PHYS1150 Problem Solving in Physics

Assignment 3

Due Date: October 19, 2024 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 problems with perseverance.

1. Find the following integrals.

Hint to part (b): Completing the square

Hint to part (c):

2. (a) Evaluate (5 marks)

(b) Evaluate (5 marks)

[Hint to part (b): cos 2 x = cos2 x − sin2 x]

3. By using the substitution: 1 + x = t/1, show that               (10 marks)

4. Evaluate where n > 1.                   (15 marks)

[Hint: If sec u + tan u = p, show that sec u − tan u = 1/p.]

5. A uniform. solid hemisphere H of mass M and radius R is fixed to a coordinate system, where the center of the hemisphere is located at the origin and the x-axis is normal to the plane surface of the hemisphere as shown in the figure below. The hemisphere is divided into numerous mass elements, each of them has the form. of a disk with infintesimal thickness, volume, and mass given by dx, dV , and dm respectively. A mass element of the hemisphere is highlighted in the figure.

The x coordinate of the center of mass of the hemisphere is defined as

where x is the coordinate of the center of the mass element and dm is the infinitesimal mass of the element. The integration includes all masses in H. Denote the density of the hemisphere as p.

(a) Express M in terms of p and R. (2 marks)

(b) Show that dm = pdR3 cos3θdθ. (3 marks)

[Hint: dm = pdV , where dV is the infintesimal volume of the mass element.]

(c) Show that xcm = 8/3R.

6. Evaluate (16 marks)

7. Let sinn x dx, where n = 0, 1, 2, 3, ··· .

(a) Show that n un un−1 = 2/π for n = 1, 2, 3, ··· . (6 marks)

(b) Show that 0 < un < un−1 for n = 1, 2, 3, ··· . (2 marks)

(c) Hence, show that

8. Evaluate the integral dx by using the following methods.

(a) Integration by parts (5 marks)

(b) Simpson’s rule (5 marks)

You are suggested to divide the interval [0, 1] into 4 subintervals of equal width. Please copy the following table to your answer book and complete it. Correct the final answer of the integral to 4 decimal places. Let f(x) = x2 ex and n = 0, 1, 2, 3, and 4.