Physical quantity • • o o Any kind of measurable quantity is called physical quantity. There are two kinds of physical quantities: Base quantities Derived quantities Derived quantity Which is an SI base unit? A current B gram C kelvin D volt example example Express the following units in terms of the SI fundamental units. a) newton (N) b) watt (W) c) pascal (Pa) d) coulomb (C) e) volt (V) prefixes • Prefixes are used to express very large and very small numbers A student measures a current as 0.5 A. Which of the following correctly expresses this result? A 50 mA B 50 MA C 500 mA D 500 MA Significant figures example Express the following numbers to three significant figures. a) 257.52 b) 0.002 347 c) 0.1783 d) 7873 e) 1.997 example A fisherman catches two striped bass. The smaller of the two has a measured length of 93.46 cm (two decimal places, four significant figures), and the larger fish has a measured length of 135.3 cm (one decimal place, four significant figures). What is the total length of fish caught for the day? example Determine the number of significant figures in each of the following. (a) 0.0015 ___ (b) (b) 0.15___ (c) (c) 1.500 ___ (d) (d) 1.0005___ (e) 1.00050___ (f) (f) 0.00010000 ___ (g) (g) 6.35106___ (h) (h) 16010-21___ example Complete the following calculations and express your answers to the most appropriate number of significant figures. example The radius of a sphere is r=12.37 m. What is the volume of the sphere correct to two significant figures? A) 7928.62 m3 B) 7920 m3 C) 7928 m3 D) 7900 m3 Dimensional analysis We use the dimensional analysis to check the correctness of the formula. We write the formula and write the units of the quantities for both sides of the formula. If the units in both sides of the equation is the same, then the formula is correct. The unit of work, the joule, may be defined as the work done when the point of application of a force of 1 newton is moved a distance of 1 in the direction of the force. Express the joule in terms of the base units of mass, length and time, the kg, m and s example Show that the expression v =v0 + at is dimensionally correct, where v and v0 represent velocities, a is acceleration, and t is a time interval. example Find a relationship between a constant acceleration a, speed v, and distance r from the origin for a particle traveling in a circle. example (a) Derive the SI base unit of force. (b) A spherical ball of radius r experiences a resistive force F due to the air as it moves through the air at speed v. The resistive force F is given by the expression F= crv, where c is a constant. Derive the SI base unit of the constant c. example A shape that covers an area A and has a uniform height h has a volume V = Ah. (a) Show that V = Ah is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form V = Ah, identifying Ain each case. (Note that A, sometimes called the “footprint” of the object, can have any shape and that the height can, in general, be replaced by the average thickness of the object.) example example example Vector and scalar quantities Scalar quantities are those that have magnitude (or size) but no direction. Vector quantities are those which have both magnitude and direction Which of the following is a scalar quantity? a) acceleration b) mass c) momentum d) velocity Which pair contains one vector and one scalar quantity? A displacement acceleration B force kinetic energy C momentum velocity D power speed Representing vector quantities A vector quantity is represented by a line with an arrow. ● The direction the arrow points represents the direction of the vector. ● The length of the line represents the magnitude of the vector to a chosen scale. Adding and subtracting vectors When adding and subtracting vectors, account has to be taken of their direction. This can be done either by a scale drawing (graphically) or algebraically Graphical method • Having chosen a suitable scale, draw the scaled lines in the direction of V1 and V2 (so that they form two adjacent sides of the parallelogram). • Complete the parallelogram by drawing in the remaining two sides. • The blue diagonal represents the resultant vectorin both magnitude and direction. Worked example Two forces of magnitude 4.0 N and 6.0 N act on a single point. The forces make an angle of 60°with each other. Using a scale diagram, determine the resultant force. example A cyclist travels a distance of 1200 m due north before going 2000 m due east followed by 500 m south-west. Draw a scale diagram to calculate the cyclist’s final displacement from her initial position Two forces of magnitude 6.0 N and 8.0 N act at a point P. Both forces act away from point P and the angle between them is 40°. Fig. 1.1 shows two lines at an angle of 40° to one another. On Fig. 1.1, draw a vector diagram to determine the magnitude of the resultant of the two forces. Subtraction of vectors When subtracting one vector from another, we just take the negative of the vector being subtracted and add this negative vector to the other vector. The diagram shows two vectors X and Y. In which vector triangle does the vector Zshow the magnitude and direction of vector X–Y? Algebraic approach Resolving vectors The force F in figure 6 has been resolved into the horizontal component equal to F cos θ and a vertical component equal to F sin θ example example example example Adding vector quantities that are not at right angles • Resolve each of the vectors in two directions at right angles – this will often be horizontally and vertically, but may be parallel and perpendicular to a surface. • Add all the components in one direction to give a single component. • Add all the components in the perpendicular direction to give a second single component. • Combine the two components using Pythagoras’ theorem, as for two vector quantities at right angles. V1 and V2 are the vectors to be added. example The diagram shows three forces P, Q, and R in equilibrium. Pacts horizontally and Q vertically. When P = 5.0 N and Q =3.0 N, calculate the magnitude and direction of R. example A boat, starting on one bank of a river, heads due south with a speed of 1.5 m s−1 . The river flows due east at 0.8 m s−1 a) Calculate the resultant velocity of the boat relative to the bank of the river. b) The river is 50 m wide. Calculate the displacement from its initial position when the boat reaches the opposite bank. example A car of mass 850 kg rests on a slope at 25° to the horizontal. Calculate the magnitude of the component of the car’s weight which acts parallel to the slope.
