Distance and Displacement Physics Notes

Distance and Displacement:
Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly different definitions and meanings.
1. Distance: Distance is a scalar quantity that refers to the length of the path covered by the object regardless of its starting or ending position. In other words, distance refers to the length of the entire path travelled by the object.
Unit of distance = meter in M.K.S. system
= centimeter in C.G.S. system

Unit of displacement = meter in M.K.S. system
= centimeter in C.G.S. system.

2. Displacement: Displacement is a vector quantity that refers to the shortest distance between the two positions of the object i.e, the difference between the final and initial positions of the object, in a given time. Its direction is from the initial to final position of the object. It is represented by the vector drawn from the initial position to its final position.

Comparison between distance and displacement:

Distance	Displacement
1. Distance is a scalar quantity. It has only a magnitude	1. Displacement is a vector quantity. It has both magnitude and direction.
2. The distance traveled by an object is equal to the length of the path covered.	2. It is the shortest distance between the initial position and final position of the object in the given time.
3. The distance traveled in a given time is either equal to or greater than displacement.	3. The displacement of an object can be equal to or less than the distance traveled but never greater than the distance traveled.
4. Distance has positive values.	4. Displacement can be either positive or negative.
5. Distance depends upon the path.	5. Displacement does not depend upon the path. It depends upon the starting and endpoints.
6. A reference point, is not used to measure distance.	6. Displacement is measured with respect to a reference point.

Generally, it can be said that the magnitude of displacement is equal to the minimum possible distance between two points or
Distance ≥ Displacement.

In a circular motion, if a body starts at point A and comes back at point A then *
Distance = 2 πr
Displacement = Zero

From point A to B
Distance = πr
Displacement = 2r

Physics Notes

Concept of Rest and Motion Physics Notes

Concept of Rest and Motion:
Rest: An object is said to be at rest if it does not change its position with respect to time, with respect to its surrounding. For example, mountain, house, tree etc. are in rest position with respect to the person standing on earth. A book lying on a table or a person sitting on a chair are also examples of rest.
*Motion: An obect is said to be in motion if it changes its position w.r.t its surroundings with the passage of time.
For example: motion of planets around the sun, motion of a train, motion of the gas particle etc.

Rest and Motion:

• Motion is a combined property of an object under study and the observer.
• Everything in the world is at rest or in motion.
• There is no meaning of rest and motion without a viewer.

Consider a scenario, you are traveling in a car, seated with your friend in the back seat. Your friend is quietly minding his own business, and according to you, your friend is at rest since his position does not change with time. But for a pedestrian standing beside the road and seeing the cargo by, your friend (as well as you) are in motion since both of you are changing position as far as the pedestrian is concerned.

So, at any point of time, an object can be at rest with respect to a second object, while being in motion with respect to a third. In other words, motion is always relative, never absolute.

It can also be said that motion and rest are relative terms.

Types of Motion:
There are two types of motion:

1. On the basis of coordinate system
2. On the basis of the nature path of moving particle.

1. On the basis of the co-ordinate system: On the basis of a co-ordinate system, motion can be divided in the following way:
(a) One-dimensional motion or 1-D motion:
The motion of an object is called one dimensional if only one of the three-co-ordinates are required to specify the position of the object in space changes w.r.t. time.

In 1-D motion, the object moves along a straight line.In 1-D motion there are two directions (backward and forward, upward and downward. In these directions the object moves and these directions are specified by +ve and -ve signs. For example, A body running on a straight road, a body thrown upwards, motion of train along a straight railway track etc.

(b) Two-dimensional motion or 2-D motion:
The motion of an object is called two-dimensional if two of the three co-ordinates required to specify the position of the object in space change w.r.t. time or if two co-ordinates are required to specify the position of the object. In 2-D motion the object moves in a plane.

For example: A billiard ball moving over the billiard table, a satellite revolving around the earth, projectile motion, an insect crawling over the floor are two-dimensional motion.

Suppose an object is moving in X-Y plane with origin ‘O’ (figure). At time t, let the object be at P, it’s coordinates are (x, y). It means to know the position of the object, we are required to know two coordinates i.e., distance along Y-axis and Y-axis.

(c) Three-dimensional motion or 3-D motion: The motion of an object is called 3-D. If all the three co-ordinates are required to specify the position Of the object in space changes w.r.t. time. Such a motion is not restricted to a straight line or a plane but takes place in space. It is the most general form of the motion. For example. Motion of flying kite, the random motion of gas-particle, a flying bird, a flying aeroplane etc.

2. On the basis of nature of motion of the particle:
(a) Translatory motion: Translatory motion is the motion by which a body shifts from one point in space to another. One example of translatory motion is motion of a bullet fired from a gun. Another example: horizontal motion of a body, a car moving in a straight line or a train moving along a straight line are some examples of translatory motion.

(b) Rotational motion: When a body rotates around a fixed axis, then it is called rotational motion. Rotational motion deals only with the rigid bodies. Rigid body is an object which can not change the position, shape and size under the influence of external force.

For example Motion of ceiling fan, a string whirled in a circular loop, the motion of a wheel about its axis are the examples of rotatory motion.

(c) Oscillatory or vibrational motion:
The motion in which a particle moves to and fro motion about a given point is known as oscillatory or vibrational motion. For example: Motion of a simple pendulum, motion of a mass attached to a spring, etc.

Physics Notes

Frame of Reference Physics Notes

The Frame of Reference:
Imagine you threw and caught a ball while you were on a train moving at a constant velocity passing a station. To you, the ball appears to simply travel vertically up and then vertically down under the influence of gravity. However, to an observer standing on the station platform, the ball would appear to travel in parabola, with a constant horizontal component of velocity equal to the velocity of the train. This is illustrated in the figure below.

Path of the ball as seen by an observer on the train and on at the station
The different observations occur because the two observers are in different frames of reference.

Thus, a frame of reference is a set of coordinates that can be used to determine positions and velocities of objects in that frame. Different frames of reference move relative to one another.

Frames of reference can be of two types:
(a) Inertial frame of reference
(b) Non-inertial frame of reference.

(a) Inertial frame of reference: A frame of reference that remains at rest or moves with constant velocity with respect to other frames of reference is called inertial frame of reference. An inertial frame of reference has a constant velocity. That is, it is moving at a constant speed in a straight line, or it is standing still. Newton’s laws of motion are valid in all inertial frames of reference. Here, a body does not change due to external forces. All inertial frames of a reference are equivalent for the measurement of physical phenomena.

There are several ways to imagine this type of motion:

• Motion of Earth
• A space shuttle moving with constant velocity relative to the earth.
• A rocket moving with constant velocity relative to the earth.

(b) Non-inertial frame of reference: A frame of reference is said to be the non-inertial frame of reference when a body, not acted upon by an external force, is accelerated. In a non-inertial frame of reference, Newton’s laws of motion are not valid. It also does not have a constant velocity and is accelerating. There are several ways to imagine this type of motion.

• The frame could be travelling in a straight line, but the speed of the object increases or decrease.
• The frame could be travelling along a curved path at a steady speed.
• The frame could be travelling along a curved path and also the speed of the object increases or decrease.
• To locate the position of a particle, the frame of reference should be universally accepted and is easily available.

The simplest reference frame is the cartesian frame of reference or cartesian coordinate system. It has three mutually perpendicular axes named as 1,7 and Z axes. The point of intersection of these axes is called origin (O) and is considered as the reference point

The X, Y and Z coordinates describe the position of the object with respect to the coordinate system. To measure time, put a clock in the system.

There are two types of frames of reference of the co-ordinate system.
1. Anticlockwise co-ordinate system

(a): Anticlockwise Reference System

2. Clockwise coordinate system:
Generally, we use clockwise coordinate system.

(b): Clockwise Reference System

Physics Notes

Kinematics Physics Notes

Generally we have seen different types of objects in our nearby environment in which we view the differences in their condition, of state, shape, position, colour, etc. A similar difference is viewed in their positions due to which some objects are stable (motionless) to us which means the body (object) does not change its positions as time passes. Whereas some objects change their positions with time.

For example, if we see the table in our class it seems stable (rest) to us; while a car moving outside the classroom seems as unstable (motion). Here the table does not change its position with respect to time where as the car moving on the road changes its position with respect to time. Therefore, in the physical world, we define speed in terms of the viewed change in the position of the object as time passes; which means:

“Speed is the change in the position of the atom (particle) or the collection of atoms (particles) as time passes.”

Movement of vehicle, speed of gaseous atoms, sound reaching from one place to another, speed of the cricket ball, flying of the birds and an aeroplane in the sky, speed of different astronomical or celestials bodies, blood circulation in the views and artery, breathing, etc., all such events and their occurences are examples of speed in the visible or invisible form.

Speed can be in the form of a simple straight line, in a plane, curved, circular or any other path, rotation, shiver or in the form of waves or can be in the independent form in the sky.

The branch of physics which deals with the study of motion of material object is called Mechanics. Mechanics can be classified into the following branches:
1. Statics: It is a branch of Mechanics which deals with the study of the material object at rest. An object can be at rest, even when a number of forces acting on it are in equilibrium. Thus, Statics is the study of the motion of an object under the effect of forces in equilibrium. Here, time factor does not play any role.

2. Kinematics: It is the branch of Mechanics which deals with the study of the motion of the object without taking into account the cause of their motion.
Here time factor plays an essential role. The term Kinematics is derived from the Greek word ‘Kinema’ meaning motion.

3. Dynamics: The study of the motion of the objects by taking into account the cause (or causes) of their change of state (rest or uniform motion) is called dynamics.

Concept of a point object: An object is considered a point object if the size of the object is much smaller than the distance it moves in a duration of time.
For example; the length of a train in comparison to the distance covered by the train is very less; therefore the train is assumed as a point object or as an atom (particle).

→ Motion: If an object changes its position with respect to its surroundings with time, then it is called in motion.

→ Rest: If an object does not change its position with respect to its surroundings with time, then it is called at rest.

→ Rest and motion are relative states. It means an object which is at rest in one frame of reference can be is motion in another frame of reference at the same time.

→ Point mass object: An object can be considered as a point mass object, if the distance travelled by it in motion is very large in comparison to its dimensions.

→ One dimensional motion: If only one of the three coordinates specifying the positions of the object changes with respect to time, then the motion is called one-dimensional motion. For instance, motion of a block is a straight line, motion of a train along a straight track, a man walking on a level or a and narrow road and object falling under gravity, etc.

→ Two-dimensional motion: If only two out of the three coordinates specifying the position of the object changes with respect to time, then the motion is called two-dimensional motion. A circular motion is a two-dimensional motion.

→ Three-dimensional motion: If all the three coordinates specifying the position of the object changes with time, then the motion is called 3-D motion. Flying kite, a flying airplane, the random motion of a gas molecule, etc. are examples of 3-dimensional motion.

→ Distance: The length of the actual path traversed by an object is called the distance. It is a scalar quantity and it can never be zero or negative. Its unit is metre.

→ Displacement: The shortest distance between the initial and final position of any object during motion is called displacement. It can be positive, zero or negative. It is a vector quantity. Its unit is metre.

→ When the motion of an object is along a straight line, there are only two directions i.e., forward and backward or upward and downward in which an object can move. Therefore, it is easier to represent the displacement in these two directions by sign and we do not need to use the vector notation.

→ Speed: Speed is the distance travelled per unit of time. It is a scalar quantity.

→ Velocity: Velocity is a vector representation of the displacement that an object or particle undergoes with respect to time.

→ Uniform motion: The motion of an object is said to be uniform if it covers equal displacement (or distance) in equal intervals of time. It is a vector quantity.

→ Uniform speed: It is defined as the ratio of the path length (distance) to the time taken by the object to cover the path.
Mathematically: Speed of the uniform motion = \(\frac{\text { Path length }}{\text { Timeinterval }}\)

→ Average velocity: The ratio of the total displacement to the total time taken is called average velocity.
Average velocity = \(\frac{\text { Totaldisplacement }}{\text { Total time taken }}\)

→ Acceleration: Acceleration is the rate of change of velocity of an object with respect to time.
Acceleration (a) = \(\frac{\text { Change in velocity }}{\text { Time interval }}=\frac{\Delta v}{\Delta t}\)

Its unit is m/s and the dimensional formula is [M⁰L¹T^-2]. It is a vector quantity.

→ Acceleration can be positive, zero or negative. Positive acceleration means velocity is increasing with time, zero acceleration means velocity is uniform while negative acceleration (retardation) means velocity is decreasing with time.

→ Uniformly accelerated motion: The motion of an object is said to be uniformly accelerated. If the same change in its velocity takes place in each unit of time.

→ Position time graph: It is a straight line inclined to the time-axis and the velocity of the uniform motion is equal to the slope of the position-time graph.

→ Velocity-time graph: It is a straight line inclined to the time axis.
(a) The acceleration of uniformly accelerated motion is equal to the slope of the velocity-time graph.
(b) The area under the velocity-time graph between the instant and t₂ is equal to the displacement of the object in the time interval (t₂ – t₁)

→ Equations of uniformly accelerated motion:
If a body starts with velocity (u) and after time t its velocity changes to v, if the uniform acceleration is ‘a’ and the distance travelled in time t is ‘ s’, then the following equations are called equations of motion.

• v = u + at
• s = ut + \(\frac{1}{2}\) at²
• v² = u² + 2as
• Distance travelled in nth second
  S_n = u + \(\frac{a}{2}\)(2n – 1)

→ Motion under effect of gravity: If an object is falling freely (u = 0) under gravity, then the equations of motion are;

• v = u + gt
• h = ut + \(\frac{1}{2}\)gt² and
• v² = u² + 2gh

Note: If an object is thrown upward then ‘g is replaced by -g in the above three equations.

→ Projectile motion: When any object is thrown from a horizontal angle θ, except 90°, then the path followed by it, is called trajectory, the object is called projectile and its motion is called projectile motion.

If any object is thrown with velocity u, making an angle θ, from horizontal, then
(a) Equation of the path of a projectile
y = xtanθ – \(\frac{g}{2 u^{2} \cos ^{2} \theta}\) x²
The path of a projectile is parabolic.

(b) Time of flight: It is defined as the total time for which the projectile remains in the air.
T = \(\frac{2 u \sin \theta}{g}\)

(c) Maximum height: It is defined as the maximum vertical distance covered by the projectile.
H = \(\frac{u^{2} \sin ^{2} \theta}{2 g}\)

(d) Horizontal range: It is defined as the maximum distance covered by the object in horizontal direction.
R = \(\frac{u^{2} \sin 2 \theta}{g}\)

(e) Horizontal range is maximum when it is thrown at angle of 45° with the horizontal
Rmax = \(\frac{u^{2}}{g}\)

(f) For angle of projection (θ) and (90-θ) the horizontal range are same.

→ Relative-velocity: The relative velocity of an object w.r.t. another moving object is the effective velocity with which the object will appear to move when the other object is considered to be at rest.

Mathematically: If VA and VB are the velocities of the two objects A and B, then the relative velocity of an object A w.r.t. the object B is given by
V_AB = V_A + (-V_B)
(a) When the two objects are moving along a straight line in the same direction, the relative velocity is
V_AB = V_A – V_B
(b) When the two objects are moving along a straight line in the opposite directions, the relative velocity is
V_AB = V_A + V_B

→ The frame of reference: A frame of reference is a set of coordinates that can be used to determine the positions and velocities of objects in that frame.

→ Translatory motion: Translatory motion is the motion by which. a body shifts from point to another w.r.t. the coordinate system. for example motion of a vehicle on straight road, horizontal motion of a body, etc.

→ Rotational motion: When a body constantly moves around an fixed axis it is called a rotational motion for example motion of ceiling fan, a string whirled in a circular loop, etc.

→ Oscillatory vibrational motion: The motion in which a particle moves to and fro motion about a given point is called oscillatory or vibrational motion, for example motion of a simple pendulum, motion of a mass attached to a spring, etc.

→ Retardation/Deceleration: Negative acceleration is called. Retardation or deceleration.

→ Uniform accelerated motion: In uniform acceleration, the magnitude and direction of an object always remain constant. This type of motion is called uniform accelerated motion.

→ Relative velocity: The relative velocity of the first body with respect to another body when both are in motion, is the rate at which the first body changes its position with respect to another body.

Physics Notes

Logarithm and Its Uses Physics Notes

Logarithm and Its Uses:
Logarithms were introduced by John Napier in the early century. It was introduced to simplify calculations. They are used by Navigators, Scientists, Engineers, and others to perform computations easily. The logarithm of a number is the exponent to another fixed value called the base, which must be raised to produce that number. In simple cases, the logarithm counts repeated multiplication.

The logarithm can be calculated for any two positive real numbers a and y, where a is not equal to
1. The logarithm of y to base a, denoted log0 (y), is the unique real number x such that
a^x = y
For example, 64= 2⁶,
then, log₂(64) = log₂(2⁶) = 6

The logarithm to base 10 (i.e., a = 10) is called the common logarithm. The natural logarithm has the number e(≈ 2.718) as its base. The binary logarithm uses base 2 (i.e., a = 2).

How to Find out the Logarithm of Any Number:
There are two parts of the logarithm of the number

1. Characteristic
2. Mantissa.

The fractional part of a logarithm is usually written as a decimal. The whole number part of a logarithm is called the Characteristic.

This part of the logarithm represents the position of the decimal point in the associated number. The decimal part of a logarithm is called the Mantissa.

The mantissa of a common logarithm is always the same regardless of the position of the decimal point in that number.

For example,
log 5270 = 3.72181
The mantissa is 0.72181 and the characteristic is 3.

(a) To Find out the Characteristic:
It is to be noted that a common logarithm is simply an exponent of base 10. Characteristic is the power of 10 when a number is written in scientific notation.

The characteristic can be determined by using the following rules:
1. For a number greater than 1 (> 1): The characteristic is positive and is one less than the number of digits to the left of the decimal point in the number.

2. For a positive number less than 1(< 1): The characteristic is negative and has an absolute value of one more than the number of zeroes (0s) between the decimal point and the first non-zero digit of the number.

The negative characteristic is shown by placing the – (bar) symbol over the number.
For example, log 0.023 = 2.36173
The characteristic is 2 (as 0.023 = 2.3 × 10^-2).
The bar over 2 indicates that only the character is negative. So, the logarithm is – 2 + 0.36173.

(b) To Find out the Mantissa:
The mantissa is the decimal part of a logarithm. The logarithm table usually contains only mantissa.

The mantissa can be determined as follows:
The first column of the logarithm table contains the number and the sixth column contains its logarithm. For example, if we want to find the logarithm of 45, then, we will find out the number 45 in the first column. Its logarithm will be 1.65321 in the logarithm table.

Suppose, we have to find out the logarithm of the number 450, but it does not appear in the logarithm table, then we will find out the number 45 in the first column. Notice that both the numbers ‘45’ and ‘450’ have the same mantissa but different characteristics. So, the logarithm of the number 450 will be 2.65321.

Examples of Logarithms of Some Numbers

Method to Find out the Antilogarithms:
The antilogarithm of the logarithm of a number is the number itself. For example, log 1268 = 3.1031, then the antilog (3.1031) = 1268.

How to Find out the Antilogarithm:

1. Separate the characteristic and the mantissa.
2. Use the antilog table to find out the corresponding value of the mantissa. Look for the row number consisting of the first two digits of the mantissa. Then find out the column number equal to the third digit of the mantissa.
3. Find out the value from the mean difference columns. The antilogarithm table also has a set of columns known as the ‘mean difference column’. Now, look at the same row to find out the column number equal to the fourth digit of the mantissa.
4. Add the values from the mean difference columns.
5. Insert the decimal point after the number of digits that corresponds to the characteristic plus one (1).

Physics Notes

Integral Calculus Physics Notes

Integration: Integration is just opposite to differentiation, if the functions F(x) and f(x) are related as :
\(\frac{d}{d x}\){F(x)} = f(x) then, ∫ f(x)dx = F(x)
∫f(x) dx is called the integral of f(x).
f(x) Δx is the area of the small segment whose height is f(x) asd width is x. When Δ x → 0 then the sum of such sLnall segments become the integral.

When the limits of the integral are definite (from x = x1 to x = x2), then,


So, ∫x1x2 f(x)dx is the process of summation in definite limits x1 of the continuous function f(x) w.r.t the variable.

Indefinite Integration:
We know that the differentiation of every constant is zero.
∴ \(\frac{d}{d x}\){F(x)} = f(x) and, \(\frac{d}{d x}\){F(x) + constant} = f(x)

Definite Integral (from a to b)

So, the integration of a constant becomes indefinite.
∫f(x)dx = F(x) + Integration constant

Definite Integration:
A definite integral has start and end values. In other words, there is an interval (say a to b). The values of the interval are placed at the bottom and top of the integral as:
∫^b_a f(x)dx (if the lower limit is a and the upper limit

To find out the definite integral, we subtract the integral at points a and b.
∫^b_a f(x)dx = [F(x)]^b_a = [F(b) – F(d)] …(i)

In other words, we can say that the definite integral between a and b is the indefinite integral at b minus the indefinite integral at a.

It is to be noted that the integration constant is not written after calculating the definite integral.

Applications of Integration:
A large number of problems can be solved through integration. Some of them are mentioned below:

• The area between curves: We can find out the area between the two functions by integrating the difference between them.
• The average value of a function.
• Arc Length: We can use integration to find the arc length of a curve. It can be used by up an infinite number of infinitely small line segments.
• Volume of solids with known cross-sectional area.
• Area defined by polar graphs: We cannot only find out the area in cartesian coordinates but also in polar coordinates.

Physics Notes

Maximum and Minimum Values of A Function Physics Notes

Maximum and Minimum Values of a function:
A function f(x) is said to have a relative maximum value at x = a, if f(a) is greater than any value immediately preceding or following.

Maximum and Minimum Points

A function f(x) is said to have a relative minimum value at x= b, if f(b) is less than any value immediately preceding or following.

The tangent to the curve in figure is horizontal (see point A and B). The slope of each tangent line, i.e., the derivative when evaluated at A or Bis zero (0).
i. e., f'(x) = 0.
At points immediately to the left of a maximum, the slope of the tangent is positive:
f'(x) > 0
At points immediately to the right of a maximum, the slope of the tangent is negative:
f'(x) < 0
In other words, at a maximum, f’ (x) changes the sign from + to – .
At a minimum, f’ (x) changes the sign from – to +.

We observe that at a maximum, at A the graph is concave upward.

The value of x at which the function has either a maximum or a minimum is called a critical value. In the figure, the critical values are x = a and x = b.

The sufficient condition for extreme values of a function at a critical value a:

• The function has a minimum value at x = a if f'(a) = 0 and f”(a) – a positive number
• The function has a maximum value at x = a if f'(a) = 0 and f”(a) = a negative number.

Physics Notes

Graphics Representation of Differentiation and Integration Physics Notes

Graphics Representation of Differentiation:
A slope field is the graphical representation of a differential equation. It is a graph of short line segments whose slope is determined by evaluating the derivative at the midpoint of the segment.

To And the derivative graphically, we must take a tangent line (the point where we have to calculate the derivative) and then find out the slope of this tangent line. To find out the instantaneous rate of change, we must repeat the process of calculating the slope.

Graphical Representation of Derivatives

Let y be a function of x so that y = f(x). This dependency of y on x is shown in the figure. On this graph, the two points P and Q are represented by (x, y) and (x + δx, y + δy) respectively. If P is closer to Q, then δx → 0. Hence, PQ becomes a straight line with
slope, tan θ = \(\frac{Q R}{P R}\)

If the angle is acute (< 90°), then tan0 or the slope is positive and if the angle is obtuse (> 90°), then tan0 or the slope is negative.

Graphical Representation of Integration:
The value of y changes with different values of x, for the function, y = f(x)

Graphical Representation of integration

The graph in figure 2.30 shows the values of dependent variables x and (x + dx). The value of dx is very small (dx → 0). So, the value of y remains constant, as the value of change in x is very small. The area of this small segment (PQRS) will be y dx. If we imagine similar segments from x = a to x – b, then the area will be the sum of area of all the segments. The process of adding up considering that the value of dx is very small is called integration.

∴ ∫^b_a y dx = area between Y-X plane and X-axis (from x = a to x = b)

Physics Notes

Differential and Applications of Calculus Physics Notes

Differential Calculus:
Differentiation is an operation that makes us able to find out a function that outputs the rate of change of one variable with respect to another variable.

Let y be a physical quantity that depends upon another quantity, say x. So, when there is a change in the value of x, then the value of y also changes. For example, the derivative of the position of a moving object with respect to time is the velocity of the object, which measures how quickly the position of the object changes when time is changed.

Let, y = f(x)
It means that y is a function of x, and the value of y changes with the change in the value of x. Here, x is said to be an independent variable because it does not depend upon any other variable, whereas y is a dependent variable as it varies with the value of x.

Let there be a small change (increase) in the value of x, so that its value changes to x + δx. So, the value of y will also change to y + δy.

When δx approaches zero, then δy also approaches zero.

In mathematical form, the rate of change of y w.r.t. the change in x is written as \(\frac{d y}{d x}\).

A derivative is the limit of the ratio of the change in a function to the corresponding change in its independent variable as the t change in the latter approaches zero.

It can also be expressed as:

Here, we should note that differentiation is a mathematical process and d/dx is an operator which acts on y to give dy/ dx.

Applications of Differential Calculus in Physics:
There are many physical quantities that’1 on other quantities. For example distance defends upon time, velocity depends upon distance and Vme, pressure of a gas depends upon volume &nd temperature, etc. .

Let us assume that the displacement \(\vec{r}\) is function of time t, i.e.,

This quantity is the velocity of the object. So, the derivative of \(\vec{r}\) w.r.t. t is the velocity.
\(\vec{v}=\frac{d \vec{r}}{d t}=\frac{d}{d t}(\vec{r})\)

Similarly, the velocity \(\vec{v}\) depends upon time t. So, the rate of change of velocity is the acceleration:

Instantaneous acceleration,
\(\vec{a}=\frac{d \vec{v}}{d t}=\frac{d}{d t}\left(\frac{d \vec{r}}{d t}\right)=\frac{d^{2} r}{d t^{2}}\)

\(\frac{d^{2} r}{d t^{2}}\) is the second derivative of displacement vector \(\vec{r}\) w.r.t time (t).

Another example : Work done (W) by a force is a function of t and power P is the rate of change of work done.
∴ P = \(\frac{d W}{d t}\)

or, P = \(\frac{d E}{d t}\) (E = Energy)
dt
The number of active atoms (N) present in radioactive substance is a function of time and the rate of decay A is
A = – \(\frac{d N}{d t}\)

Physics Notes

Properties of Cross Product Physics Notes

Properties of cross product:
1. Vector products do not show commutative property:

2. Vector Product of Two Parallel Vectors:
Let \(\vec{A}\) and \(\vec{B}\) be two parallel vector. So, the angle between the two vectors, θ = 0°.

From vector products,

So, the vector product of two parallel vectors is a zero product.

3. Vector Product of Equal Vectors: Equal vectors are also parallel. So, the angle between them will also be zero. So, θ = 0°.
∴ \(\vec{A} \times \vec{A}\) = AA sin 0° = 0
So, the vector product of equal vectors is a zero vector.
∴ î × î = ĵ × ĵ = k̂ × k̂ = 0 …(iv)

4. Vector Product of Perpendicular:
Vectors: Let us assume that \(\vec{A}\) and \(\vec{B}\) are two vectors perpendicular to each other. So, the angle between them will be 90°, i.e., θ = 90°.
Then,

Here, the direction of ñ is according to the right-hand rule, in the direction perpendicular to the plane of \(\vec{A}\) and \(\vec{B}\). So, in this way,


If î, ĵ, and k̂ are in the cyclic order (clockwise) then the vector product is positive and if not in cyclic then the product will be negative.

5. Vector Product in the form of a Determinant: If the vectors are expressed in terms of unit vectors î, ĵ and k̂ in the X, Y, and Z directions, then the vectors can be expressed as:

Using equations (iv) and (v), we have

It can be re-written in the form of determinants

The magnitude of cross product:

6. Cross product of two vectors is distributive:
i. e., \(\vec{A} \times(\vec{B}+\vec{C})=\vec{A} \times \vec{B}+\vec{B} \times \vec{C}\)

7. Cross product of two vectors is associative i.e.,
\((\vec{A}+\vec{B}) \times(\vec{C}+\vec{D})=\vec{A} \times \vec{C}+\vec{A} \times \vec{D}+\vec{B} \times \vec{C}+\vec{B} \times \vec{D}\)

8. Examples of Some Physical Quantities Obtained from Vector Product
(a) Let \(\vec{L}\) and \(\vec{B}\) respectively be the adjacent sides of a parallelogram, then the area of it will be:
\(\vec{A}=\vec{L} \times \vec{B}\)

(b) Torque (\(\vec{τ}\)), displacement (\(\vec{r}\)) and force (\(\vec{F}\)) are associated as:
\(\vec{\tau}=\vec{r} \times \vec{F}\)
i.e., torque (\(\vec{τ}\)) is the cross product of displacement and force.

(c) Angular momentum (J), displacement (\(\vec{r}\)) and linear momentum (\(\vec{p}\)) are associated as:
\(\vec{J}=\vec{r} \times \vec{p}\)
i. e., angular momentum is equal to the cross-product of displacement and linear momentum.

(d) Angular velocity (ω), linear velocity (v) and displacement (r) are associated as :
\(\vec{v}=\vec{\omega} \times \vec{r}\)
i.e., linear velocity is equal\o the cross-product of angular velocity and displacemerí.

(e) The vectors of adjacent sidef a triangle are and b the area of triangle:
\(\vec{A}=\frac{1}{2}|\vec{a} \times \vec{b}|\)

Physics Notes

Rules for determining the Direction of the Vector Product Physics Notes

Rules for determining the Direction of the Vector Product:
1. Right-hand thumb rule: The right hand rule states that the orientation of the vector’s cross product is determined by placing \(\vec{A}\) and \(\vec{B}\) tail to tail, flattening the right hand, extending it in the direction of A and then curling the fingers in the direction that the angle makes with A The thumb then points in the direction of \(\vec{A} \times \vec{B}\).

(i) Vector Product

2. Right-Hand Screw Rule: The right-hand screw rule can be used when a direction must be determined, based upon the rotational direction or vice-versa.

The axis of the screw is placed as shown in the figure. (ii). When the screw is moved by a small
angle in the direction from \(\vec{A}\) to \(\vec{B}\), then the direction in which the screw moves forward gives the direction of the resultant vector.

(ii) Vector Product

Physics Notes