## MAS112 Vectors and Mechanics

 Both semesters, 2021/22 20 Credits Lecturer: Dr Rekha Jain Timetable Reading List Aims Outcomes Assessment Full Syllabus

The module begins with the algebra of vectors, essential for the study of many branches of applied mathematics. The theory is illustrated by many examples, with emphasis on geometry including lines and planes. Vectors are then used to define the velocity and acceleration of a moving particle, thus leading to an introduction to Newtonian particle mechanics. Newton's laws are applied to particle models in areas such as sport, rides at theme parks and oscillation theory.

Prerequisites: A-level mathematics (or equivalent)
Corequisites: MAS110 (Mathematics Core I); MAS111 (Mathematics Core II)

The following modules have this module as a prerequisite:

 MAS280 Mechanics and Fluids MAS314 Introduction to Relativity MAS315 Waves MAS324 Quantum Theory MAS413 Analytical Dynamics and Classical Field Theory

## Aims

• Introduce students to applied mathematics through the theory and application of vectors;
• To develop the students' knowledge of mathematical modelling by applying Newton's laws to particle models in areas such as sport, rides at theme parks and oscillation theory.

## Learning outcomes

• Demonstrate knowledge of vector algebra up to (and including) triple products;
• Apply vector methods to simple problems in 2D and 3D geometry;
• Understand the formulation of simple mathematical models and their limitations;
• Obtain the velocity and acceleration of a particle, given its position vector as a function of time;
• Demonstrate knowledge of relative motion and motion under a constant acceleration, including projectiles;
• Understand motion in a circle with uniform angular speed;
• Calculate work done by a force, kinetic energy, power and use these quantities in solving problems;
• Investigate the motion of a particle moving in a vertical circle;
• Use their knowledge to model applications of circular motion;
• Solve differential equations including linear first order and second order with constant coefficients and interpret these solutions;
• Use and apply results from mathematical models based on differential equations.

40 lectures, 20 tutorials

## Assessment

Semester 1 coursework (10%) Semester 2 coursework (10%)

One formal 2 hour written examination at the end of semester 2. All questions compulsory (80%).

## Full syllabus

Vector geometry
Vectors as displacements of space. Addition, subtraction, multiplication by a scalar. Position vector. Cartesian basis, co-ordinates. Scalar product. Vector product. Triple products. Applications throughout geometry especially lines and planes.

Kinematics
The path of a particle given its position vector as a function of time. Differentiation of vectors with respect to a scalar; velocity, acceleration. Motion in a circle with constant speed. Relative motion.
Motion with constant acceleration
Motion in a straight line. Motion near the Earth's surface under gravity. Projectiles (no air resistance). Examples from sport.
Newton's laws
Force, momentum. Newton's laws of motion. Newton's law of gravitation, gravitational acceleration. Planetary orbits, Kepler's third law. Impulse, conservation of momentum. Types of force. Resistance proportional to speed : one-dimensional case.
Circular motion
Kinematics of circular motion. Conical pendulum. Normal contact force. Friction. Penny on turntable. Penny on turntable with banking. Vertical circular motion.
Work and energy
Kinetic energy. Work. Work-energy equation. Gravitational potential energy. Conservation of mechanical energy (KE+PE). Power. Hooke's law. Elastic potential energy (EPE). KE + PE + EPE = constant.
Oscillations
Oscillations : horizontal spring, simple harmonic motion, amplitude, frequency. Vertical spring. Damped oscillations: damping factor, weak, strong and critical damping. Forced oscillations, resonance.

Type Author(s) Title Library Blackwells Amazon
B Hirst Vectors in two or three dimensions 516.182 (H) Blackwells Amazon
C Atkin Mechanics 531 (M) Blackwells Amazon
C Burghes Further mechanics 531 (B) Blackwells Amazon
C Collinson and Roper Particle mechanics 531.16 (C) Blackwells Amazon
C Dyke and Whitworth Guide to mechanics 531 (D) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.