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# Physics Presentation

## by hassaan usmani on May 17, 2008

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## Physics PresentationPresentation Transcript

• Physics Presentaion
• Members of the Group
• Ammar Maqsood
• Hassaan Ahmed Usmani
• Naeem Nassirudin
• Vector Analysis
• Scalars Vectors
• A scalar quantity is one which can be described fully by just stating its magnitude .
• Some examples are
• Mass
• time
• length
• temperature
• density
• speed
• energy and volume
• A vector quantity is one which can only be fully described if its magnitude and direction stated.
• Some examples are
• displacement
• velocity
• acceleration
• force
• momentum
• magnetic density and electric intensity.
• Why vectors are important?
• Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity , the magnitude of which is speed . For example, the velocity 5 meters per second upward could be represented by the vector (0,5). Another quantity represented by a vector is force , since it has a magnitude and direction.
• Properties of a Vector
• A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A .
• The magnitude of A is |A| ≡ A
• We can represent vectors as geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector
O A |A| ≡ A
• Types of Vector
• Null Vector
• Vector with zero magnitude
• Position Vector
• Vector starting from origin
• Free Vector
• Vector starting from anywhere but origin
• Unit Vector
• Vector with magnitude of 1
• Equal Vectors
• Two vectors equal in magnitude and direction
• Opposite Vectors
• Two vectors equal in magnitude but opposite in direction.
• Laws of vector algebra
• A + B = B + A (Commutative law of addition)
• A +( B + C )=( A + B )+ C (Associative law of addition)
• m A = A m (Commutative law of Multiplication)
• m(n A )=(mn) A (Associative law of Multiplication)
• (m+n) A =m A +n A (Distributive law)
• m( A + B )=m A +m B (Distributive law)
• The addition of two vectors yields another vector known as Resultant vector.
• For example if vector A and vector B are added their sum will be equal to (A+B).
• Methods of Addition of Vectors
• There are 3 methods for addition of Vectors.
• For vectors precisely along X or Y axis.
• Parallelogram law
• Vector P and Q are drawn from same origin.
• Straight lines are drawn parallel to both vectors so as to form a parallelogram.
• The resultant (P+Q) is represented by the diagonal of the parallelogram that passes through the origin.
P Q + P P Q Q +
• Triangle law (head to tail rule)
• The result of two vectors could be determined by drawing a triangle.
• Vector Q and P are drawn in such a way that the tail of vector Q touches head of vector P .
• The resultant (P+Q) is represented by the third side of triangle from tail of P to head of Q .
P Q + P Q P Q +
• Subtraction of vectors
• The subtraction of two vectors can be treated as the addition of a negative vector.
• ( P - Q )= P +(- Q )
• The vector ( P - Q ) can then be determined by any of the two methods.
P Q - P -Q P-Q P P -Q -Q = OR
• Resolution of a vector
• Vector R could be considered to be the resultant of two vectors.
• R = A + B
• Here the vectors A and B are known as the components of vectors.
• Resolution of a vector
• It is useful to find the components of a vector R in two mutually perpendicular directions. This process is known as resolving a vector into components.
• The magnitude of the two components can be written in the form Rcos  and Rsin 
R 0 Rcos0 Rsin0 R  Rcos  Rsin 
• Triangle of forces
• If three forces acting on a point can be represented in magnitude and direction by three sides of a triangle taken in order, then the three forces are in equilibrium.
• The converse is also true:
• Three forces acting on a point are in equilibrium, they can be represented in magnitude and direction by sides of a triangle taken in order.
• Triangle of forces
• By the triangle of vectors, the resultant of P and Q is represented in magnitude and direction by side of OC of the triangle OAC. If third force R is equal in magnitude to (P+Q) but in opposite direction, then the point O is in equilibrium and also R could be represented by the side CO of the triangle.
o P Q R A C O R P Q P+Q A C O P R Q
• Polygon of Forces
• When more than three coplanar forces act on a point , the resultant (or vector sum) of forces can be found by drawing a polygon of forces .
• Polygon of Forces
• If forces acting on a point can be represented in magnitude and direction by sides of a polygon taken in order ,then the forces are in equilibrium.
o S T P Q R Q R S P T A B C D E
• Resultant of a number of forces
• If point O is being acted upon by a number of coplanar forces such as A,B,C,D,E do not form a closed polygon then the forces are not in equilibrium
• The resultant in magnitude and direction is represented by R.
A B C D E A B C D E R o
• When vector is multiplied by another vector in some cases a scalar quantity is obtained whereas in some other cases a vector quantity obtained .There are two types of product.
• Scalar product
• Vector product
• Scalar product or Euclidean inner product
• INTRODUCTION
• WHY SCALAR PRODUCT IS USE
• Geometrically, the scalar product is useful for finding the direction between arbitrary vectors in space.
• involve the components of the two vectors and since the magnitudes A and B can be calculated from the components using:
A . B = A B COS θ
• SPECIAL CASES
• Scalar product is maximum
• Angle b/w vectors is zero ( θ = 0°)
• Vectors are in same direction
• Scalar product is minimum
• Angle b/w vectors is right ( θ = 90°)
• Vectors are perpendicular to each other
• Scalar product is negative
• Angle b/w vectors is 180 ( θ = 180°)
• Vectors are opposite in direction
X Y X X Y Y
• SCALAR PRODUCT
• LENGTH/MAGNITUDE OF A VECTOR
The Dot Product of a vector with itself is always equal to its magnitude squared
• PARALLEL VECTORS
• PERPENDICULAR VECTORS
When A and B are parallel to each other, their Dot Product is identical to the ordinary multiplication of their sizes When A and B are perpendicular to each other, their Dot Product is always Zero
• SCALAR PRODUCT
• Since i and j and k are all one unit in length and they are all mutually perpendicular, we have
• i. i = j. j = k. k = 1 and i. j = j. i = i. k = k. i = j. k = k. j = 0.
• SCALAR PRODUCT IN COMPONENT FORM
• Two vectors in component forms are written as
• a = ax i + ay j + az k b = bx i + by j + bz k
• In evaluating the product, we make use of the fact that multiplication of the same unit vectors is 1, while multiplication of different unit vectors is zero. The dot product evaluates to scalar terms as :
• a . b =( ax i + ay j + az k ).( bx i + by j + bz k ) ⇒ a . b = ax i . bx i + ay j . by j + az k . bz k ⇒ a . b = axbx + ayby + azbz
• LAWS OF DOT PRODUCT
• Commutative law
• A . B = B . A
• Distributive law
• A . ( B + C ) = A . B + A . C
B BCOS θ A B ACOS θ A B A R=A+B C
• EXAMPLES OF SCALAR PRODUCT
• WORK DONE
• Definition
• Example:
• The man is pulling the block with a constant force a so that it moves along the horizontal ground . The work done in moving the block through a distance b is then given by the distance moved through multiplied by the magnitude of the component of the force in the direction of motion.
• EXAMPLES OF SCALAR PRODUCT
• POWER
• Definition
P = F . V
• ELECTRIC FLUX
• Definition
Electric flux = E . A
• EXAMPLES OF SCALAR PRODUCT A reader has used the dot product to help him analyze the movements of a tagged right whale. He had the x and y coordinates for a set of positions of the whale and wanted to calculate the angles turned through between successive sections of its journey. He found each section as a vector by calculating the differences between the pairs of x and y coordinates of the endpoints. Then he used the dot product of each successive pair of vectors to find the angle between those two legs of the whale's journey. A right whale fluke © the New England Aquarium
• CROSS PRODUCT Q.What is Cross Product ? Hassaan Ahmed Usmani
• What is Cross Product
• In mathematics, the Cross Product is a binary operation on two vectors in three-dimensional space that results in another vector which is perpendicular to the two input vectors.
• However the Dot Product produces a scalar result.
• Cross Product
• The cross product of two vectors a and b is denoted by a × b .
• The cross product is given by the formula
• Where θ is the measure of the angle between a and b , a and b are the magnitudes of vectors a and b , and is a unit vector perpendicular to the plane containing a and b .
• The right hand rule
• By using the right hand rule we can find out the direction of the vector.
• Cross Product of standard basic vectors
• Cross Product of standard basic vectors
• Geometric Meaning
• The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides
• Examples of cross product
• Area
• The magnitude of the cross product a b is the area of the parallelogram with sides a and b.
• Examples of cross product
• Volume :
• To find the volume of a paralleliped with sides a, b, c:
• we get
• Torque
• When force is applied to a lever fixed to a point, some of the force goes towards rotation while the rest goes towards stretching the lever.
• The magnitude of the torque is also proportional to the length of the lever, and has a direction depending on which direction the lever pivots.
• POINT,LINE AND PLANE By Naeem Nassiruddin
• What is Line?
• A line is a series of points that extends without end in two directions.
• A line is made up of an infinite number of points.
• The line below is named line AB or line BA.
A B
• Line
• Three points may lie on the same line. These points are Collinear.
• Points that DON’T lie on the same line are Not collinear .
R T S U V
• Line
• Horizontal lines have a slope of zero.
• Vertical lines are said to have infinite slope.
• Skew Line
• Skew lines only happen in space. They are non coplanar lines that never intersect. Unlike parallel lines, however, they don't always have a set distance between them, nor do they always have the same direction.
• Two lines are skew if they are not both contained in a single plane.
• Postulates of Geometry
• Postulate 1
• Two points determine a unique line .
• Postulate 2
• If two distinct lines intersect, then their intersection is a point .
• Postulate 3
• Three no collinear points determine a unique plane .
Q P T l m A B C
• Postulate 4
• If two distinct planes intersect, then their intersection is a line .
M N D E
• Equation of line in 3D
•
•
• What is Plane?
• A plane is a flat surface that extends without end in all directions.
• Points that lie in the same plane are coplanar .
• Points that do not lie in the same plane are non coplanar.
B A C
• Quadrants of Plane y x 5 -4 -2 1 3 5 5 -4 -2 1 3 5 -5 -1 4 -5 -1 4 -3 -5 2 2 -5 -3 Quadrant I (+, +) Quadrant II ( – , +) Quadrant III ( – , – ) Quadrant IV ( + , – )
• Equation of Plane
• Equation of plane in 3D This time, the locus is a plane .
•
•
• Distance b/w point to plane
•
•
• Thank You !! Any Questions??????