Advanced mathematics 2

Frontmatter

1. Graphical and numerical methods

Abstract

When the variables x and y satisfy an equation of the form y = mx + c, it can be said that there is a linear relation between them. The graph of y against x will then be a straight line with gradient m. The constant c gives the value of y at the point at which the line crosses the y-axis, i.e. c is the intercept of the line on the y-axis.

C. W. Celia, A. T. F. Nice, K. F. Elliott

2. Geometry in three dimensions

Abstract

Consider the straight line which is parallel to the vector b and which passes through the point A whose position vector is a. Any vector parallel to the vector b can be expressed in the form tb, where t is a scalar, i.e. a real number. If P is any point on this straight line, the vector

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C. W. Celia, A. T. F. Nice, K. F. Elliott

3. Kinematics

Abstract

Consider a particle moving along a straight line BOA (Fig. 3.1). The position of the particle is given by its distance from O, distances to the right of O being taken to be positive and distances to the left of O being taken to be negative. The particle is at the point P, where OP = s, at time t and at the point Q, where OQ = s + δs at time t + δt. Thus, in moving from P to Q, the particle travels a distance δs in time δt. The average velocity of the particle over this interval of time δt is

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C. W. Celia, A. T. F. Nice, K. F. Elliott

4. Statics of particles

Abstract

In Chapter 3 the motion of a point in two dimensions was discussed without any consideration being given to the means by which this motion was produced. The next two chapters introduce ‘Newtonian mechanics’ i.e. a study of the physical world as governed by Newton’s Laws of Motion, formulated by Isaac Newton in the seventeenth century.

C. W. Celia, A. T. F. Nice, K. F. Elliott

5. Dynamics of a particle

Abstract

In Chapter 3 the motion of a particle was dealt with without considering how that motion was produced. In Chapter 4 the forces acting on a particle were considered. Now the forces acting on a particle and the motion they produce will be considered.

C. W. Celia, A. T. F. Nice, K. F. Elliott

6. Work, power and energy

Abstract

When a force moves its point of application it is said to do work. The work done by a constant force is defined as the product of the magnitude of the force and the distance moved by the point of application in the direction of the force.

C. W. Celia, A. T. F. Nice, K. F. Elliott

7. Moments

Abstract

The effect of a force on a particle can be calculated when the mass of the particle and the magnitude and direction of the force are known. In order to find the effect of a force on a rigid body, it is essential to know at which point of the body the force is applied.

C. W. Celia, A. T. F. Nice, K. F. Elliott

8. Equilibrium of a rigid body

Abstract

When a rigid body is in equilibrium suspended by a string, it is acted on by two forces only. The tension in the string acts vertically upwards and the weight of the body acts vertically downwards. These two forces must be equal in magnitude and they must act along the same line, or else they would form a couple and rotate the body. This shows that the centre of gravity of the body must be in the same vertical line as the point of suspension.

C. W. Celia, A. T. F. Nice, K. F. Elliott

9. Momentum and impulse

Abstract

When a particle of mass m is moving with velocity u, the product mu is known as the momentum of the particle, or more precisely, as its linear momentum. This is a vector in the same direction as u, along a straight line passing through the particle. Momentum is not a free vector, but is a localised vector. Since mass is measured in kg and velocity in m s⁻¹, we measure momentum in kg m s⁻¹.

C. W. Celia, A. T. F. Nice, K. F. Elliott

10. Motion in a circle and simple harmonic motion

Abstract

Let the point P be moving in a circle of radius r with centre at the origin O. At time t let the radius OP make an angle θ with the x-axis. Then the angular velocity ω of P about O is given by

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C. W. Celia, A. T. F. Nice, K. F. Elliott

11. Probability

Abstract

Suppose that three playing cards, an ace, a king and a queen are to be placed in line. The first place can be filled in three ways, with the ace or the king or the queen. For each of these three ways, the second place can be filled in two ways, and then the last card takes the third place. The cards can therefore be arranged in 3 × 2 × 1 ways, and these six ways are shown in Fig. 11.1.

C. W. Celia, A. T. F. Nice, K. F. Elliott

Backmatter