Category: Notes

Graphical Representation of Motion Physics Notes

Graphical Representation of Motion:
1. A graphical representation is a pictorial representation of the relation between two sets of data of which one set is of a dependent variable and other set is of independent variables. Now here dependent variable is shown on Y-axis and independent variable is shown on X-axis.

2. To describe the motion of an object we can use line graphs. In this case line graph shows dependence of one physical quantity, such as distance (position), velocity, acceleration, on the another quantity such as time.

3. Now the slope of graph and area from figure provides other physical quantities. The slope of graph is calculated by tanθ which is the ratio of physical quantities taken on Y-axis to the X-axis. While by the area of figure, the product of quantities taken from the X axis and Y-axis which are dependent upon the figure and its shape.

Now here we showed the three types of graphical representation of motion.
Dependent Quantities

Independent Quantity

1. Position-time graph
2. Velocity-time graph
3. Acceleration-time.

Position-Time Graph:

1. The change in the position of an object with time can be represented on the position-time graph.
2. In this graph, time is taken on the X-axis and position is taken along the Y-axis.
3. The Position-time (P – T) graph of a moving body can be used to calculate the speed of the body as they specifically represent velocity.
4. The slope of the tangent at any point of position¬time graph denotes the instantaneous velocity of the object at that point or tan θ = v

(a) When the object is at rest
Its position will not change with time. Let the object be stationary at position x(t) = x0 from the origin. Then the position (x) time (t) graph for the stationary object is a straight line AB parallel to the time-axis.

Object at rest
For example: (i) A train standing at the railway track or line.
(ii) A bus standing on the road side.
Here θ = 0°
then tan θ° = v
so v = 0 {v ∴ velocity}

(b) When the object moves with a uniform motion:
Uniform motion: Uniform motion is defined as equal displacement occurring during successive equal time periods (some times called constant velocity).

The graph of position time for uniform motion in a straight line that the slope or velocity is constant.

(a) When the object is moving with a positive velocity then the slope of the position-time graph will be tan 0 and is positive then v is constant but positive and acceleration (a) = 0.

Slope of tan θ = \(\frac{y \text { axis }}{x \text { axis }}=\frac{x}{t}\) = velocity
velocity = v = constant
but a = \(\frac{d v}{d t}=\frac{d}{d t}\) (const)
a = 0

(b) When the object is moving with a negative velocity then the slope of position time graph will be negative. Hence θ = constant but θ > 90°, negative tan θ then v is constant but acceleration (a) = 0.

(c) When the object moves with a variable velocity or Non-Uniform Motion: Figure (a) depicts the non-uniform motion in which the graph is not a straight line in the position-time graph. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P [tanθ₂ > tanθ₁]. Hence, the velocity at Q is more than the velocity at P.

In this way, the velocity of the particle is increasing with time. Hence, the acceleration of the particle will be positive. Similarly, figure (b) shows the non-uniform motion in the position-time graph. But here when the slopes are calculated at points P and Q; the slope at Q is less than the slope at P; which shows the decreasing velocity and depicts negative acceleration.

Non Uniform Motion

Velocity-Time Graph:
With the help of a velocity-time graph we get the knowledge about the acceleration and displacement of the particle.
1. Uniform velocity motion: Consider that an object is moving with uniform velocity v. Since the object is in uniform motion, the magnitude of it’s velocity at t = 0, t = 1s, t = 2s …. will always be v therefore graph between time and the velocity of the object will be as shown in the figure

Displacement of the object in a given time interval
Consider the two points A and B on the v – t graph corresponding to instants t₁ and t₂ respectively.
Then
Area ABB’A’= v(t₂ – t₁) …(1)

As given by equation (1), the displacement of the object in the time interval between t₁ and t₂ is
x₂ – x₁ = v(t₂ – t₁) ….(2)

It means the displacement of an object in the time interval (t₂ – t₁) is numerically equal to the area under velocity-time graph between the instant t₁ and t₂.

Note: It may be pointed out that this geometrical method of finding the displacement of an object holds good even in the case, when the object is moving with negative velocity. In such a case, the area below the velocity-time graph is taken as negative and corresponding to this, the displacement will also be negative.

2. Constant/uniform acceleration motion:
The slope of a velocity slope of the graph – tanθ – y/x time graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
The slope of a velocity-time graph will be given by the following formula.
Slope of graph = tanθ = y/x time = \(\frac{v_{2}-v_{1}}{t_{2}-t_{1}}=\frac{\Delta v}{\Delta t}\)

Since \(\frac{\Delta v}{\Delta t}\) is the definition of acceleration, the At the slope of a velocity-time graph must be equal to the acceleration of the object.
Slope = acceleration

This means that when the slope is steep, the object will be changing velocity rapidly. When the slope is shallow, the object will not be changing its velocity as rapidly. This also means that acceleration will be negative if the slope is negative. Directed downwards the acceleration will be negative and if the slope is positively directed upwards the acceleration will be positive.

Uniform Acceleration Motion

3. Non-Uniform Acceleration Motion: In the figure, the velocity-time graph shows not a straight line motion. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P which means that the acceleration at Q is more than the acceleration at P. In this way this graph shows non-uniform motion whose acceleration is increasing. Similarly, figure (b) shows the non-uniform motion in the

Non Uniform Acceleration Motion

velocity-time graph. But when the slopes are calculated at points P and Q, the slope at Q is less than the slope at P; which shows that the acceleration is decreasing.

Circular Sine Velocity-Time Graph:
In the figure, the velocity-time graph is a line graph. This type of motion generally depicts simple harmonic motion. In this type of motion velocity changes with time in the form of a sine function. By this graph we can easily understand positive and negative displacement. In the given graph from O to T and 2T to 3T time interval velocity is positive. Whereas, T to 2T time interval velocity is negative and the area (S₂) in the graph for this interval is also negative. Therefore, displacement from 0 to 3T time interval is:

Circular Sine Velocity Time Graph
S = S₁ – S₂ + S₃
Whereas distance covered in the time interval in 0 to 3 T is
S = S₁ + S₂ + S₃

Acceleration-Time Graph:
Similar to position-time graph and velocity-time graph acceleration-time graph also gives various information regarding the motion of a particle. In figure (a) acceleration is constant with time which depicts uniform accelerated motion. In figure (b) acceleration is changing with time which shows non-uniform accelerated motion. The change in the acceleration is a straight line or not in straight-line motion both shows non-uniform accelerated motion.

(a) Uniform Acceleration Motion

The area between the two-time intervals in the acceleration-time graph tells us about the change in velocity. As in the figure the shaded area between the time interval t₁ and t₂ tells about the change in velocity;
Δv = Area of shaded part.

(b) Non-Uniform Acceleration Motion

Physics Notes

Acceleration Physics Notes

Acceleration:
Generally, the velocity of a moving object changes with time. Sometimes the magnitude of velocity increases and sometimes it decreases. Sometimes the magnitude remains constant but the direction changes as in circular motion. The rate of change in velocity is defined as acceleration.

Therefore, “The rate of change of velocity of an object with respect to time is known as acceleration.”

In terms of the formula:
Acceleration = \(\frac{\text { Change in Velocity }}{\text { Time Taken }}\)
\(\vec{a}=\frac{\Delta \vec{v}}{\Delta t}\)

The unit of acceleration in M.K.S. system is meter/second 2 (m/s²) Its dimensional formula is [M⁰L¹F²]. Acceleration is a vector quantity. Similar to velocity it is also divided as follows:

Types of acceleration:
1. Average Acceleration: “The ratio of the total change in velocity to the total time taken is called average acceleration”.

If Δv is the change in velocity in it time interval, then;

Average acceleration (a) = \(\frac{\text { Total Change in Velocity }}{\text { Total Time Taken }}\)
= \(\left[\frac{\Delta v}{\Delta t}\right]\)

2. Instantaneous Acceleration:
Instantaneous acceleration is defined as “acceleration at any given point or at any instant of time.” If at Δt time interval velocity is Δv then according to the above definition; to calculate instantaneous acceleration
Δt → 0. Hence,

Here, \(\frac{d v}{d t}\), differentiation of v with respect to time t which can be known mathematically.

Here \(\frac{d^{2} x}{d t^{2}}\) double differentiation of x w.r.t. t which can he calculated mathematically.

Therefore, instantaneous acceleration is the differentiation of velocity with respect to time and is double differentiation of displacement w.r.t. time.

For any moving object at definite time intervals, if the change in velocity is also the same then this is known as uniform acceleration. And if the changes are different in velocity then this is non-uniform acceleration. In the same accelerated motion, average acceleration and instantaneous acceleration are the same.

If in any circular motion (path), the magnitude of the velocity of the moving object does not change but the direction of the moving object changes continuously, they this type of motion is also called accelerated motion.

Acceleration can be positive, negative or zero. If acceleration is positive its velocity increases. If acceleration is zero then the object moves with a constant speed (velocity). And if the acceleration is negative then the velocity of the object decreases. Hence, negative acceleration is called retardation.

Physics Notes

Speed and Velocity Physics Notes

Speed and Velocity:
Speed: The speed of something is the rate at which it moves or travels. Speed is defined as the rate of movement of a body expressed either as the distance traveled divided by the time taken or the rate of change of position with respect to time at a particular point. It is a scalar quantity that refers to “how fast an object is moving.” “The time rate of distance is called speed.
Speed = \(\frac{\text { distance }}{\text { time }}\)
The unit of speed is m/s. The dimensional formula of speed is [M⁰LT^-1]

Velocity: Velocity is the vector quantity that refers to “the rate at which an object changes its position.” Imagine a person moving rapidly one step forward and one step back-always returning to the original starting position. While this might result in a frenzy of activity. It would result in zero velocity. Because the person always returns to the original position.

The time rate of displacement is called velocity
velocity = displacement time

The following points may be noted about the speed of an object moving along a straight line:

• Speed is a scalar quantity. The magnitude of the velocity of the object is called its speed.
• The speed of an object in a particular direction is called the velocity of the object.
• The speed of an object has the same unit as that of velocity.

Types of Speed:
(a) Uniform speed: If a body covers equal distance in equal time intervals, howsoever small these intervals of time maybe is called uniform speed.

(b) Variable speed: If a body covers equal distances in unequal intervals of time or unequal distances in equal intervals of time, howsoever small these intervals of time maybe is called variable speed.

(c) Average speed: Average speed of a body is defined as the total distance travelled divided by the total time taken.
v_av = \(\frac{\Delta x}{\Delta t}=\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\)

(d) Instantaneous speed:
The speed of a body at any instant of time is called instantaneous speed. When we say about speed then it means about instantaneous speed. For this time interval should be very less or Δt → 0

The instantaneous speed of the vehicle is measured by the speedometer.

Types of velocity:
(a) Uniform velocity: It is defined as the ratio of the displacement to the time taken by the object to cover the displacement.
Uniform velocity = \(\frac{\text { displacement }}{\text { timeinterval }}\)

(b) Variable velocity: A body is said to be in variable velocity if it covers equal displacement in unequal intervals of time.

(c) Average velocity: Average velocity of a body is defined as the change in position or displacement (Δx) divided by time interval (Δt) in which that displacement occurs.
\(\overrightarrow{v_{a v}}=\frac{\Delta \vec{x}}{\Delta t}=\frac{\overrightarrow{x_{2}}-\overrightarrow{x_{1}}}{t_{2}-t_{1}}\)

(d) Instantaneous velocity: The instantaneous velocity of a body is the velocity of the body at any instant of time or at any point of its path.

\(\vec{v}=\frac{d \vec{x}}{d t}\)
Velocity can be positive, negative or zero.

By studying speed and velocity we come to the result that at any time interval average speed of an object is equal to or more than the average velocity but instantaneous speed is equal to instantaneous velocity.

Physics Notes

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