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Page 1: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

University of KwaZulu-Natal, Pietermaritzburg

Examination

November 1st, 2012

INTERNAL EXAMINERS: DR JM T NGNOTCHOUYE

DR H MAMBILI MAMBOUNDOU

EXTERNAL MODERATION: DR K ARUNAKIRINATHAR

Subject (course code): Math 142 (Applied Mathematics 1B)Time: 3 HoursTotal Marks: 110 Marks

INSTRUCTIONS

1. Fill in the following:

Student Number:

Signature:

2. A complete paper has sixteen pages, including this coverpage. Check that you have them all.

3. There are nine questions, with marks given to the right.Answer all questions and show all working.

4. Write in ink. Rough work can be done in pencil on thereverse side of each page.

5. You may leave your answers in terms of π or in surd form.

For Marker Only

1 /24

2 /10

3 /10

4 /6

5 /12

6 /12

7 /13

8 /7

9 /14

Σ /110

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Applied Mathematics 1B: Examination 2012. 2

Question 1 [24 Marks]

(a) Consider the shaded area below.

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x

y

y = ex

y = −x + 1

1

(3, e3)

(3,−2)

(i) Determine by integration the surface A of the area. (3)

(ii) Find the x− coordinate of the centroid of the area. (3)Hint: An anti derivative of xex is (x − 1)ex.

(ii) Find the volume of the object of revolution obtained by rotating the area aboveabout the y− axis. (2)

Page 3: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 3

Question 1 (Continued)

(b) The homogeneous plate with uniform thickness T below has three parts. Part 1 is zincwith density ρz = 7135 kg/m3 ; part two is aluminum with density ρa = 2712 kg/m3

and part 3 is iron with density ρi = 7850 kg/m3. Determine the center of mass of theobject. (4)

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3m

4m

4m5m

6m

12m

x

y

Zinc

Aluminum

Iron

(c) Find the x− coordinate of the centroid of the composite line below. (3)Hint. The centroid of a quarter circle of radius R in first quadrant of the x− y planeis (2R

π, 2R

π).

3

3

4

(0, 2, 3)

z

x

y

Page 4: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 4

Question 1 (Continued)

(d) The light beam depicted below is supported by a pin support at A and a roller supportat B.

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�������������������������������������������������������� ������������������

2m

6m

10m

200N/m

100N/m

100N

A B

45◦

(i) Find the equivalent force F associated with the distributed load (1)

(ii) Find the x-coordinate of the centroid of the area under the loading curve in areference frame with the origin at A. (1)

(iii) Find the reactions at the supports A and B. (4)

Page 5: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 5

Question 2 [10 Marks]

(a) Determine the area-moment of inertia Iy of the composite surface below. (5)HINT: The moment of inertia of rectangle of base b and height h about vertical axispassing through its centroid is 1

12hb3; for a triangle it is 1

36hb3; and for a disk of radius

R, it is 1

4πR4.

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x

y

8 cm

2 cm15 cm

20 cm

22 cm

30 cm

C

1. Find the x coordinate of the centroid of the area. (2)

2. Find the area-moment of inertia of the area about an horizontal Cy′ axis passingthrough C. (3)

Page 6: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 6

Question 3 [10 Marks]

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A

B

25◦

The mass of block B is mB = 20 kg. The tension in the cable is T = 50N. The coefficientsof static and kinetic friction for the surfaces are µs = 0.15 and µk = 0.10, respectively.

(a) Assume there is no motion and that the friction force exerted on box A is the maximumstatic friction force. Find the friction and normal forces acting on box B in term themass mA of box A. (3)

(b) Find the maximum value of the mass of box A for which the two boxes move to theright. (2)

(c) Now let mA = 35 kg. Assuming that the two boxes are released from rest, what istheir velocity when they have moved 40 cm to the right? (5)

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Applied Mathematics 1B: Examination 2012. 7

Question 4 [6 Marks]

The 2 kg mass rotates around the vertical pole in a horizontal circular path. The angleθ = 45◦ and the length of the string is L = 1.5m. Find the tension in the string T and themagnitude of the velocity v of the mass. (6)HINT: Write Newton second law in the vertical and normal direction.

m

θ L

Page 8: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 8

Question 5 [12 Marks]

(a) A projectile is launch at v0 = 20m/s at the elevation 30◦ above the horizontal. Thesurface on which it lands is described by the equation y = −0.005x. Determine thex− coordinate of the point of impact and the flying time of the projectile. (5)

(b) The position in terms of polar coordinates of an object is given as a function of timeas

r = 2t2 + 2et,θ = 4 − 3 sin(π

2t).

Find the radial acceleration ar and the transverse acceleration aθ of the object at timet = 2 s. (3)

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Applied Mathematics 1B: Examination 2012. 9

Question 6 [12 Marks]

Consider the two 300g pendulums A and B, of length l = 1m. Pendulum A is released withno velocity from position A1. Pendulum A hits pendulum B which is initially at rest. Afterthe impact pendulum B swings through an angle θ.

1. Pendulum A swings from position A1 to position A2. Use conservation ofenergy to find the velocity vA2

of pendulum A, just before it collides with pendulumB. (2)

2. Pendulum A hits pendulum B. Use conservation of momentum to find the veloc-ities vA3

and vB3of the two pendulums after impact if the coefficient of restitution is

e = 0.8. (6)

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Applied Mathematics 1B: Examination 2012. 10

Question 1 [continued]

3. Pendulum B swings from B3 to B4. Use conservation of energy for pendulum Bto determine the height y4 and then the angle θ. (4)

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Applied Mathematics 1B: Examination 2012. 11

Question 7 [13 Marks]

A 8-kg crate is initially at rest when it is subjected to an an horizontal force given as afunction of time by F = 82 + 5t2 (in Newton.) The coefficients of kinetic friction betweenthe crate and the inclined surface is µk = 0.25.

(a) Draw the free-body diagram of the crate showing all forces acting on it. (1)

(b) Determine the magnitude of the linear impulse exerted on the crate from t = 0 tot = 5 s. (5)

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Applied Mathematics 1B: Examination 2012. 12

Question 7 [continued]

(c) Using the principle of impulse and momentum, find the velocity of the crate at t = 5s. (2)

(d) Suppose that the force F is only applied from t = 0 to t = 5 second. Using theprinciple of work and energy, find out how further the crate travels up the inclinedbefore it stops (5)

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Applied Mathematics 1B: Examination 2012. 13

Question 8 [7 Marks]

At t = 0, the velocity of a 20-Kg object is V= 3i + 5j - 2k (in m/s). The total force actingon it from t = 0 to t = 10 second is

F =(

(t − 1)3 − 2t + 4)

i + t2j - 4tk

(a) Use the principle of impulse and momentum to determine the magnitude of the ob-ject’s velocity at t = 10 s. (5)

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Applied Mathematics 1B: Examination 2012. 14

Question 8 [continued]

(b) What is the average force on the object during that interval of time (2)

Question 9 [14 Marks]

(a) At the instant shown, disc A has an angular velocity ωA = 7 + 0.5t (in rad/s). As-suming that there is no relative motion between the discs and if RA = 500-mm,RB =750-mm and RC =1200 mm

(i) Find the angular velocities of discs B and C as functions of time. (2)

(ii) Find the angular accelerations of discs A, B and C. (3)

Page 15: University of KwaZulu-Natal, Pietermaritzburg Examination ... · University of KwaZulu-Natal, Pietermaritzburg Examination November 1st ... at the supports A and B ... around the

Applied Mathematics 1B: Examination 2012. 15

Question 9 [continued]

(ii) Through what angles do discs B and C turn from t = 0 to t = 4. second ? (2)

(b) The total external forces acting on a 5-Kg object is given as a function of time by∑

F = 5 i - 7t k. At time t = 0 the object’s position and velocity are r = 0 and v =0.

(i) Use Newton’s second law to determine the object’s position and velocities asfunctions of time. (2)

(ii) Find the object’s moment M0 about the origin as a function of time. (3)

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Applied Mathematics 1B: Examination 2012. 16

Question 9 [continued]

(iii) Using the principle of angular impulse and momentum, find the change in theobject’s angular momentum from t = 0 to t = 8 second. (2)