Posted: November 27th, 2015

Mathematics

Question 1

Find the general solution of the differential equation

Question 2

The points A and B have coordinates (3,1,2) and (0,-1,2), respectively, with respect to a three-dimensional Cartesian coordinate system with origin O. The vectors OA-? and OB–? are denoted by a and b, respectively.

In this question you should express all vectors in the form a1i + a2j + a3k, where i, j and k are Cartesian unit vectors.

Calculate the cross product of a and b.

Question 3

Two blocks, of masses m1 and m2, are connected by a model string, as shown below. Both blocks lie on a plane that is inclined to the horizontal at an angle a. The part of the plane supporting mass m1 is smooth, so that there is no frictional force. The upper part of the plane is rough, and the coefficient of static friction between the block of mass m2 and the plane is µ. The system is in equilibrium, and the string is taut.

This question follows Procedure 2 for solving statics problems, on page 129 of Unit 2, and the steps of this procedure appear in the parts of the question.

(a) “Choose axes

Draw a diagram, marking clearly your choice of coordinate axes.

(b) “Draw force diagrams

Modelling the blocks as particles, draw two force diagrams showing all

the forces acting on the two particles.

(c) “Apply laws

Apply appropriate laws to find two vector equations, one scalar

equation and one inequality.

(d) “Solve equations

Assuming that the system remains in equilibrium, show that

(m1 + m2)tana = µm2.

(e) “Interpret solution

If the mass of the first block is m1 = 2kg, the coefficient of friction for the second block is and tan , for what range of m2 will

the system remain in equilibrium?

Question 4

A heavy pole of mass M and length L is freely hinged to a vertical wall at a point O. The weight of the pole is supported by a wire attached to the end, which goes over a pulley and is connected to a particle of mass m. The wire makes an angle ? with the horizontal direction, as shown in the following diagram.

Model the wire as a model string and the pulley as a model pulley. Three forces act on the pole: the weight of the pole W, the tension force due to the wire T, and a force P due to the hinge. These forces acting on the pole are shown in the force diagram below. Choose horizontal and vertical unit vectors and an origin at O, as shown in the diagram.

Assume that the pole is in equilibrium. By taking torques about O, calculate the angle ? of the wire in terms of the parameters given above.

Question 5

A stone of mass m is thrown vertically upwards with speed u, and travels upwards under the influence of gravity and air resistance. Use the quadratic model of air resistance with the stone modelled as a sphere of effective diameter D.

This question follows Procedure 1 on page 197 of Unit 3, and the steps of this procedure appear in the parts of the question.

(a) “Draw picture

Draw a picture and mark on it any relevant information.

(b) “Choose axes

Choose an axis for this problem.

(c) “State assumptions

State any assumptions made.

(d) “Draw force diagram

Draw a force diagram.

(e) “Apply Newton’s second law!

Apply Newton’s second law to obtain the equation

,

where v is the speed of the stone, x is the distance travelled by the stone, g is the magnitude of the acceleration due to gravity, and

k = c2D2/m is a constant.

(f) “Solve differential equation!

Solve the differential equation and apply the initial condition to find

the time t in terms of the speed v and the constants given above.

(g) “Interpret solution!

Use your equation to show that the time take to reach the maximum height attained by the stone, tmax, is given by

.

This model predicts that as the initial velocity u increases, there is an upper limit for the time to reach maximum height. Calculate this upper limit for a beach ball with diameter 0.5m and mass 0.1kg (use

c2 = 0.2 and g = 9.81).

Question 6

A block of mass m is moving on a rough slope with coefficient of sliding friction µ!. Initially the block is moving up the slope with speed u.

(a) Draw a force diagram showing all the forces acting on the block.

(b) Choose axes with i pointing up the slope and j perpendicular to the slope, as shown in the following diagram.

Write the forces acting on the block in terms of the unit vectors i

and j.

(c) When Newton’s second law is applied to this mechanical system, it is found that the acceleration of the block up the slope, a, is given by a = -g(sina + µ! cosa). Use this result to show that the distance x travelled by the block until it comes to rest is given by

u2

x = .

2g(sina + µ! cosa)

(d) This result is to be used in the design of a ramp that will slow down blocks that enter at a fixed speed. You are given the following data.

Length of ramp, x 0.75m

Initial speed, u 3ms-1

Acceleration due to gravity, g 9.81ms-2

Coefficient of sliding friction,

Using Maxima, or otherwise, find to the nearest degree the angle a that the slope needs to have to slow the block to rest at the top of the slope. Include in your answer your input to Maxima, the output that Maxima produced, and your interpretation of this output. [3]

(Hint: Use the Maxima command find_root to solve the equation in part (c). As an example of usage, the Maxima expression find_root(x^2=2,x,1,2) will find the value of x between 1 and 2 for which x2 = 2; that is, it will return an approximation to

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