Question

2. A circular pipe of radius b is to be welded to a circular pipe of radius a b in such a way that the centrelines of the two pipes intersect at right angles, as shown below. (a) Suppose that the centreline of the pipe with the larger radius a coincides with the x axis, while the centreline of the pipe with the smaller radius b coincides with the y axis. Write down equations for the surface of each pipe in Cartesian coordinates. [2 marks] (b) The pipes are to be welded continuously along the intersection of the two pipes, shown in yellow in the figure above. Find a parametric expression for this curve in the form r(@) = (b cos 0, f(0), b sin ) for 0 <0 < 27, where fe is to be determined as a part of your solution. You must also show that this parameterisation satisfies the equation for the surface of the smaller pipe in your answer to (a). [3 marks (c) Use your answer to (b) to write down an expression for the length L of the weld in terms of a and b. Simplify your expression as much as possible. You may leave the integral unevaluated. [3 marks (d) Find the value of the ratio L/l, where l is the circumference of the smaller pipe, for b/a = 3/4. Express your answer as a number to three decimal places. You may use your calculator, Wolfram Alpha or other mathematical software to evaluate the integral. [2 marks]

Answer 1

Answers Page no-o Given that, A circular pipe of ladius b is to be welded to a n circular pipe of ladius t u a>b i The centrelines of the two pipes inturect at eight angles, (a) The given surface is i cylindrical with the axis - x and axis-yo - the equation of cylinder is, 1 xht Z² 62 ght z² az (6) The pipes are to be welded continuosly along the intersection of the two pipes. The paranetic expression for this curve in the form r(O) = (b caso , fem), .. para moteric equation for Surface. Zat Y = √ azzz =/6²t² ret) =(X,Y,Z) =</62tr, sant, t>

(c) Length of yellow welt - Lotto Page No - ③ dret runt en dit ) i de t2 tl at 15 - d² t (d) ķ erment in the patt") 4 ; 2=45. ✓ Horror +1) dt T L/_t tat2 X 21b rl²t² 166² 9 t -942 Calculator is not able to evaluate this integral]. e