ADVANCED LEVEL MATHEMATICS
MECHANICS AND PROBABILITY
Why do you hate applied mathematics? Is it because you did not study its basics in additional mathematics at the ordinary level? Now calm down! You are where everything will be chewed for you.
All you need is to open your mouth and swallow. I guest this
sound like joke but I must confess that, it is manner falling from your creator
in real time. You will change your grade from E to A in mathematics within a short space of
time. Are you ready to work? Okay let’s go!
This part of mathematics is meant for
students who choose to study applied mathematics at this level. To ease work, vector is introduced at an
early stage throughout this subject and exercises solved using vectors whenever
the need arises.
Each topic treated is followed by many
solved examples to enhance understanding and also motivate the learner to go
for more challenging questions. A selection of more challenging questions and
answers are given at the end of each topic. You will also have some multiple
choice questions on selected topics.
The objectives of each topic are clearly stated to give learners a clear and precise route map for more research. Diagrams are used in most cases to simplify a concept.
You should note that it is difficult to solve a problem correctly in math mechanics without the use of a diagram. It is advisable to always use diagrams to answer a question. Most of the questions here are from Cameroon General Certificate of Education Board (CGCE) and the University of London Examination Board (U of L).
A few questions
are taken from other examination boards like University of Cambridge Local
Examinations Syndicate(C), Joint Matriculations Board (JMB), Welsh joint
Education Committee (WJEC), etc, to help students meet up with familiar
questions and welcome new challenges if they have to study out of the country.
Below is an outline of the content of what we
shall be working on in the next 6 weeks. Hope you are up to the task. On your
mark goooo!….
CONTENT
1) VECTORS , SYSTEM OF FORCES AND RESOLUTION.
Ø Topic’s objectives.
Ø Resultant of vectors.
Ø Vector representation.
Ø Unit vectors.
Ø Addition of vectors.
Ø The magnitude of a vector.
Ø Resolving vectors.
Ø Scalar product.
Ø Scalar product from vectors
in component form.
Ø Object on a table.
Ø Object on inclined planes.
Ø Horizontal force applied
on objects on inclined plane.
Ø Rough planes.
Ø Forces acting on a ladder
leaning against a wall and a floor.
Ø Horizontal rod suspended
by strings.
Ø String and a particle.
Ø Horizontal rod smoothly
hinged into a wall.
Ø Solved exercises.
Ø Summary.
Ø Examination questions.
Ø Multiple choice questions.
2)
FORCES IN EQUILLIBRIUM.
Ø Topic’s objectives.
Ø Equilibrant.
Ø Three forces in
equilibrium.
Ø Friction.
Ø Angle of friction.
Ø Equilibrium on inclined planes.
Ø Moment.
Ø Couple.
Ø Effect f coplanar forces.
Ø General conditions for
equilibrium.
Ø Solved exercises on the
above.
Ø Summary.
Ø Examination questions and
answers.
Ø Multiple choice questions.
3)
VELOCITY AND ACCELERATION
>Topic’s objectives.
>Motion in a straight line.
>Equation of motion with constant acceleration.
>Distant –time graph.
>Velocity-time graph.
>Vertical Motion Under gravity.
>Solved exercises on the above.
>Summary.
>Examination questions and answers for practice.
>
Multiple choice questions
4)
NEWTON’S LAWS OF MOTION.
>Topic’s objectives.
>Newton’s first law and applications.
>New second law and applications.
>Weight and mass.
>Newton’s third law.
>Motion of connected particles.
>Solved exercises on the above.
>Summary.
>Examination questions and answers for practice.
>Multiple Choice questions.
5) MOMENTUM: DIRECT IMPACT.
>Topic’s objectives.
>Momentum.
>Impulse.
>Law of conservation of Momentum.
>Direct
Impact.
>Impulse and loss of kinetic
energy.
>Solved exercises on the above.
>Recoil of a gun.
>Summary.
>Examination questions and
answers for practice.
>Multiple Choice questions.
6)
PROJECTILES.
>Topic’s objectives.
>Velocity and speed at a given time.
>Velocity and speed at a given point.
>Distance at a given time.
>Equation of path (trajectory) of a particle.
>Parabolic flights feature.
>Time of flight.
>Horizontal range.
>Maximum horizontal range.
>Greatest height.
>Solved exercises on the above.
>Summary.
>Examination question and answers for practice.
>Multiple choice questions.
7)
WORK ENERGY AND POWER.
>Topic’s objectives.
>Work done by a constant force.
>Work done against gravity.
>Work done against and friction.
>ENERGY.
>The principle of conservation of energy.
>Equivalence of work and energy.
>Work done by force acting in a direction different from
that of the motion of the particle.
>POWER.
Pump raising and ejecting water.
>Vehicles in motion.
>Solved Exercises.
>Summary.
>Examination questions and answers for practice.
>Multiple choice questions.
8)
ELASTICITY
>Topic’s objectives.
>Elastic strings.
>Hooke’s law.
>Energy store in elastic string.
>Conservation of energy for elastic string.
>Potential energy of an elastic spring.
>Gravitational potential energy.
>Elastic potential energy.
>Energy and power.
>Energy equation.
>Elastic impact.
>Impact with the surface.
>Solved exercises.
>Summary.
>Examination questions and answers for practice.
>Multiple choice questions.
9)
RELATION MOTION.
Ø Topic’s objective.
Ø Resultant velocity.
Ø Crossing a river.
Ø Relative velocity.
Ø Solved exercises.
Ø Summary.
Ø Examination questions.
Ø Multiple choice questions
1O) APPLICATION OF CALCULUS.
Ø Topic’s objective.
Ø Straight line motion.
Ø Calculating velocity and
distance when acceleration is given as a function of velocity.
Ø Calculating velocity and
distance if acceleration is given is given as a function of displacement.
Ø Calculating acceleration
given velocity as a function of displacement
Ø Variable force.
Ø Forces in terms of I,j,and
k.
Ø Opposing Medium.
Ø Impulse of a variable
force.
Ø Work done by a variable
force.
Ø Further applications.
Ø Vector applications.
Ø Collision of moving
particles.
Ø Minimum distance.
Ø Solved exercises.
Ø Summary.
Ø Examination questions.
Ø Multiple Choice Questions.
11)CIRCULAR MOTION.
> Topic’s
objectives.
>The radian.
>Angular velocity.
>Relationship between linear and angular velocity.
>Constant acceleration.
>Angular- velocity graph.
>Uniform motion in a circle with constant speed.
>Conical pendulum.
>Circular motion in a spherical bowl.
>Solved exercises.
>Summary.
>Examination questions.
>Multiple Choice questions.
12)
CENTER OF GRAVITY
>Topic’s Objectives.
>Centre of gravity of a finite
number of particles.
>Centre of mass.
>Centre of gravity of some two
dimensional bodies.
>Centre of gravity of a compound body.
>Suspended lamina.
>Toppling.
>Use of calculus to find centre of
gravity.
>Solved exercises.
>Summary.
>Examination questions.
>Multiple Choice questions.
13) PROBABILITY
> Topic’s Objectives.
>Definition of terms.
>Classical definition of probability.
>P(AnB).P(AuB).
>Mutually exclusive events.
>Conditional probability.
>Independent events.
>Selection with or without
replacement.
>Probability tree.
>The “at least” and “at most”
situations.
>Use of an infinite geometric series
in probability.
>Solved exercises.
>Summary.
>Examination questions.
> Multiple choice questions.
These are the topics you
will learn. They may not be thought following the order in which they appear
above. You are free to ask questions where you do not understand. You can share
this website with your classmates to ease study. Use your android phones. The
side is compatible with all smart phones. You may use your laptop or desktop to
study solutions to problems presented here.
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