CHAPTER 31 STRAIGHT LINE GRAPHS

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1 CHAPTER 31 STRAIGHT LINE GRAPHS EXERCISE 134 Page Assuming graph paper measuring 20 cm by 20 cm is available, suggest suitable scales for the following ranges of values: (a) Horizontal axis: 3 V to 55 V (b) Horizontal axis: 7 m to 86 m (c) Horizontal axis: 5 N to 150 N Vertical axis: 10 Ω to 180 Ω Vertical axis: 0.3 V to 1.69 V Vertical axis: 0.6 mm to 3.4 mm (a) Horizontal axis: 1 cm = 4 V (or 1 cm = 5 V), vertical axis: 1 cm = 10 Ω (b) Horizontal axis: 1 cm = 5 m, vertical axis: 1 cm = 0.1 V (c) Horizontal axis: 1 cm = 10 N, vertical axis: 1 cm = 0.2 mm 2. Corresponding values obtained experimentally for two quantities are: x y Plot a graph of y (vertically) against x (horizontally) to scales of 2 cm = 1 for the horizontal x-axis and 1 cm = 1 for the vertical y-axis. (This graph will need the whole of the graph paper with the origin somewhere in the centre of the paper.) From the graph find: (a) the value of y when x = 1 (b) the value of y when x = 2.5 (c) the value of x when y = 6 (d) the value of x when y = 5 A graph of y against x is shown plotted below. (a) When x = 1, the value of y = 1 (b) When x = 2.5, the value of y = 8 (c) When y = 6, the value of x =

2 (d) When y = 5, the value of x = 4 3. Corresponding values obtained experimentally for two quantities are: x y Use a horizontal scale for x of 1 cm = 1 2 unit and a vertical scale for y of 1 cm = 2 units and draw a graph of x against y. Label the graph and each of its axes. By interpolation, find from the graph the value of y when x is

3 Graph of y/x The graph of y against x is shown plotted above. From the graph, when x = 3.5, y = Draw a graph of y 3x + 5 = 0 over a range of x = 3 to x = 4. Hence determine (a) the value of y when x = 1.3 and (b) the value of x when y = 9.2 If y 3x + 5 = 0 then y = 3x 5 A table of values is shown below: x y

4 A graph of y against x is shown plotted below. (a) When x = 1.3, the value of y = 1.1 (b) When y = 9.2, the value of x =

5 5. The speed n rev/min of a motor changes when the voltage V across the armature is varied. The results are shown in the following table: n (rev/min) V (volts) It is suspected that one of the readings taken of the speed is inaccurate. Plot a graph of speed (horizontally) against voltage (vertically) and find this value. Find also (a) the speed at a voltage of 132 V and (b) the voltage at a speed of 1300 rev/min. A graph of voltage against speed is shown below. The 1010 rev/min reading should be 1070 rev/min at 140 V 517

6 (a) At a voltage of 132 V, the speed = 1000 rev/min (b) At a speed of 1300 rev/min, the voltage = 167 V 518

7 EXERCISE 135 Page The equation of a line is 4y = 2x + 5. A table of corresponding values is produced and is shown below. Complete the table and plot a graph of y against x. Find the gradient of the graph. x y y = 2x + 5 hence y = 0.5x The missing values are shown in bold. x y A graph of y/x is shown below. Gradient of graph = AB BC = = = Determine the gradient and intercept on the y-axis for each of the following equations: (a) y = 4x 2 (b) y = x (c) y = 3x 4 (d) y = 4 519

8 (a) Since y = 4x 2, then gradient = 4 and y-axis intercept = 2 (b) Since y = x, then gradient = 1 and y-axis intercept = 0 (c) Since y = 3x 4, then gradient = 3 and y-axis intercept = 4 (d) Since y = 4 i.e. y = 0x + 4, then gradient = 0 and y-axis intercept = 4 3. Find the gradient and intercept on the y-axis for each of the following equations: (a) 2y 1 = 4x (b) 6x 2y = 5 (c) 3(2y 1) = 4 x (a) Since 2y 1 = 4x then 2y = 4x + 1 and y = 2x and gradient = 2 and y-axis intercept = 1 2 (b) Since 6x 2y = 5 then 2y = 6x 5 and y = 3x 5 2 and gradient = 3 and y-axis intercept = (c) Since 3(2y 1) = 4 x then 12(2y 1) = x and 24y 12 = x and 1 1 y = x and gradient = 1 24 and y-axis intercept = Determine the gradient and y-axis intercept for each of the equations and sketch the graphs: (a) y = 6x 3 (b) y = 2x + 4 (c) y = 3x (d) y = 7 (a) Since y = 6x 3, then gradient = 6 and y-axis intercept = 3 A sketch of y = 6x 3 is shown below. 520

9 (b) Since y = 2x + 4, then gradient = 2 and y-axis intercept = 4 A sketch of y = 2x + 4 is shown below. (c) Since y = 3x, then gradient = 3 and y-axis intercept = 0 A sketch of y = 3x is shown below. (d) Since y = 7, then gradient = 0 and y-axis intercept = 7 A sketch of y = 7 is shown below. 5. Determine the gradient and y-axis intercept for each of the equations and sketch the graphs: (a) 2y + 1 = 4x (b) 2x + 3y + 5 = 0 (c) 3(2y 4) = 3 x (d) 5x 2 y 7 3 = 0 521

10 (a) Since 2y + 1 = 4x then 2y = 4x 1 and y = 2x 1 2 Hence, gradient = 2 and y-axis intercept = 1 2 A sketch of y = 2x 1 is shown below. 2 (b) Since 2x + 3y + 5 = 0 then 3y = 2x 5 and y = 2 5 x 3 3 Hence, gradient = 2 and y-axis intercept = or A sketch of y = 2 5 x is shown below. 3 3 (c) Since 3(2y 4) = 3 x then 6y 12 = 3 x and 6y = 3 x + 12 from which, y = 18 x + 2 Hence, gradient = 1 18 and y-axis intercept = 2 A sketch of y = 18 x + 2 is shown below. 522

11 y 7 (d) Since 5x 2 3 = 0 then 6(5x) (6) y 7 (6) 2 3 = (6)0 i.e. 30x 3y 14 = 0 i.e. 30x 14 = 3y and y = 10x 14 3 Hence, gradient = 10 and y-axis intercept = 14 or A sketch of y = 10x 14 3 is shown below. 6. Determine the gradient of the straight line graphs passing through the coordinates: (a) (2, 7) and ( 3, 4) (b) ( 4, 1) and ( 5, 3) (c) 1 3, 4 4 and 1 5, 2 8 (a) From page 261 of textbook, gradient = y x y = = x = (b) Gradient = y2 y = = x x =

12 (c) Gradient = y x y x = = = = = = State which of the following equations will produce graphs which are parallel to one another: (a) y 4 = 2x (b) 4x = (y + 1) (c) x = 1 (y + 5) 2 (d) y = 3 2 x (e) 2x = 1 (7 y) 2 (a) Since y 4 = 2x then y = 2x + 4 (b) Since 4x = (y + 1) then y = 4x (c) Since x = ( 5) y + then 2x = y + 5 and y = 2x (d) Since 1+ y = x then 2 + y = 3x and y = 3x (e) Since 2x = ( y) 1 7 then 4x = 7 y and y = 4x Thus, (a) and (c) are parallel (since their gradients are the same), and (b) and (e) are parallel 8. Draw on the same axes the graphs of y = 3x 5 and 3y + 2x = 7. Find the coordinates of the point of intersection. Check the result obtained by solving the two simultaneous equations algebraically. 2 7 The graphs of y = 3x 5 and 3y + 2x = 7, i.e. y = x + are shown below

13 The two graphs intersect at x = 2 and y = 1, i.e. the coordinate (2, 1) Solving simultaneously gives: y = 3x 5 i.e. y 3x = 5 (1) y = 2 x i.e. 3y + 2x = 7 (2) 3 (1) gives: 3y 9x = 15 (3) (2) (3) gives: 11x = 22 from which, x = 2 Substituting in (1) gives: y 6 = 5 from which, y = 1 as obtained graphically above 9. Plot the graphs y = 2x + 3 and 2y = 15 2x on the same axes and determine their point of intersection. The graphs y = 2x + 3 and 2y = 15 2x, i.e. y = x are shown plotted below The point of intersection of the two graphs is at x = 1.5 and y = 6, i.e. at the coordinates (1.5, 6) 10. Draw on the same axes the graphs of y = 3x 1 and y + 2x = 4. Find the coordinates of the point of intersection. A table of values is drawn up for each equation. x x y = 3x y = 2x

14 Graphs of y/x are plotted for both equations on the same axes as shown below. The coordinates of the point of intersection are: (1, 2) 11. A piece of elastic is tied to a support so that it hangs vertically, and a pan, on which weights can be placed, is attached to the free end. The length of the elastic is measured as various weights are added to the pan and the results obtained are as follows: Load, W (N) Length, l (cm) Plot a graph of load (horizontally) against length (vertically) and determine: (a) the value length when the load is 17 N, (b) the value of load when the length is 74 cm, (c) its gradient, and (d) the equation of the graph. A graph of load against length is plotted as shown below (a) When the load is 17 N, the length = 89 cm (b) When the length is 74 cm, the load = 11 N (c) Gradient = AB BC = = = 2.4 (d) Vertical axis intercept = 48 cm 526

15 Hence, the equation of the graph is: l = 2.4W The following table gives the effort P to lift a load W with a small lifting machine: W (N) P (N) Plot W horizontally against P vertically and show that the values lie approximately on a straight line. Determine the probable relationship connecting P and W in the form P = aw + b. A graph of W against P is shown plotted below. Gradient of graph = AB BC = = = 0.15 Vertical axis intercept = 3.5 Hence, the equation of the graph is: P = 0.15W

16 13. In an experiment the speeds N rpm of a flywheel slowly coming to rest were recorded against the time t in minutes. Plot the results and show that N and t are connected by an equation of the form N = at + b. Find probable values of a and b. t (min) N (rev/min) A graph of N against t is shown plotted below. Gradient of graph, a = AB BC = = = 20 Vertical axis intercept, b = 412 Hence, the equation of the graph is: N = 20t

17 529

18 EXERCISE 136 Page The resistance R ohms of a copper winding is measured at various temperatures t C and the results are as follows: R ohms t C Plot a graph of R (vertically) against t (horizontally) and find from it (a) the temperature when the resistance is 122 Ω and (b) the resistance when the temperature is 52 C. A graph of resistance R against temperature t is shown below. From the graph: (a) the temperature when the resistance is 122 Ω is 40 C (b) the resistance when the temperature is 52 C is 128 Ω 530

19 2. The speed of a motor varies with armature voltage as shown by the following experimental results: n (rev/min) V volts Plot a graph of speed (horizontally) against voltage (vertically) and draw the best straight line through the points. Find from the graph (a) the speed at a voltage of 145 V and (b) the voltage at a speed of 400 rev/min. A graph of speed against voltage is shown below. From the graph: (a) at a voltage of 145 V, the speed = 850 rev/min (b) at a speed of 400 rev/min, the voltage = 77.5 V 531

20 3. The following table gives the force F newtons which, when applied to a lifting machine, overcomes a corresponding load of L newtons. Force F newtons Load L newtons Choose suitable scales and plot a graph of F (vertically) against L (horizontally). Draw the best straight line through the points. Determine from the graph (a) the gradient, (b) the F-axis intercept, (c) the equation of the graph, (d) the force applied when the load is 310 N and (e) the load that a force of 160 N will overcome. (f) If the graph were to continue in the same manner, what value of force will be needed to overcome a 800 N load? A graph of F against L is shown below. From the graph: (a) the gradient = AB BC = = = 0.25 (b) the F-axis intercept = 12 N 532

21 (c) the equation of the graph is: F = 0.25L + 12 (d) the force applied when the load is 310 N is 89.5 N (e) the load that a force of 160 N will overcome is 592 N (f) If the graph were to continue in the same manner, the force needed to overcome a 800 N load is 212 N. From the equation of the graph, F = 0.25L + 12 = 0.25(800) + 12 = = 212 N 4. The following table gives the results of tests carried out to determine the breaking stress σ of rolled copper at various temperatures, t: Stress σ (N/cm 2 ) Temperature t( C) Plot a graph of stress (vertically) against temperature (horizontally). Draw the best straight line through the plotted coordinates. Determine the slope of the graph and the vertical axis intercept. A graph of stress σ against temperature t is shown below. The slope of graph = AB BC = = = Vertical axis intercept = 8.73 N/cm 2 533

22 5. The velocity v of a body after varying time intervals t was measured as follows: t (seconds) v (m/s) Plot v vertically and t horizontally and draw a graph of velocity against time. Determine from the graph (a) the velocity after 10 s, (b) the time at 20 m/s and (c) the equation of the graph. A graph of velocity v against time t is shown below. From the graph: (a) After 10 s, the velocity = 22.5 m/s (b) At 20 m/s, the time = 6.5 s 534

23 (c) Gradient of graph = AB BC = = = 0.7 Vertical axis intercept at t = 0, is v = 15.5 m/s Hence, the equation of the graph is: v = 0.7t The mass m of a steel joist varies with length L as follows: mass, m (kg) length, L (m) Plot a graph of mass (vertically) against length (horizontally). Determine the equation of the graph A graph of mass against length is shown below. Gradient of graph = AB BC = = = 26.8 Vertical axis intercept is m = 0 Hence, the equation of the graph is: m = 26.8L 535

24 7. The crushing strength of mortar varies with the percentage of water used in its preparation, as shown below. Crushing strength, F (tonnes) % of water used, w% Plot a graph of F (vertically) against w (horizontally). (a) Interpolate and determine the crushing strength when 10% water is used (b) Assuming the graph continues in the same manner, extrapolate and determine the percentage of water used when the crushing strength is 0.15 tonnes. (c) What is the equation of the graph? A graph of F against w is shown below. (a) When w = 10%, crushing strength, F = 1.25 tonnes (b) When F = 0.15 tonne, percentage of water used, w = 21.6% (c) Gradient of straight line = AB BC = = = Vertical axis intercept =

25 Hence, the equation of the graph is: F = 0.095w In an experiment demonstrating Hooke s law, the strain in a copper wire was measured for various stresses. The results were: Stress (pascals) Strain Plot a graph of stress (vertically) against strain (horizontally). Determine (a) Young s Modulus of Elasticity for copper, which is given by the gradient of the graph, (b) the value of strain at a stress of Pa, (c) the value of stress when the strain is A graph of stress against strain is shown below. (a) Gradient of straight line = AB BC ( ) = = = Hence, Young s Modulus of Elasticity = gradient = Pa (b) From the graph, when stress is Pa, strain = (c) From the graph, when the strain is , stress = Pa 537

26 9. An experiment with a set of pulley blocks gave the following results: Effort, E (newtons) Load, L (newtons) Plot a graph of effort (vertically) against load (horizontally) and determine (a) the gradient, (b) the vertical axis intercept, (c) the law of the graph, (d) the effort when the load is 30 N and (e) the load when the effort is 19 N. A graph of effort E against load L is shown below. (a) Gradient of straight line = AB BC = = = 1 5 or 0.2 (b) Vertical axis intercept = 6 (c) The law of the graph is: E = 0.2L + 6 (d) From the graph, when the load is 30 N, effort, E = 12 N (e) From the graph, when the effort is 19 N, load, L = 65 N 538

27 10. The variation of pressure p in a vessel with temperature T is believed to follow a law of the form p = at + b, where a and b are constants. Verify this law for the results given below and determine the approximate values of a and b. Hence determine the pressures at temperatures of 285 K and 310 K and the temperature at a pressure of 250 kpa. pressure, p kpa temperature, T K A graph of pressure p against temperature T is shown below. Plotting the values of p against T produces a straight line, hence will be of the form p = at + b Taking points A and B on the straight line gives: For point A, (300, 267) 267 = 300a + b (1) For point B, (273, 244), 244 = 273a + b (2) 539

28 (1) (2) gives: 23 = 27a from which, a = = 0.85 Substituting in (1) gives: 267 = 300(0.85) + b from which, b = (0.85) = 12 Hence, the law of the graph is: p = 0.85T + 12 When T = 285 K, pressure, p = 0.85(285) + 12 = kpa When T = 310 K, pressure, p = 0.85(310) + 12 = kpa When p = 250 kpa, then 250 = 0.85(T) + 12 from which, = 0.85T and temperature, T = = 280 K 540