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Friday, October 9, 2015
Vectors
Introduction:
a
...
A unit vector is given by,
π΄β
π΄Μ =
π΄
b
...
Such components are called Rectangular components of a vector
...
And
Magnitude of π΄π¦ πΜ or y-component of π΄β = π΄ sin π
c
...
Position vector
It is a vector which describes the location of some points with respect to their
origin
...
Unit vector
π = βπ2 + π 2
It is a vector which has magnitude one and it is used to describe the direction of a
given vector
...
g
...
and π
The vector addition by rectangular components involve the following steps:
1
...
3
...
Find x any y components of all the vectors
...
Find y-component π π¦ of the resultant vector by adding y-components
...
The angle π of the resultant vector is given by,
π = tanβ1
π π¦
π π₯
ο If π π₯ πππ π π¦ are positive then the angel is written as it
...
π© = π΄π΅ cos π
Where π is angle between π¨ and π©
...
i
...
π¨
...
i
...
π¨
...
i
...
Μ=π
Μ
...
π
o The scalar product of vector with itself is equal to the square of its
magnitude
...
e
...
π¨ = (π΄)2 cos 0Β° = π΄2
o The scalar product of unit vector with itself is equal to 1
i
...
Μ
...
πΜ = π
o The scalar product of two parallel vectors is equal to the product of their
magnitude
...
e
...
π© = π΄π΅ cos 0Β° = π΄π΅
o Scalar product of two vectors in terms of their rectangular components:
π¨
...
π΅π₯ +π΄π¦ π΅π¦ +π΄π§ π΅π§
π΄π΅
Vector or Cross product
The vector or cross product of two vectors is defined by:
π¨ Γ π© = π΄π΅ sin π πΜ
Where πΜ is a unit vector and it is perpendicular to the plane containing π¨ πππ π©
...
π¨ Γ π© = π΄π΅ sin 0Β° πΜ = π
i
...
o The cross product of unit vector with itself is also zero
...
π is position vector of moment arm
...
Torque is a vector
quantity and it SI unit is ππ
...
Equilibrium
A body is said to be in equilibrium when it is at rest or moving with uniform
velocity
...
i
...
Ξ£π = 0
Second condition of equilibrium
When the sum of all the torques acting on the body is zero then the second
condition is satisfied
...
When the second condition of equilibrium is satisfied then there is no angular
acceleration and body will be in rotational equilibrium
...
Numerical:
Date of completing: Monday, October 12, 2015