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MATH10222 CALCULUS
创建日期：2024-04-12 14:57:12
所属分类：数学mathematics
标签： MATH10222
CALCULUS

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MATH10222 Advised Time to Complete: One and a Half Hours This exam will be worth 40% 

of the final mark on this course unit. You will also need to complete a separate online assessment worth 40%. 

CALCULUS AND APPLICATIONS Practice take-home test Exam Released: 09:00 (BST) End of Submission Window: 13:00 (BST) Answer the ONE question in Section A and TWO of the THREE questions in Section B. If more than TWO questions are attempted in Section B, then credit will be given for the FIRST TWO answers. This is a take-home open book exam. Your solutions should be written on white paper using blue or black ink (not pencil), on a tablet using a stylus, or typeset. Your solutions should include complete explanations and all intermediate derivations, and notation should be consistent with that in the Lecture Notes. © The University of Manchester, 2021 Page 1 of 4 P.T.O. MATH10222 SECTION A Answer the ONE question A1. An ODE question from the first half of the course. [13 marks] [End of Question A1; 13 marks total] Page 2 of 4 P.T.O. MATH10222 SECTION B Answer TWO of the three questions B2. A particle P of constant mass m is on a plane that is inclined at an angle 0 ≤ α < pi/2 to the direction of the unit vector i. At time t = 0, P is at the origin of the coordinate system and is projected with a speed U at an angle 0 ≤ θ ≤ pi to the plane as shown in the figure below. The particle moves in a uniform gravitational field −gj. (i) State Newton’s second law. [1 marks] (ii) Resolve the weight of the particle into two components, one parallel and one perpendicular to the plane. By considering the motion of P perpendicular to the plane, show that the particle hits the plane when t = τ , where τ = 2U sin θ g cosα . By considering the motion parallel to the plane, show that the displacement R of the particle P (measured up the plane), is R = U2 g cos2 α (sin(2θ + α)− sinα) , when t = τ . [10 marks] [Hint: You may use sinA cosB + cosA sinB = sin(A+B), and 2 sin2A = 1− cos(2A), for any angles A and B.] (iii) Find the maximum and minimum values of R for varying θ. Hence show that, for fixed U , the particle can always be projected farther down the slope than up the slope (before hitting the plane). [2 marks] α −mgjθ U i j [End of Question B2; 13 marks total] Page 3 of 4 P.T.O. MATH10222 B3. For a particle P in motion in a central field of force, you are given that the equations to be satisfied are r¨ − rθ˙2 = f(r) , and r2θ˙ = H0 , in the usual notation, where H0 is the constant angular momentum and f(r) defines the form of the field centered at O. By making the substitution u = 1/r, show that the governing equations above can be reduced to d2u dθ2 + u = − 1 H2 0 u2 f(1/u) . [5 marks] Hence, for the two cases of f(r) = −H2 0 /r3 and f(r) = 0, obtain the path of the particle given that it starts a distance a from O (along the line θ = 0) and is projected in such a way that du dθ = 0 , when θ = 0. Describe the shape of the two paths that you obtain. [8 marks] [End of Question B3; 13 marks total] B4. A uniform (thin) rod of total mass M and length 3l lies along the interval [−2l, l] of the y-axis of a Cartesian coordinate system. A particle P of mass m is situated a distance x along the x-axis. (i) Consider an element of the rod of length dy and give an expression for the gravitational force it exerts on P . [4 marks] (ii) By integration over all such elements of the rod, find the total gravitational force exerted on the particle P . [7 marks] (iii) If P is ‘far away’ from the rod, give the leading-order result for the gravitational attraction. In one sentence, interpret this simplified result. [2 marks] Hint: You may use ∫ x (x2 + y2)3/2 dy = y x √ x2 + y2 . [End of Question B4; 13 marks total] END OF EXAMINATION PAPER Page 4 of 4

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