1. For Scenario A, produce the following mathematical models for this system to allow its response to a range of rumblings to be studied in simulations:
a) In order to prevent movement during construction, the building is erected standing on steel supports that will be removed suddenly when the concrete has set. In these circumstances, at and shortly after the moment of release the rubber acts mainly as a damper and the value of k can be neglected. Develop a differential equation for the velocity v of the building modelling the behaviour of the building immediately after release.
b) Produce a differential equation for the behaviour of the building after construction linking its absolute position (i.e. its position relative to its original position) in the event that the ground moves according to some function F(t). Hence produce a Laplace transfer function linking the absolute position of the building (output) to the absolute position of the ground (input).
Scenario A
You are part of a team that is being consulted over the design of a new concert hall over an underground railway (tube) line. Your civil engineering colleagues have completed an outline design in which the main body of the office block is to be a reinforced concrete building. In an attempt to provide some barrier transmitting underground rumbling into the concert space, the building is to stand on rubber buffers (see below) sited in a concrete basement cast into the local bedrock. The building is restrained from moving horizontally because of its proximity to nearby buildings and has freedom to move in the vertical plane only.
Rubber has internal properties that combine springiness and a measure of damping so for all practical purposes the building can be modelled as a fixed mass M supported by a spring system of combined stiffness k and subject to a damping force equal to a constant c times its instantaneous velocity. Variations in the composition of the rubber can vary the effective values of k and c over a very wide range.
The underground rumblings involve small movements of thousands of tons of rock, so the mass of the building has virtually no effect on the displacement of its base. Consequently, in the event of a train passing, the base of the rubber mount can be assumed to move a distance x which is a function of time. Please note that this is not the same as subjecting the base to a force which is a function of time.

1. An automobile traveling at 25 km/hour has kinetic energy equal to 1 x 104J.
a. What is the mass of the car?
b. If the velocity of the car is increased to 45 km/hour, what is its kinetic energy?
c. If the velocity is increased to 80 km/hour, what is its kinetic energy?
2. A rock with a mass of 1.0 kg is at the top of a hill and has a potential energy of 50 J.
a. How high is the hill?
b. If the rock rolls to the bottom of the hill, what is its kinetic energy at the bottom of the hill?
3. A certain pendulum has a potential energy of 20 J when it is at the top of its swing.
a. What is the kinetic energy?
b. When the pendulum reaches a certain point in its downward motion it has a potential energy of 5 J. What is its kinetic energy?
c. At the bottom of the swing its potential energy is zero. What is its kinetic energy?

a) What is a Binary Eutectic System? Explain in detail and give an example eutectic reaction.
b) What are the eutectoid and the peritectic reactions? Explain and give example reaction for each one.
c)
(12 points: 4 pts each)
Consider 4.0 kg of a 99.55 wt % Fe – 0.45 wt % C alloy that is cooled to a temperature just below the eutectoid.
Show your calculation details in your answer and draw lines on the phase diagram to label the phases and compositions.
a) How many kilograms of proeutectoid ferrite form?
b) How many kilograms of eutectoid ferrite form?
c) How many kilograms of cementite form?
(10 points: 2 pts each)
State whether the following questions are True or False.
Type your answer (True or False) inside the box below for each question:
a) Ductility is a solid material’s ability to deform under tensile stress; this is often characterized by the material’s ability to be stretched into a wire.
Answer:
b) The ratio of the shear stress to the shear strain in the elastic range of torsion test is known as the Young’s Modulus or Modulus of Elasticity, E.
Answer:
c) The Engineering Stress, σ is defined as the ratio of the applied load to the instantaneous cross-sectional area of the test specimen.
Answer:
d) The area under the true stress–true strain curve up to fracture is known as the material’s toughness.
Answer:
e) The melting point of a solid is the temperature at which it changes state from liquid to gas at atmospheric pressure.
Answer: