VECTORS – 4 (MASTER ADVANCED PHYSICS)

MASTER ADVANCED PHYSICS 

(FOR ALL TOP EXAMS) 

VECTORS – 4 

RESOLUTION OF VECTORS 

It is a good idea to break one vector into what is called rectangular components.

What does this mean?

Suppose you have a 2 dimensional vector and we want to have 2 one dimensional vectors, one on the x axis and the other on the y axis.

These two one dimensional vectors sum up and give the original vector.

So, if we have a vector of 10 units and at 60 degrees to the X axis, then

The component on the X axis is A Cos (theta) and the component of the Y axis is A Sin (theta)

Let us check whether this is true.

WE have seen that R2 = a 2 + b2 + 2ab Cos(theta)= a sin2 (theta) + BCos2(theta) + 2ab Cos(theta)

= 10(Sin2(theta) + Cos2(theta) ) + 0 (as Cos 90 is zero

= 10 which is true...

So, we can break a vector into two components along X axis and along Y axis.

If we have say 10 forces acting on a body, and if we break EACH one into components in X axis and y axis, then finally we will have

10 vectors along x axis

10 vectors along y axis.

Each can be added very simply. As it is one dimensional.

And finally we will have after adding, ONE value of the vector along x axis A and one along y axis B.

So, resultant can be got by

R2 = A2 + B2

R = root of a2 + b2

The final angle would be Tan b/a where b= Y component and a= X component.

So these are the basic concepts and also formulae.

PRODUCT OF VECTORS

Dot product – A DOT B = AB COS (THETA)

Cross product – A CROSS B = AB SIN (THETA)

We will see these concepts by linking to physics in later chapters.

VECTORS – 3 (MASTER ADVANCED PHYSICS)

MASTER ADVANCED PHYSICS 

(FOR ALL TOP EXAMS) 

VECTORS – 3

Addition, Summing Up or Finding the Resultant of Vectors 

When two vectors are in the same direction 

In such a case we simply add the magnitudes and the direction is the same for the resultant vector too. This is plain and obvious because two vectors, let s say two forces acting on a body in the same direction with two magnitudes have to be simply added arithmetically only. 

When two vectors are in exactly the opposite direction 

In such a case we simply subtract the magnitudes and the direction for the resultant vector is the direction of the one which has the larger magnitude. 

This is plain and obvious because two vectors, let s say two forces, 5 units and 10 units, acting exactly in the opposite directions on a body, will have a resultant, 10 – 5 = 5 Magnitude and the direction will be the one for the vector that has 10 units, 10 being larger than 5. 

When two vectors are inclined at some angle 

Here we do the geometric way. We take the first vector let us say, 5 units, as 5 cm, as a horizontal line on paper. The other vector also we do the same thing but we put the tail of the second vector at the head of the first vector with the angle of 60 degrees. 

We have now first vector as AB, and the second vector as BC. So, the resultant would be AC. 

Measure both the magnitude, that is, the length of AC and the direction as the angle from the horizontal.

This is also called the Triangular law of vector addition.


Another way to get resultant is through Parallelogram law of vector addition.

Here we put BOTH vectors at the same point. 5 cm, 10 cm, and 60 degrees, to take one example and COMPLETE the parallelogram as shown.

We can easily show that with this geometrical way, we can also use algebra and find, magnitude and direction by the following formulae.

By Pythagoras theorem and Trigonometry,

R2 = A2 + B2 + 2 AB Cos theta.

Tan (alpha) = B Sin (theta)/ A + B Cos (theta)

VECTORS – 2 (MASTER ADVANCED PHYSICS)

MASTER ADVANCED PHYSICS 

(FOR ALL TOP EXAMS) 

VECTORS – 2

Few Basic Concepts in Vector Algebra 

We have seen that a vector is inherently a quantity that has both magnitude and direction. We cannot add vectors like normal addition. We need to consider both magnitude and direction and find ways to get the final effect or what is called a resultant. 

Before we go into that, let us understand a few basic concepts that are needed to find resultants. 

Negative vector – 

A negative vector is simply the same vector with the same magnitude but exactly the opposite direction. 

Like and unlike vectors – 

Like vectors are those that may differ in magnitude but have the same direction. 

Unlike vectors have different magnitude but have directions exactly opposite to each other. 

Equality of vectors – 

If two vectors have BOTH same magnitude and the same direction, they are called equal vectors. 

Unit vector – 

A vector having unit magnitude, means one, but the same direction as a given vector, is called UNIT VECTOR of the given vector. 

So, we if we have a vector of 5 units and a direction of 60 degrees from horizontal, then the unit vector is simply ONE unit at 60 degrees from horizontal. 

It is convenient to have 3 unit vectors in 3 directions, x, y, and z as shown. 

Then we have a special symbol , 

I cap, j cap, k cap for unit vectors along x, y, and z axes respectively. 

Multiplication of a vector by a real number - 

We simply multiply the real number with the magnitude of the vector and the direction remains the same. 

For example, if we have a vector with 5 units magnitude and 60 degrees from the horizontal as direction. Multiplication of 3 with the number is simply 

3 multiplied by 5. So, we have 15 units of magnitude and the same 60 degrees direction. 

VECTORS – 1 (MASTER ADVANCED PHYSICS)

MASTER ADVANCED PHYSICS

(For all Top Exams) 

VECTORS – 1

Vectors and Scalars 

What are vectors? Vectors are those quantities that have both magnitude and direction. Scalars are those quantities that have ONLY magnitude. 

Let us take two examples to make our concept of vectors totally clear. 

Let us take force and temperature. 

Now suppose there are two or more forces acting on a body. Is it not important IN WHICH DIRECTION they are acting? 

If two equal forces act in exactly opposite directions, the body will not move. 

If they act in the same direction, the body will move with both forces added up. 

The force might act at different angles too. Is it not? 

Definitely, by nature force has both magnitude and direction. Such quantities are called as vectors. 

Other examples of vectors are Velocity, Momentum, Displacement etc. In each of them, the direction becomes important. 

Now, let us take temperature. In any direction, the temperature remains constant. There is nothing like a direction. Such quantities, where direction is not inherent to the quantity and only magnitude is there, we call scalars. 

Other examples of scalar quantities are mass, distance, speed, work, energy..etc 

What is Vector Algebra? 

Now, since vector quantities have both magnitude and direction, how do we ADD or SUM UP or find what is called the RESULTANT of two vectors. 

In real life, suppose we have many forces acting on a body, at different angles, how do we predict beforehand, what the RESULTANT force would be, and hence finally how the body would move? 

It cannot be just addition of magnitudes because DIRECTION also is involved. So, we have concepts of vector algebra and with these concepts, we can add/sum up vector quantities keeping direction in mind also. 

We also have PRODUCTS, dot product and cross product, some special concepts that we will see in the later units. 

To do all these things, we need some simple and clear definitions and symbols that we use in vector algebra. In the next section we will study them. 

KINEMATICS - 3 (MASTER ADVANCED PHYSICS)

KINEMATICS - 3 (MASTER ADVANCED PHYSICS)

MASTER ADVANCED PHYSICS

(For all Top Exams) 

Video – 3 

KINEMATICS - 3

Motion under Gravity 

We must remember that even in straight line motion, we have positive and negative directions for velocity, acceleration and displacement 

a=g = 9.8 m/s/s 

1. Body dropped from a height – h 

Equations 

U= 0, v= v, S= H, t= t (time of flight) 

a= +g 

So, V= g*t 

H= ½ gt2 

v2= 2*g*H 

Time of fall = t = root of 2h/g 

Velocity with which it reaches the ground = v= root of 2gh 

2. When a body is thrown upwards with a velocity – u 

Equations – upwards – positive , downwards – negative 

U= u, v= 0 , a= -g, s= H, t= t 

0= u-gt 

h = ut – ½ gt2 

U2 = 2*g*h 

So, 

Maximum height reached = h= u2/2g 

Time of ascent= time of descent = u/g 

Total time in flight = 2u/g 

3. Body projected from a tower of height H with velocity – u 

Equations – upward is positive 

Downward is negative 

U= u, s= -h(height of the tower is displacement) , t= t , v= -v, g= g 

U2- V2= -2gh 

So, 

Velocity with which it reaches the ground = v = root of u2 +2gh 

Total time taken to reach the foot of tower = time to reach the top + time of fall 

= u/g + root of (u2 +2gh) / g