Notes for motion in a plane chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.
How to Locate Points in Space?
Positions in space are designated relative to coordinate systems. The Cartesian coordinate system is a particularly convenient coordinate system in which position are designated by distances (x, y, z) along three perpendicular axes that intersect at a point called the originIn a polar coordinate system positions in a plane are designated by a length r from the origin, and an angle q usually measured from the positive x - axis. From simple trigonometry we see that the relationships between the polar coordinates and the Cartesian coordinates are
A frame of reference is another name for the particular coordinate system with respect to which we are making observations of physical phenomena.
A scalar quantity requires only a number for its complete description. Mass, volume, density, pressure and temperature are all examples of scalar quantities. The mathematics of scalar quantities is the ordinary algebra of numbers.
Vector quantities requireÂ both magnitude and direction for its complete description. Velocity, acceleration, force and momentum are the examples of vector quantities. A vector can be represented graphically by a directed line segment. The length of the line segment represents the vector's magnitude and its angle with respect to some coordinate system specifies its direction.
We will represent vectors byÂ bold face type letters, with an arrow over the letter such as . When written by hand the same representation may be used.
The magnitude of a vector will be represented by italic type letters such as a, b, c etc.
If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space.
If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space.
The negative of a vector is defined as a vector having the same magnitude as the original vector but the opposite direction.
When two vectors Â and Â have opposite directions, whether their magnitudes are the same or not, we say that they are antiparallel.
Vector Algebra
Addition of Vectors
(i) Geometrical Method
Two vectors Â Â and Â may be added geometrically by drawing them to a common scale and placing then head to tail. The vector connecting the tail of the first to the head of the second is the sum vector Â .
Vector addition is commutative and obeys the associative law.
(ii) Analytical MethodÂ (Parallelogram law of vector addition)
If the two vectors Â and are given such that the angle between them is . The magnitude of the resultant vector of their vector addition is given by
and its direction is given by angleÂ Â with
vectorÂ
It is a common error to conclude that if Â the magnitude ofÂ Â should be just equal to the magnitude ofÂ Â plus the magnitude ofÂ Â In general, the conclusion is wrong; one can see that c < a + b. The magnitude of the vector sumÂ Â depends on the magnitudes ofÂ Â and ofÂ Â and on the angle betweenÂ andÂ Â Only in the special case in whichÂ andÂ Â are parallel; the magnitude ofÂ Â equal to the sum of the magnitudes ofÂ andÂ Â By contrast, when the vectors are anti-parallel the magnitude ofÂ Â equals the difference of the magnitudes ofÂ andÂ .
Subtraction of Vectors
To subtractÂ fromÂ , reverse the direction ofÂ to get - ; then add - toÂ
Resolution of a Vector
The (scalar) components, a_{x} and a_{y} of any two dimensional vectorÂ Â are found by dropping perpendicular lines from the ends ofÂ Â onto
the coordinate axes. The components are given by
andÂ
whereÂ Â is the angle from the positive direction of the x axis to the direction ofÂ Â The algebraic sign of a component indicates its direction along the associated axis.
Given the components, we can reconstruct the vector from
andÂ
Application 1
Find the components of the vectors and as shown in the figure, if a = 2 unitsÂ and b = 3 units.
Solution
units
Â units
Â units
Â units
Application 2
Find the magnitude and direction of the resultant vector Â where Â and are vectors shown in figure.
Solution
Â units
Â units
= 4 units
units
Note
There is one slight complication in using equationÂ Â to findÂ SupposeÂ m andÂ m; thenÂ Â But there are two angles having tangents of -1, namely 135Â° and 315Â° (or -45Â°).Â In general, any two angles that differ by 180Â° have the same tangent. To decide which is correct, we have to look at the individual components. Because a_{x} is positive and a_{y} is negative, the angle must be in the fourth quadrant; thus = 315Â° (or -45Â°) is the correct value. But if instead we have a_{x} = -2 m and a_{y} = 2 m, then the correct angle is 135Â°. Similarly, when a_{x} and a_{y} are both negative, the tangent is positive, but the angle is in the third quadrant. One should always draw a sketch to check which of the two possibilities is the correct one.
Unit Vectors
A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point, that is, to describe a direction in space. Unit vectors provide a convenient notation for many expressions involving components of vectors
In an x-y coordinate system we can define a unit vector Â that points in the direction of the positiveÂ x - axis and a unit vectorÂ Â that points in the direction of the positive y-axis. Then we can express the relationship between component vectors and components as
The vectorÂ Â can be written in terms of its components as
When two vectorsÂ andÂ Â are represented in terms of their components, we can express the vector sumÂ Â using unit vector as follows
=
In three dimensional coordinate system a vectorÂ Â may be expressed as
whereÂ Â is the unit vector along the z-axis.
Application 3
IfÂ andÂ
(a) Find the components of
(b) Find the magnitude ofÂ Â and its angle with x-axis.
Solution
(a)Â
ThusÂ ;
(b)
Multiplication of Vectors
(i) Multiplication of a vector by a scalar
The product of a scalar b and a vectorÂ Â is a new vector whose magnitude is bÂ Â and whose direction is the same as that of if b is positive, and opposite to that ofÂ Â if b is negative. To divideÂ Â by b, multiplyÂ byÂ (1/b).
(ii) Multiplication of a vector by another vector
(a) Scalar or Dot Product
The scalar (or dot) product of two vectorsÂ andÂ Â is writtenÂ Â and is the scalar quantity c given by
In whichÂ Â is the angle between the directions ofÂ andÂ .Â The scalar product may be positive, zero, or negative, depending on the value of . A scalar product is the product of magnitude of one vector and the component of the second vector along the direction of the first vector.
Some properties of Dot product
1) It is commutative, i.e.Â =Â
2) It is distributive over addition and subtraction i.e.Â
3) IfÂ andÂ areÂ perpendicularÂ = 0.
4)Â IfÂ andÂ Â are parallel then =a.b
5) Square of a vector is defined asÂ = a^{2}Â (scalar) and notÂ
6)Â In unit-vector notation we have,Â
Thus,Â
Application 4
If ; Find
Solution
(b) Vector or Cross product
The vector (or cross) product of two vectors and is written as and is a vector whose magnitude c is given by
In which is the smaller of the angles between the direction of and . the direction of is perpendicular to the plane defined by and and is givenÂ by right hand rule.
=
and
where is the unit vector.
The vector Â is directed perpendicular to the plane formed byÂ and Â The direction of vectorÂ Â may be obtained by using the Right Hand Thumb Rule. Stretch all the fingers and thumb of your right hand such that they are perpendicular to each other.Â Now alignÂ your hand such that its plane is perpendicular to the plane formed by vectorsÂ and . Align the stretched fingers along the direction of the vector written first in order i.e., in this case . Curl the fingers of your hand towards the second vector through the smaller angle. Then, the direction of the thumb gives the direction of the cross product.
Properties of Cross Product
1) It is not commutativeÂ .Â In factÂ
2)Â It is distributive over addition and subtraction
3) IfÂ , thenÂ and ifÂ ,Â
4) IfÂ ,Â ,Â Â be the unit vectors along in the positive directions of x, y, and z axes thenÂ
;Â ;Â Â (maintain cyclic order)
5)Â The cross product may also be expressed by the determinant
where
andÂ
Application 5
If ; Find
Solution
SinceÂ ;Â
;Â
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