Applied Mathematics B Study Guide

Transcription

1 Science, Engineering and Technology Portfolio School of Life and Physical Sciences Foundation Studies (Applied Science/Engineering) Applied Mathematics B Study Guide Topics Kinematics Dynamics Work, Energy & Power Differential Equation Applications Momentum Vector Calculus and Projectile Motion Circular Motion and Simple Harmonic Motion Contents Topic 1: Kinematics Uniformly Accelerated (i) displacement, (ii) velocity, average velocity, (iii) acceleration, (iv) kinematic formulae, State the kinematic formulae under constant acceleration. Describe position, velocity, average velocity and acceleration and state their units. Solve kinematic problems incorporating the formulae along the horizontal. (iv) Exercise 1 (v) vertical motion under gravity Solve kinematic problems dealing with vertical motion due to gravity. Graphical Representation: (i) displacement-time graphs, (ii) velocity-time graphs, (iii) area-displacement, (iv) gradient -acceleration Differentiation: (i) position relative to a fixed point, (ii) velocity as a rate of change, (iii) acceleration as a rate of change Investigate drawing displacement-time graphs. Describe the key features of a velocity-time graph. Solve kinematic problems incorporating a velocity-time graph. Describe velocity and acceleration in terms of rates of change with respect to time. Define position relative to a fixed point. Determine the velocity and acceleration using differentiation. (iii) Exercise 3 (iv) Exercise 3 (iii) Exercise 4

2 Topic 2: Dynamics (i) Newton's laws of motion (ii) Types of forces (iii) Constant acceleration (iv) Resultant force Define Newton's law of motion. Determine the types of forces. Find the resultant force along a plane and inclined planes. Resolve forces on smooth surfaces. (i) Connected masses Solve dynamic problems involving connected masses on smooth surfaces. Define the coefficient of friction. (i) Frictional surfaces Determine the resultant force along (ii) Coefficient of friction rough planes and inclined planes. Solve dynamic problems on frictional surfaces. (i) Connected masses on frictional surfaces Solve dynamic problems involving connected masses on frictional surfaces. Topic 3: Work, Energy and Power (i) Definition of work Define work done. Describe done gravitational potential energy and how it (ii) Work done by a relates to work done. Solve problems constant force (iii) Potential energy dealing with work done by a constant force and potential energy. (i) Kinematic energy (ii) Work energy equation (i) Conservation of energy (ii) Potential and kinetic energy in a gravitational field (i) Power (ii) Constant force Define kinetic energy. Describe the work-energy equation. Solve problems involving the work-energy equation. Describe the conservation of energy principle. Solve problems dealing with the conservation of energy in a gravitational field Define power. Solve problems concerning power under a constant force.

3 Topic 4: Differential Equations equations - kinematics Outline how problems in kinematics can be represented as solutions to first order differential equations. Solve kinematic problems using first equations. 4 hours equations - growth and decay equations - water flow, - Newton's Law of cooling (i) Applications of second equations - kinematics (i) Applications of second equations - dynamics (i) Application of second equations - beam deflections.... Describe the law of natural decay. Solve first equations related to growth and decay. Describe Newton's Law of Cooling. Set up and solve differential equations which relate to Newton's Law of Cooling and water flow. Describe how acceleration can be represented as a space derivative. Solve differential equations of the first and second order which involve kinematics. Investigate differential equations which incorporate variable force. Solve differential equations involving dynamics. Describe and outline second order differential equations which incorporate beam deflections. Solve differential equations relating to beam deflections. (i) Exercise 5 (i) Exercise 6 Topic 5: Momentum (i) Principle of conservation of momentum (ii) Impulse Describe the principle of conservation of momentum. Define impulse. Solve problems involving impulsive forces. (i) Inelastic collisions (i) Elastic collisions Describe what is meant by an inelastic collision. Solve problems which incorporate the principle of work -energy. Describe an elastic collision. Solve problems which involve elastic collisions.

4 Topic 6: Vector Calculus and Projectile Motion (i) Vector equations Describe a vector equation. Solve (ii) Parametric parametric equations to obtain the equations (iii) Equations of path cartesian equation of path. (i) Position, velocity and acceleration (ii) Differentiation of Define the position, velocity and acceleration. Solve problems which require the differentiation of. Find the maximum and minimum speed from a velocity vector. (i) Integration of Final Examination Solve problems which require the integration of. Projectile Motion (i) cartesian equation of trajectory, (ii) time of flight, (iii) maximum height, (iv) maximum range, (v) angles of projection, Describe the path of projectile motion. Use vector calculus to derive the position and velo city for horizontal and vertical motion. Establish the equation of path. Determine the time of flight, maximum height reached and maximum range of a projectile. Solve problems dealing with projectile motion. Solve projectile motion questions involving determining angles of projection through a given point. (iii) Exercise 4 (iv) Exercise 4 (v) Exercise 4 Topic 7: Circular Motion and Simple Harmonic Motion 1 hour Circle (i) Angular displacement, velocity and acceleration. (ii) period Dynamics of Circular (i) tension in a string, (ii) banked curves, (iii) frictional surfaces, Final examin ation (iv) conical pendulum Simple Harmonic (i) equations, (ii) amplitude, (iii) period (i) Hooke's law (ii) Work done in stretching an elastic string Dynamics of Simple Harmonic (i) horizontal surfaces, (ii) frictional surfaces, (iii) vertical springs Describe angular velocity and centripetal acceleration. Determine the period of circular motion. Solve Circular motion dynamic problems relating to tension in a string, banked curves and frictional surfaces. Investigate uniform circular motion in a Conical pendulum. Solve problems relating to circular motion in a conical pendulum incorporating dynamics. Define the simple harmonic second order differential equation. Derive the velocity and position equations for simple harmonic motion. Solve problems involving simple harmonic motion: the period, amplitude, extreme positions and the velocity through the mean position. Define Hooke's law for a spring or elastic string. Describe elastic potential energy and work done in stretching an elastic string. Solve problems using Hooke's law and finding work done. Solve simple harmonic questions involving dynamics on smooth and rough surfaces. Solve simple harmonic questions involving the dynamics of a vertical spring (iv) Exercise 1 (iii) Exercise 2 (iii) Exercise 3

5 Assessment When Task Topics Duration % of course End of mid-semester Test Work,Energy & Power Kinematics. Dynamics Differential equation Applications 30% During semester Project Differential equation Applications 3-4.weeks 20% End of semester Examination All topics 50%