# MATH S222 MATHEMATICAL MODELS WITH APPLICATIONS

## MATH S222 MATHEMATICAL MODELS WITH APPLICATIONS

Assignment Booklet I
This booklet contains TMA01 covering Units 1 to 4. The cut-off date is on 12 April 2021. The total mark of this TMA is 100. Each question is marked:
Question No.
Covers
Marks
Q1
Unit 1 25
Q2
Unit 2 25
Q3
Unit 3 25
Q4
Unit 4 25
You are required to use Mathcad to answer Question 2. Submit a printout of Mathcad solutions to show your working.
If you submit your TMA01 via post, be sure to fill in TMA FORM. In the TMA Form, you must fill in the assignment number, your student number, date sent to tutor, and your corresponding address. You are advised to keep a copy of your answers. In the case that your assignment is lost due to any postage errors, you can re-submit the duplicated copy to your tutor.
Please note that MATH S222 will count the highest three out of the four TMAs towards the final assessment score. The weighting for each TMA is 33.33%.
2
Question 1 (Unit 1) – 25 marks
(a) A block of mass M lies at rest on a rough inclined plane with an inclination of  to the horizontal. The coefficient of static friction between the block M and the plane is . The block M is connected by a model string to another block of mass m hanging vertically as shown in the following figure. The magnitude of the acceleration due to gravity is g.
(i) Model the block as a particle. Let tension in the string be T, its weight W, normal reaction and frictional force on it be N and F respectively. Draw a force diagram showing all the forces acting on block M. 
(ii) Write each of the forces in terms of its components in the coordinate system i and j shown. 
(iii) Use the conditions for equilibrium, together with the property of a model pulley, to find scalar equations relating the magnitudes of the forces. 
(iv) Show that if the block is on the point of moving up the plane, show that: 
m = M(sin + cos)
(v) The block of mass m and the pulley are now removed. A horizontal force P is applied onto the block of mass M as shown in the following figure to pull it down the plane.
If the block is now on the point of moving down the plane, show that the magnitude of P is given by:  Mgμcosθ−sinθμsinθ+cosθ
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(b) Two ladders, AB and AC, of equal length and mass m1 and m2 respectively, are smoothly jointed at A. B is joined by a string to the mid-point D of AC. The whole system then rests with B and C on a smooth horizontal plane with BCA = 60o as shown in the following figure. The magnitude of the acceleration due to gravity is g.
(i) Draw a force diagram showing all the forces acting on the system. 
(ii) Using the coordinate system shown, express each of the forces in terms of the unit vectors i and j. 
(iii) Find the vertical reactions at C and B in terms of m1, m2 and g. 
(v) The tension in the string can be found by considering only the forces acting on ladder AC. Draw a force diagram showing the forces acting on AC. Hence, by considering the torques at A, show that the tension in the string is: 
(m1 – 0.5m2)g.
(c) A block of weight W in the form of a cube of side 2a stands on a horizontal plane, the coefficient of friction between the block and the plane being  as shown in the following figure. A gradually increasing horizontal force P is applied to a vertical face of the cube, at right angles to it, and is applied to a vertical plane through the centre of mass of the cube and at a height of x from the ground.
Let R be the normal reaction and F be the frictional force.
(i) Draw a force diagram showing all the forces acting on the block. 
(ii) Using the conditions of equilibrium, find scalar equations relating the magnitudes of the forces R and F. 
(iii) Show that if <0.5, the cube will slide first without tilting. 
4
Question 2 (Unit 2) – 25 marks
A man as shown in the following figure stands on the edge of a cliff of height 20m above the sea, and throws a pebble of mass 0.02kg vertically upwards with speed 10ms-1. On its return he does not catch the pebble, which continues past him and falls into the sea.
(a) Assume that the only force acting on the pebble is its weight due to gravity and ignore air resistance and the height of the man. Taking the magnitude of the acceleration due to gravity g to be 9.81ms-2, show that the maximum height above the cliff that the pebble can attain is 5.1m. Find the speed of the pebble just before it hits the sea. 
(b) The model in (a) is revised by taking air resistance into account. Model the pebble as a sphere of diameter 0.02m, and assume that the quadratic model of air resistance applies.
(i) Taking the origin O at the point of projection and the x-axis pointing vertically upwards, draw a force diagram showing all the forces acting on the pebble, and express each force in vector form. 
(ii) By using the formula a = vdvdx for the acceleration a and the position x of the pebble as a function of its speed v, find the maximum height of the pebble that the pebble can attain in its upward motion. 
(iii) Take a new origin at the highest point and a new x-axis pointing vertically downwards. Draw a force diagram showing all the forces acting on the pebble. Write down the equation of motion of the pebble while it is moving downwards. 
(iv) Use Mathcad worksheet “Clifton Suspension Bridge” to calculate the speed and time of the pebble when it hits the sea. Has the pebble reached its terminal velocity before hitting the water? Is the quadratic air resistance model valid for this problem? 
5
Question 3 (Unit 3) – 25 marks
You may ignore air resistance and other frictional forces in this question.
(a) A block P of mass m is attached to two springs whose other ends are attached to fixed points A and C. The upper spring has a stiffness of 2k and natural length lo, and the lower spring has a stiffness of k and natural length 2lo. The point A is a distance 4lo above C. The magnitude of the acceleration due to gravity is g.
Model the block as a particle and the springs as model springs. Take the origin at A, with the displacement of P from A being x with the x-axis shown.
(i) Draw a force diagram indicating all the forces acting on the particle. 
(ii) Express all the forces in vector form, explaining how you derive the expressions for the spring forces. 
(iii) Determine the equilibrium position xeq of the particle in terms of m, k, lo and g. 
6
(b) A third spring, of natural length 2l0 and stiffness k, is now attached to the particle, and its other end is fixed at a point B which is a distance l0 below A.
Use the same coordinate system as in (a).
(i) Draw a force diagram indicating all the forces acting on the particle. 
(ii) Express in vector form the force due to the additional spring. 
(iii) Show that the equation of motion of the particle is: 
ẍ+4kmx=6klom
(iv) Find the general solution of the differential equation you found in (b)(iii). 
(v) The particle is initially released from rest at a distance 2lo below A. Determine the solution of the differential equation that satisfies these initial conditions. 
(vi) Write down the period and the amplitude of the oscillations of the particle during its subsequent motion. 
(vii) Sketch of the graph of x(t) as a function of time t, clearly identifying the amplitude, starting position and average position. 
7
Question 4 (Unit 4) – 25 marks
(a) Consider the same system as Question 3(b):
Use the coordinate system as shown as the datum for the gravitational potential energy of the particle P. The particle P is now at a distance x below A.
(i) Write down the gravitational potential energy of the particle P at this instant. 
(ii) Write down the kinetic energy of the particle P at the same instant. 
(iii) Determine the potential energy stored in each spring at the same instant. 
(iv) Write down an equation representing the conservation of mechanical energy for the system. By differentiating this equation with respect to time t, verify that your answer is equivalent to the equation of motion derived in Question 3. 
(v) Determine the lowest and highest points of its motion, and hence write down the amplitude of the motion. 
8
(b) A block of mass m initially at rest slides down from a 30o inclined plane. After sliding a distance d, the block encounters a spring with spring constant k and natural length L as shown in the following figure. The block can be modelled as a particle, and the plane can be assumed to be frictionless. The magnitude of the acceleration due to gravity is g.
(i) Take the datum for the gravitational potential energy of the block as shown. After it slides a distance x (where x>d) down the plane, show that the total mechanical energy E(x) of the block is given by: 
E(x)=−12mgx+12mẋ2+12k(x−d)2
(ii) If d=L and k=2mgL, use (b)(i) to find the maximum compression of the spring in terms of L, when the block comes instantaneously to rest. 
(iii) In reality, there is always frictional force in the plane, what is its effect on the maximum compression of the spring as obtained in (b)(ii)? Give reason(s). 
(iv) Sketch a graph showing the relationship of the total mechanical energy E(x), the gravitational potential energy Ug, the spring potential energy Us, and the kinetic energy KE of the block with the distance travelled x with and without the effect of the frictional force. 

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