric relationships are often found to be useful: 1. The **cosine formula**, for a triangle with sides a, b, c, where b makes angle with a (Fig. 1.8) $c² = a² + b² -2abcos θ$. 2. **Pythagoras Theorem**: for a right-angled triangle (Fig. 1.9) $c² = a² + b²$. ### 1.7 Resolution of Vectors - A single vector can be replaced by two components acting along two perpendicular directions. These are the resolved components along the two directions. - The magnitude of the resolved component of a vector along a specified direction is the product of the vector and the cosine of the angle which the vector makes with the direction. For a vector A lying in an x-y plane and which makes an angle θ with the x-axis (Fig. 1.10) the resolved components are $A_x = A cos θ$ along the x-direction, and $A_y = Acos (90-θ) = A sin θ$ along the y-direction. The relationship between A and the resolved component is found by Pythagoras theorem $A² = A_x² + A_y²$ or $A = √{A_x² + A_y²}$ and the angle $θ = tan^{-1} [A_y/A_x]$. ### **True or False-Type Questions** 1. **Answer True(T) or False (F) to the following questions.** a. Both a spring balance and a chemical balance are used to measure the mass of an object. b. Either may be used to measure the weight. c. Hooke's law forms the basis of operation of the spring balance. d. The chemical balance works on the principle of moments. e. If the acceleration due to gravity changes, the reading of a spring balance will change but that of a chemical balance will not. f. A micrometer screw gauge can be used to measure the internal diameter of a tube. g. For a ruler graduated only in centimetres, measurements can be made accurately to the nearest millimetre. 2. **Each of the quantities in the following list is indicated as a scalar or a vector quantity. Answer True(T) if the classification is correct and False (F) if incorrect.** a. Pressure [vector] b. Electric potential [scalar] c. Impulse [vector] d. Heat capacity [scalar] e. Altitude (vector] f. Momentum [vector] g. Electric potential difference [vector] h. Magnetic induction [scalar] i. Acceleration due to gravity [vector] j. Power (scalar] k. Electric field [scalar] l. Electric current [vector] 3. **In the following list the unit of each quantity is indicated in brackets. Answer True (T) if the indicated unit is correct and False (F) if incorrect.** a. Work [kgm²s²] b. Power [Js] c. Gravitational potential [Jkg-¹) d. Pressure [kgm's²] e. Specific latent heat [JkgK-1] f. Density [kgm³] g. Elastic modulus [Nm] h. Electric field [NC-¹] i. Potential difference [JC] j. Resistivity [m²] ### **Answers to True or False-Type Questions** 1. a (F) The chemical balance is used to measure mass; the spring balance to measure weight. b (F) See (a) above. c (T) The extension of the spring is directly proportional to the weight of the body. d (T) This involves two bodies and a lever. e (T) Weight, but not mass, depends on gravity. f (F) This is best done with a vernier callipers. g (F) The precision is limited to one - half of the smallest division on the ruler, or 0.5cm (5 millimetres). 2. a (F) g (F) b (T) h (F) i (T) d (T) j (F) c (F) k (F) r l (F) 3. a (T) Work = [Force x distance] = [mass x acceleration x distance] = [kg x ms²xm] or [kgm²s²] b (F) Power = [work/time] = [Js¹] c (T) d (T) Pressure = [force/area] = [kgms-2/m²] = [kgm's²] e (F) Specific latent heat = [Jkg-¹]. Note that the specified unit is that of specific heat capacity. f (T) g (F) Elastic modulus = [Nm²). The unit indicated is that of elastic constant. h (T) i (T) Note that [JC-¹] is also the volt [V]. j (F) Resistivity = (om]. ### **SSCE/JME - Type Objective Questions** 1. Which of the following are derived units? I. Metre II. Coulomb III. Kilogram IV. Ampere V. Joule A. I and III only B. II and V only C. II. IV and V only D. All of them. 2. Which of the following are derived quantities? I. Thrust II. Temperature III. Area IV. Pressure A. I and IV only B. II. III and IV only C. I. III and IV only D. I, II, III and IV 3. The derived dimension [ML272] is a dimension of 1. acceleration II torque III. energy A. I only B. III only C. I and II only D. II and III only 4. Which of the following are not fundamental units? I. Kelvin II. Newton III. Second IV. Radian A. I and III only B. II a d IV only C. I and II only D. 1. 11 and IV only. 5. The unit of momentum is A. Js¹ B. Ns C. Ns¹ D. Nms 6. The dimension of power is A. ML2T-3 B. MLT-2 C. ML2T-2 D. ML-2T3 7. In which of the following physical quantities are the units correctly indicated? I. Weight [N] II. Energy [N m III. Momentum [kgms¹] IV. Acceleration [Nkg¹| A. I and II only B. III and IV only C. 1. 11 and III only D. I. II. III and IV 8. The watt is equivalent to A. Nms¹ B. Js C. kgm²s-2 D. Ns 9. Which of the following quantities has the same unit as the kilowatt - hour? A. Force x acceleration B. Force x velocity C. Force x distance D. Force x time. 10. Which of the following are vector quantities? I. Work II. Displacement III. Acceleration IV. Electric field intensity V. Magnetic induction A. I. II and III only B. II. III and IV only C. III, IV and V only D. II. III. IV and V only 11. Which of the following are scalar quantities? 1. Torque II. Electric potential III. Kinetic energy IV. Momentum A. I and IV only B. II and III only C. I. II and III only D. II. III and IV only 12. Which of the following is a set of vectors? A. Weight, displacement and moment B. Velocity, volume and upthrust C. Density, capacitance and distance D. Mass. force and impulse 13. Which of the quantities in question 12 is a set of scalars? 14. Which of the following readings gives the correct precision of the length of a rod using vernier calli pers? A. 4.1 cm B. 4.13 cm C. 4.120 cm D. 4.125 cm 15. Which of the following readings gives the correct precision of the length of a rod using a metre rule? A. 75 mm B. 75.0 mm C. 75.00mm D. 75.01 mm 16. The reading of the vernier callipers shown above is A. 2.38 cm B. 2.80 cm C. 2.88cm D. 3.60 cm 17. The reading on the micrometer screw gauge shown below is A. 4.16 mm B. 4.60 mm C.4.66 mm D. 4.70 mm 18. Which of the following instruments is most suitable for measuring the outside diameter of a narrow pipe a few millimetres in diameter? A. Pair of callipers B. Metre rule C. Micrometre screw gauge D. Tape rule. 19. A student measures the internal and external diameters of a cylindrical vessel as 100 (+1) mm and 104 (+1) mm respectively. The possible error in the determination of the thickness of the vessel is A. ±2mm B. ± 1 mm C. ± 0.5 mm D. 0 20. The least possible error in using a scale graduated in millimeters is A. 0.1 mm B.0.5mm C. 1.0mm D. 2.0 mm 21. A boy walks 10 m due west and then 10 m due south. His displacement is A. 10 m, S30°W B. 10 m, S60°W C. 102m, S45°W D. 10/2 m, S60°W 22. Agirl walks 12 m northwards, 5 m eastwards and 7 m southwards. Her total displacement is A. 5 m, north B. 5 m, east C. 7.07 m, S45°W D. 7.07 m, N45°E 23. A man walks 5 km south and then 3 km in the direction 60° west of south. His distance from the starting point is A. 7.00 km B. 7.50 km C. 8.00 km D. 10.72 km 24. A boat travelled 5 km in crossing a canal from point P to point R (see figure above). The effective displacement of the boat along the edge PQ is A. 3 km B. 4 km C.5km D. 8 km 25. N In the diagram shown above, KM represents the magnitude and direction of a force which is the resultant of forces represented in magnitude and direction by A. NK and KL B. NKand LK C. KN and KL D. KN and LK 26. A body is acted upon by the two forces 6 N and 8 N as shown in the diagram above. The resultant of the forces is A. 10 N in the direction N 37°E B. 10 N in the direction N53° E C. 10 N in the direction N 37°W D. 10 N in the direction N 53° W 27. Two perpendicular forces have a resultant of 13 Ν. If one of the forces is 5 N, the other force is A. 8 N B. 9 N C. 12 N D. 18 N 28. Two forces. 12 N and 16 N inclined at an angle to each other, have a resultant which is parallel to the 16 N force. The value of cos e is A. 1.0 B. 3/4 C. 1/2 D.0 29. A body of mass Mrests on a plane inclined at an angle a to the horizontal. The component of the weight of the body along the normal to the plane is A. Mgsin a B. Aígcos a C. Mgtan a D. Mg/sin a 30. A body pulls a box along a horizontal table with a rope inclined to the horizontal at angle 60º. If the tension in the rope 40 N. what is the effective horizontal force? [sin 60° = 3/2. cos 60°=1/2, tan 60° = 3. cot 60°= 1/3] A. 20 N B. 203 N C. 403 N D. 40/3 N ### **Solutions to Objective Questions** 1. (B) The five frequently occurring fundamental units in physics are the metre, kilogram, second, kelvin and ampere. All others are derived from a combination of these five. The coulomb is [As] while the joule is [N.m] or [Kgm²²]. [Note: The mole and the candella are also fundamental units, which, however, do not occur as frequently as the five listed above). 2. (C) The fundamental quantities are length, mass, time, temperature and electric current. All others are derived from these. Thrust is force which has the unit [N] or [kgms], Area has unit [m²] and pressure is force/area, with unit [Nm³]. 3. (D) M, L, T, stand for mass, length and time respectively. Hence [MLT] = [M][LT][L] = [mass] [acceleration][distance] = [force] [distance] = unit of torque or work (energy). 4. (B) See (1) above. 5. (B) Momentum = mass x velocity = [kgms¹] = [kgms²][s] = [Ns]. 6. (A) Power = energy/time. From (3) above, energy = [ML2T-2], hence power = [ML2T-2/T], or [ML2T-3]. 7. (D) Weight has unit of force = [N] Energy has unit of force x distance = [Nm] Momentum has unit of mass x velocity = [kgms'] Acceleration has unit of force/mass = [Nkg'] & (A) Watt = [Js'] = [Nms'] 9. (C) Kilowatt-hour = [power x tine)] = [energy] = [force x distance] 10. (D) All except work, are vector quantities. 11. (B) Electric potential and energy are scalars. 12. (A) In groups B to D, volume, density, capacitance, distance and mass are scalars. 13. (C) See (12) above. 14. (B) With the vernier callipers, measurements can be made to the nearest 0.01cm. 15. (B) The smallest division on the meter rule is 1mm. Precise measurements can be made to the nearest 0.5 mm 16. (C) The main scale reading (left edge of vernier scale) is 2.80. The eighth division on the vernier scale coincides with a division on the main scale. Hence the reading is 2.80 +0.08, or 2.88 cm. 17. (C) The reading on the main scale is 4.50 mm. The 16th division on the circular scale coincides with the horizontal line on the main scale. Hence the reading is 4.50 +0.16, or 4.66 mm. 18. (C) The micrometer screw gauge is the most suitable when the diameter is of the order of a few millimetres. 19. (A) The internal diameter could range from 99 mm to 101mm and the external diameter from 103 mm to 105 mm. The thickness could therefore range from 103-101, or 2 mm to 105-99, or 6 mm, i.e. thickness will be measured as 4 mm +2 mm. 20. (B) The scale can be read to the nearest 0.5 mm. 21. (C) Taking X as the starting point and Z as the end point, (see diagram), the displacement is XZ = 10√2m since XYZ is an isosceles triangle and.θ = 45°. The bearing of Z from X is S 45°W. 22. (D) The girl starts from Sand ends at F (see diagram). FP=SP=5m, hence SF = 5√2 = 7.07 m and the angle FSP = 45°, i.e. Fis 7.07 m from S in the direction N45°E. 23. (A) The man starts from Sand ends at F (refer to diagram). θ=180°-60°=120° Cosine rule: SF2=52+32-2(5)(3)cos 60이 =34-30cos120°=34+15 =49, or SF=7km. 24. (B) Referring to the diagram above. effective displacement of the boat along the edge PQ is PV. Since PXR is a right-angled triangle with VR = 3 and PR = 5 km, PX must be equal to 4 km. 25. (C) See revision notes. 26. (A) The resultant of the forces is R = \82+ 62 = 10 N which makes angle) with the north where() = tan¹ (6/8)=37°. i.c. R lies in the direction N37º E. 27. (C) From the diagram. /2 = 1.32-52-169-25=144. or ド= 12 N. ### Revision Check-list Under this chapter a candidate should be able to - distinguish between fundamental and derived quantities. - classify familiar units as fundamental or derived units. - derive the units of common physical quantities in terms of the fundamental units. - distinguish between scalars and vectors. - classify common physical quantities as vectors or scalars. - specify appropriate istruments for measuring the dimensions of common objects. - interprete correctly the readings on a pair of vernier callipers and a micrometer screw gauge. - specify the margin of error involved in the measurement taken by any common measuring instrument. - determine the resolved component of a vector along any specified direction. - determine the resultant of two vectors. - determine the difference of two vectors. ### 2.1 Types of Motion - Motion is the time-dependent change in position associated with a body. Motion is usually classified into four broad types: a. **Translational**: Movement of the whole body from one position to another, e.g a falling stone. b. **Rotational**: The turning of a body around a fixed point, e.g. the blade of an electric fan. c. **Oscillatory (or vibratory)**: To-and-fro movement about a central position, e.g. the bob of a pendulum. d. **Random**: Irregular movement which follows no definite pattern, e.g movement of players on a football field. - In reality, objects in motion often exhibit more than one type of motion at the same time. The wheels of a moving car, for instance, undergo both translational and rotational motion. ### 2.2 Linear Motion - This is movement along a straight line. The following terms are used to describe such a motion: - **Displacement (s)**: change in position along a specified direction. - **Velocity (v)**: change in displacement per unit time. - **Speed**: the magnitude (but not direction) of velocity - **Acceleration (a)**: change in velocity per unit time. - **Uniform Motion**. motion at constant velocity. - **Uniformly Accelerated Motion**: motion at constant acceleration. - If a body undergoing uniform motion covers a distance s in time, the velocity is $v = \frac{s}{t}$ along the direction of displacement - If a body changes its velocity uniformly from an initial value u to a final value v within a time interval t (uniform acceleration), the acceleration is $a = \frac{(v-u)}{t}$ or $v=u+at$ - Other equations of motion which relate the quantities u, v,a,l and s are: $s= ut + \frac{1}{2} at²$ and $v² = u² + 2as$ - Units: x= [m] - [s], u = [ms], v = [ms].a= [ms]. ### 2.3 Graphical Methods - The slope of a distance-time plot gives the velocity of motion (Fig. 2.1). - The slope of a velocity-time plot gives the acceleration (Fig. 2.2) - The area under a velocity-time plot gives the total distance covered (Fig. 2.3) ### 2.4 Freely Falling Bodies - A freely falling body does so under uniform acceleration This acceleration is roughly constant (about 9.80 ms²) at all points near the earth's surface and it is called the acceleration due to gravity (g). - For a body in free fall, the equations of motion (2.2-2.4) are applicable provided: - (i) a is set equal to g for downward motion - (ii) a is set equal to -g for upward motion. ### 2.5 Newton's Laws of Motion - Motion occurs in response to a force, which may be defined as a push or pull. Newton's laws of motion are: - **First Law**: 1. body remains in its state of rest or uniform motion in a straight line unless a net external force acts upon it - **Second Law**: 1 body which is subjected to a net external force (F) acquires an acceleration (a) which is directly proportional to the force and inversely proportional to the mass of the body. - **Third Law**: To every action (force) there is an equal and opposite force of reaction. - The first law introduces the concept of inertia, i.e. the reluctance of a body to change its state of rest or of uniform motion. - The second law is written mathematically as a = F/m, or $F=ma$ where m is the mass of the body. - To apply Newton's second law to a freely falling body, we set a = g and F = W (weight of the body) in eq. (2.5) to give $W=mg$ i.e. the weight of the body is the product of its mass and the acceleration due to gravity. The weight is also called the force of gravity on the body. - The third law involves two separate bodies. Thus if body A exerts a force F_A on B, then B also exerts a force F_B = -F_A on A. - Units: m = [kg], F= [newtons], abbreviated to [N], W=[N]. In terms of the fundamental units. [N] = [kg.ms²]. ### 2.6 Impulse and Momentum - The impulse on a body is the product of the force (F) and the time (1) for which it acted, i.e. $Impulse = Ft$ - The momentum of a body is the product of the mass (m) and velocity (v) of the body, i.e. $Momentum = mv$ - For a body which is uniformly accelerated from an initial velocity u to a final velocity v in time t, $a = \frac{(v-u)}{t}$ and the force producing the acceleration is, from eq. (2.5). $F = m \frac{(v-u)}{t} = \frac{(mv-mu)}{t}$ - Since mu = initial momentum and m = final momentum, eq. (2.9) states that the rate of change of momentum is directly proportional to the net external force acting on the body. This is an alternative statement of Newton's second law. - We can re-write eq. (2.9) as $Ft = mv-mu$ - In other words, the impulse on a body is equal to the change in the body's momentum. - The principle of conservation of momentum states that the total momentum of two colliding bodi-es remains constant if no external forces act on the bodies. - If masses m₁, m₂ move with velocities u,, u, before collision and with velocities v₁, v₂ after collision (Fig. 2.4): total momentum before collision = mu₁ + m₂uz total momentum after collision = m+m₂2 Conservation of momentum gives: total momentum before collision = total momentum after collision i.e. mu₂ + m₂42 = m₁₁ + m22 - Units: Impulse = [N.s] = [kg.ms'] = unit of momentum ### 2.7 Motion in a Circle - Consider a body which moves in a circular path of radius r. centred at O. Let us suppose that the body moves at a constant speed (of magnitude v) while its direction of motion changes continuously as it moves round the circular path. i.c. the body is accelerating (Fig. 2.5). For such motion: - Angular velocity (ω) = $\frac{angular displacement (\theta)}{time (t)}$ - Tangential velocity (v) = linear velocity in a tangential direction to the circular path. - The relationship between v and ω is $v=rω$