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PHYS1150 Problem Solving in Physics

Assignment 6

Due Date: November 26, 2022 at 11:00 pm

The problems in this paper have a wide range of diﬃculties. They are specially designed to strengthen students’ concepts and to develop the problem solving skills. Students are expected to drill the prob- lems with perseverance.

1. If u = tan− 1 ╱ ^ 夕^夕、, ﬁnd and夕(u) . Verify that z + 夕夕(u) = sin 2u.

,z ,2 z

3. (a) If f (zo 夕) = 0 and 9(夕o 之) = 0, show that

1 ,2 之 (b) If a2 z2 + b2 夕2 = c2 之2 , show that

,f ,9 d之 ,f ,9

,夕 ,之 dz ,z ,夕 .

1 ,2 之 1

b2 ,夕2 c2 之 .

(8 marks)

(8 marks) (8 marks)

(8 marks)

4. Given that u(zo 夕) and u(zo 夕) are diﬀerentiable functions, where z = r cos y and 夕 = r sin y .

Show that the equation set

(12 marks)

．(．) =

(*) ．

．‘ ,夕 =

becomes

,夕

．,z

,u 1 ,u

．(．) ,u 1 ,u

5. Given that u(zo 夕) is a diﬀerentiable function, where z = r cos y and 夕 = r sin y . Show that (12 marks)

,2u ,2u ,2u 1 ,u 1 ,2u

,z2 ,夕2 ,r2 r ,r r2 ,y2

6. Given that u(zo 夕) and u(zo 夕) are diﬀerentiable functions. Show that (12 marks)

+ = 1 and

,u ,z ,u ,夕

,z ,u ,夕 ,u

7. Given that 之 = f ╱ o r 2 a(．)s、, where a is a constant. Let z = and 夕 = r 2 a(．)s . (a) Show that r(之) = ╱ + 一夕(f)、 and s(之) = ╱ ．一夕(f)、. (8 marks)

(b) Show further that r(2)2(之) +s(2)2(之) = ╱ + 一夕(2)f2、. (8 marks)

8. (a) If 之(zo 夕) is a homogeneous function of degree 2, i.e. 之 = z2 f ╱ z(夕)、, show that (8 marks)

z z(之) + 夕夕(之) = 2 之 and z2(之) + 2 z夕之,夕 + 夕2夕(2)2(之) = 2 之 (b) If u = ln ╱ z(z)2(4) + 夕(夕)2(4)、, ﬁnd z + 夕夕(u) and z2 + 2 z夕 ,2夕 + 夕2夕(2)u2 . (8 marks)