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Unformatted text preview: 0.01[(10 cos t)2 − (10 sin t)2 ] · 10 dt 40 + 10 cos 2t dt =
0 2π =
0 = 40t + 5 sin 2t 2π 0 = 80π Note that BOTH sides of the fence will be painted, so the total area is A = 2 × 80π = 160π ≈ 503 (m2 ). In the question, we suppose 1 litre of paint covers 100 m2 , and so the amount of paint used is about 5 litre. 2 187 3. An 80 kg man carries a pail of water weighing 10 kg up a helical (spiralling) staircase with radius 6 m. The man makes exactly three complete revolutions and reaches a height of 30 m. Suppose a total of 2 kg of water leaks steadily during the man’s ascent. Express the work done by the man against gravity g ms−2 as a line integral vector ﬁeld F and the vector equation of the curve C . Evaluate the work done by computing this line integral. Preliminaries
C F • dr by writing down the Force and the Work Done DEFINITION Work The work done by a variable force F(x) directed along the xaxis from x = a to x = b is
b W= a L Fs x d dx . (2) Similarly, the work along a curve C is given by the line integral F(x, y, z ) • dr. C The Stairs in Vatican Museums 188 Solution Method I. (Direct Computation) The work done is given by the line integral W=
C F • dr, where, • F is the gravitational force contributed by the weight of the man and the pail of water; • C is the path traced out (which is a helix) by the man as he climbed up the staircase. Since C is a helix, and so its vector equation is given by r(t) = 6 cos ti + 6 sin tj + λtk, where λ is some constant. Remark: The vector equation means x and y forms a circle, and z is increasing with t, so it is a helix. Check more details from wikipedia: http://en.wikipedia.org/wiki/Helix. The solution contains two steps: Step 1 What is the force F? Since F is given by gravitation, the vector ﬁeld is of the form F(x, y, z ) = 0i + 0j − (Wm + Wp )gk where Wm = 80kg is the mass of the man and Wp is the mass of the pail of water. Since the pail leaks 2kg of water throughout the ascent, Wp is a linear function of z , i.e., Wp is of the form Wp (z ) = a + bz, for some constant a and b. Notice that Wp (0) = 10, Wp (30) = 8, and we can ﬁnd the value of a and b: • Use the slope of the straight line: slope = 30 − 0 z z−0 = =⇒ z = −15(Wp − 10) =⇒ Wp = 10 − . Wp − 10 8 − 10 15 ⎧ ⎨ a = 10 0 ≤ t ≤ 6π, • Use the Method of Undeterminated Coeﬃcients: Wp (0) = 10 =⇒ a + b × 0 = 10 =⇒ 1 ⎩ b=− Wp (30) = 8 =⇒ a + b × 30 = 8 15 Hence, F(x, y, z ) = 0i + 0j − (Wm + Wp )gk = 0i + 0j − (90 − 189 z )gk. 15 Step 2 What is the curve r(t)? 30 25 20 15 10 5 0 6 4 2 0 −2 −4 y −6 −6 −4 −2 x 0 2 4 6 We only need to determine the coeﬃcient λ in the vector equation of r(t): r(t) = 6 cos ti + 6 sin tj + λtk, 0 ≤ t ≤ 6π, Note that the question says that the height (the z compon...
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 Spring '09
 MA1505

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