Exercises 2 (Plotting in rectangular coordinates)

  1. The strain \(\varepsilon\) induced in a wire when subjected to a series of stress \(\sigma\) values produced the following results:
\(\sigma\) (M Pa) \(10.8\) \(21.6\) \(33.3\) \(37.8\) \(45.9\)
\(\varepsilon\) (\(\times 10^{-5}\)) \(12\) \(24\) \(37\) \(42\) \(51\)

Show that the stress is related to the strain by a law of the form \(\sigma=E\varepsilon\), where \(E\) is a constant. Determine the law for the wire under test.

  1. During a test on a simple lifting machine, the following results were obtained showing the applied force, \(F\), for the load, \(L\), lifted:
\(F\) (N) \(19\) \(37\) \(50\) \(93\) \(125\) \(149\)
\(L\) (N) \(40\) \(120\) \(230\) \(410\) \(540\) \(680\)

It is thought that the equation relating \(F\) and \(L\) is of the form \(F = kL + c\) where both \(k\) and \(c\) are constants. Assuming that the law holds true, find the force necessary to lift a load of \(1\,\mathrm{kN}\).

  1. The variation in pressure, \(p\), within a vessel at a temperature, \(T\), follows a law of the form \(p = aT + b\). Verify that the data below relates the data by this law and determine the law.
\(p\) (kPa) \(248\) \(253\) \(257\) \(262\) \(266\) \(270\)
\(T\) (K) \(273\) \(278\) \(283\) \(288\) \(293\) \(298\)

  1. Determine graphically the solution to the simultaneous equations: \[\begin{align*} 2.5x + 0.45 - 3y &= 0\\ 1.6x + 0.8y - 0.8 &= 0 \end{align*}\]

  2. (After Chapter 4: Quadratic equations) Plot graphs of:

    1. \(y = 2x^2\)
    2. \(y = 2x^2 - 4\)
    3. \(y = 2x^2 - 2x + 0.5\)
    4. \(y = 2x^2 + x - 6\)

  3. (After Chapter 9: Long division and factorisation) Plot the graph of \(y = 4x^3 - 4x^2 - 15x + 18\) for values of \(x\) from \(-3\) to \(+3\) and using the graph, determine the roots of the polynomial.

  4. (After Chapter 8: Exponential functions) On the same axes and to the same scale plot the equations \(y = 1.5e^{-1.18x}\) and \(y = 1.1(1 - e^{-2.3x})\). Determine the solution to the equations from your graph.