Second order Differential equations
An equation is termed second order when it contains the second derivative
The solution of a second-order differential equation depends on the type of solution which satisfies its
auxiliary equations. There are three typ
Vectors
Lines in 3-D
In 3-D, lines which are not parallel may or may not meet. Non-parallel lines which do not meet are said
to be skew.
The vector equation of a line
T
h
e
lines with vector equation r=a +sp and r=b + tq have the same direction if p is a
*strings
Tension on a rope always acts towards the center of the rope
i.
A loose string is said to be slack. There is no tension acting on a slack rope.
ii.
When a rope is being pulled there is tension in the rope
*The difference between a rope and a rode
Momentum and impulse
Momentum
The moment of a body of mass m having a velocity of v is mv. If the units of mass and velocity are kg and
ms-1 respectively, then the units for momentum are newton-seconds (Ns)
Changes in momentum- if the velocity of a body c
Matrices and linear space
Identity matrices
An identity matrix is any matrix, all of whose elements in the leading diagonal are 1, and all of the other
elements are zeros.
1) When you multiply I by any matrix of the same order, I behaves as unity; IM=MI=M
Polar coordinates
Examples
1
Sketching curves given in polar coordinates
The normal way to sketch a curve expressed in polar coordinates is to plot points roughly using simple
values of .
Tips for sketching curves
1. Look for any symmetry
i.
If r is a fun
Roots of polynomial equations
Quadratic equations
If and are roots of a quadratic equation ax2 + bx + c =0 where a0
The equation can be written as (x - )(x - )
Therefore we have;
X2 x x + = 0 which is the same as x2 x ( + ) + = 0
2
This can be written as
Differentiation and integration
Implicit differentiation
1
Reduction formulae
We need reduction formulae to facilitate the integration of functions whose integrals cannot otherwise
be found directly
2
We usually obtain a reduction formula by changing the
Matrices and linear space (1st)
Linear vector space
Let V be a set on which two operations, addition and scalar multiplication, have been defined. If u and v
are in V, the sum of u and v is denoted by u + v, and if c is a scalar, the scalar multiple of u
Simple harmonic motion
The acceleration is always towards the origin, O
Since the acceleration is directed towards O, the particles speed will decrease as it approaches
P. if the particle reaches P, the acceleration will cause the particle to travel back
Circular motion
Linear and angular speed
*the linear speed of a body, i.e. its speed in a straight line, is usually measured in ms -1 or similar units
When a body is moving on a circular path, its often useful to measure its speed as the rate of change of
Rotation of a rigid body
If a rigid body is rotated about a fixed axis through o with angular speed rads s-1, the kinetic energy of
the body is given by:
K.E=
1
2
m r2
Where I=
I 2
and is the moment of inertia
Moment of inertia
Moment of Inertia (Mass Mo
Equilibrium of a rigid body under coplanar forces
Centre of mass
The centre of mass of an object is where all the weight of the object appears to act through
For a 2-D object, to describe the position of the centre of mass you need to split the object int
Resolving forces
* remember SOH CAH TOA
*remember F=ma
Resolving horizontally and vertically (*resolving in one way will lose you the forces in the
other direction)
When you write a resolving equation, you should always indicate, either in words or by usi
Determinants
For 2 x 2 matrices
For 3 x 3 matrices
take each element in the first row, cover its column and row, and find the determinant of the 2 x
2 matrix that is lef
notice the signs alternate (+, -, +, -)
Rules for the manipulation of determinants
Ch
Further work on distributions
Probability density function (P.D.F)
A continuous random variable X is given by its probability density function, which is specified for the
range of values for which x is valid. The function can be illustrated by a curve, y=
Inference using normal and t-distribution
Z- Tests
1
2
3
T-tests
This is testing the mean when the population X is normal but the variance 2 is unknown and the
sample size n is small (less than 30)
The t-distribution
The distribution of T is a member of a
Bivariate data
Regression and correlation
Scatter diagrams
Data connecting two variables are known as bivariate data. When pairs
of values are plotted, a scatter diagram is produced.
Dependent and independent variables
If one of the variables has been con
The X2-tests
There are two main situations where an X2 (pronounced kye squared) significance test is used:
An X2 goodness-of-fit-test
This is used when you have some practical data and you want to know how well a particular statistical
distribution, e.g.
1.1a Representations of Function, I
In this lesson, we will review the notion of a function.
Functions are a central object of Calculus. Functions to Calculus are like atoms to chemistry, data to
statistics, or brain to neuroscience. Functions allow us to
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HONORS PHYSICS PROBLEM SET
CIRCULAR MOTION & GRAVITATION
Frequency, Period & Rotational Speed
1. A satellite in a circular orbit has a period of 10 h. What is the satellites frequency in revolutions
per day? (2.4 rev/day)
2. What is the period of revoluti
Renaissance and Reformation Essay Questions
Choose one of the following questions and answer with a 5 paragraph essay.
1. Why was the Church of England different from many other Protestant sects
established during the Reformation? Explain how it was estab
Molarity and Dilutions Worksheet
1. What is the molarity of a solution that has 6 moles of KI in 12L of water?
a. If that solution is diluted to 18L what is the new molarity?
2. What is the molarity of a solution that has 70g of KCl in 1800ml of water?
a.