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GNDU Question Paper-2021
BA/Bsc
1
st
Semester (Batch 2024-28) (CBGS)
PHYSICS: Paper-A
(Mechanics)
Time Allowed: Three Hours Max. Marks:35
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. Derive expression for the acceleration of a particle moving in three dimensional space
in spherical polar co-ordinates.
2. (a) Define homogeneity of time. Prove that the law of conservation of energy follows
from the homogeneity of time.
(b) Define solid angle. Prove that the solid angle subtended by a sphere at its centre is
steradian.
SECTION-B
3. State and prove Kepler's law of planetary motion.
4. (a) Define a conservative force. Give one example. How a conservative force is related
to potential energy?
(b) Show that the total energy E is constant of motion under the central force field.
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SECTION-C
5. (a) What are inertial and non-inertial frames of reference? Is earth an inertial frame of
reference?
(b) Show that the laws of conservation of linear momentum and energy are invariant
under Galilean transformations.
6. (a) What is Coriolis force? Name its two geographical consequences.
(b) Discuss the effect of Coriolis force on the free fall of a body from a height h above the
surface of earth.
SECTION-D
7. (a) Define elastic collision and in-elastic collision.
(b) Define centre of mass system. Prove that in centre of mass system, the magnitude of
the velocities of the particles remains unaltered in elastic collision.
8. Derive Euler's equations of rotation of rigid body about a fixed point.
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GNDU Answer Paper-2021
Bachelor of Computer Application (BCA) (Hons.)
1
st
Semester (Batch 2024-28) (CBGS)
PHYSICS: Paper-A
(Mechanics)
Time Allowed: Three Hours Max. Marks:35
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. Derive expression for the acceleration of a particle moving in three dimensional space
in spherical polar co-ordinates.
Ans: Introduction
When we study the motion of a particle in three-dimensional (3D) space, we need a system
that can describe its position easily. Normally, we use Cartesian coordinates (x, y, z).
However, when the particle moves around a central point (such as a planet revolving around
the Sun or an electron moving around the nucleus), spherical polar coordinates become
much more convenient.
In spherical polar coordinates, the position of a particle is described using three quantities:
Distance of the particle from the origin.
Polar angle (angle made with the positive z-axis).
Azimuthal angle (angle measured in the x-y plane from the positive x-axis).
Unlike Cartesian coordinates, these directions themselves change continuously as the
particle moves. Therefore, deriving acceleration becomes more complicated.
Diagram of Spherical Polar Coordinates
Z-axis
↑
|
P(r,θ,φ)
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/|
/ |
r / |
/θ |
/ |
O---------------------→ X-axis
\
\
\ φ
\
Y-axis
Where:
O = Origin
OP = r (radius vector)
θ = Angle with Z-axis
φ = Angle in X-Y plane
Step 1: Position Vector
The position vector of the particle is
where
= Radial unit vector
= Distance from the origin
Step 2: Velocity
Differentiate the position vector with respect to time.
Using the product rule,
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Since the unit vector also changes direction,
Therefore,
This shows that velocity has three components:
Radial velocity
Polar (θ) velocity
Azimuthal (φ) velocity
Step 3: Acceleration
Acceleration is the time derivative of velocity.
After differentiating every term carefully and considering the changing unit vectors, we
obtain the standard expression:
Meaning of Each Component
1. Radial Component
This represents acceleration towards or away from the origin.
= Change in radial speed
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The remaining terms are centripetal accelerations, caused by motion along curved
paths.
2. Polar Component
This acts in the direction of changing θ.
It includes:
Angular acceleration in θ.
Coupling between radial and angular motion.
Effect of rotation in the φ-direction.
3. Azimuthal Component
This represents acceleration around the z-axis.
It depends on:
Angular acceleration in φ.
Change in radius.
Simultaneous change of θ and φ.
Why is this Formula So Complex?
In Cartesian coordinates, the directions x, y, and z never change. But in spherical
coordinates:
The particle's position changes.
The direction of
changes.
The direction of
changes.
The direction of
also changes.
Therefore, while differentiating velocity, we must differentiate both the magnitudes and
the directions of the unit vectors. This is why additional terms appear in the acceleration
formula.
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Applications
The acceleration formula in spherical polar coordinates is widely used in:
Motion of planets around the Sun.
Satellite motion around Earth.
Spacecraft trajectory calculations.
Atomic and quantum physics.
Electromagnetic field analysis.
Celestial mechanics.
Conclusion
The acceleration of a particle in spherical polar coordinates has three componentsradial,
polar, and azimuthal. Unlike Cartesian coordinates, the unit vectors themselves change with
time, making the derivation more involved. The final expression combines the effects of
radial motion, angular motion, and the changing directions of the coordinate system. This
form of acceleration is especially useful for studying objects moving along curved or circular
paths, making it an essential concept in advanced mechanics, astronomy, and engineering.
2. (a) Define homogeneity of time. Prove that the law of conservation of energy follows
from the homogeneity of time.
(b) Define solid angle. Prove that the solid angle subtended by a sphere at its centre is
steradian.
Ans: 2. (a) Homogeneity of Time and Law of Conservation of Energy
What is Homogeneity of Time?
The homogeneity of time means that time is the same everywhere and at every moment.
In simple words, the laws of physics do not change with time.
For example:
If you throw a ball upward today, it follows the same laws of motion.
If you perform the same experiment tomorrow, next year, or even after 100 years
(under the same conditions), the result will remain the same.
This shows that nature does not care about the date or time. The physical laws are
constant. This property is called homogeneity of time.
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How Does It Lead to the Law of Conservation of Energy?
The Law of Conservation of Energy states:
Energy can neither be created nor destroyed. It can only change from one form to
another.
The connection between homogeneity of time and conservation of energy was explained by
the famous mathematician Emmy Noether through Noether's Theorem.
Let us understand it in a simple way.
Imagine you have a perfectly working pendulum.
If the laws of physics changed every minute, then the pendulum could suddenly start
moving faster or slower without any reason.
This would mean energy is being created or destroyed randomly.
But in reality, this never happens.
Since the laws of physics remain exactly the same at every instant of time (homogeneity of
time), the total energy of an isolated system also remains constant.
For example:
Suppose a ball is dropped from a height.
At the top, it has Potential Energy.
While falling, Potential Energy decreases.
At the same time, Kinetic Energy increases.
Although the form of energy changes, the total energy remains constant.
Therefore,
Homogeneity of Time → Physical laws remain unchanged → Total Energy remains
constant → Conservation of Energy.
Diagram
Ball at Height
●
Potential Energy (Maximum)
│
│ Falling
▼
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●
Potential Energy ↓
Kinetic Energy ↑
│
▼
Ground
Kinetic Energy (Maximum)
Total Energy = Constant
Key Points
Homogeneity means "same everywhere in time."
Physical laws do not depend on when an experiment is performed.
Because the laws remain unchanged, energy cannot appear or disappear on its own.
Therefore, energy is always conserved.
2. (b) Solid Angle and Solid Angle Subtended by a Sphere
What is an Angle?
Before understanding a solid angle, first recall an ordinary angle.
A normal angle is formed between two lines and is measured in radians.
A solid angle is the three-dimensional version of an angle.
Instead of two lines, it is formed by a three-dimensional object as seen from a point.
Definition of Solid Angle
A solid angle is the angle subtended by a surface at a point in three-dimensional space.
It is represented by the symbol Ω (Omega).
Its SI unit is the steradian (sr).
The formula for solid angle is
where
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Ω = Solid angle
A = Area of the surface
r = Radius from the observation point
Understanding the Formula
Imagine you are standing at the center of a sphere.
Every part of the sphere surrounds you equally.
The larger the surface area, the larger the solid angle.
Proof that the Solid Angle of a Sphere at its Centre is 4π Steradian
For a complete sphere,
Surface area of sphere
Using the solid angle formula,
Substituting the value of A,
Cancelling
,
Hence,
steradian
Therefore,
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The solid angle subtended by a complete sphere at its centre is steradian.
Diagram
***********
**** ****
*** ***
** ● O **
** (Centre) **
** **
*** ***
**** ****
*******
Every point on the sphere is seen from O.
Surface Area = 4πr²
Solid Angle = A/r² = 4π sr
Important Points for Examination
Homogeneity of Time
Time has the same properties at every instant.
Physical laws do not change with time.
It leads to the conservation of energy.
Explained mathematically by Noether's Theorem.
Conservation of Energy
Energy cannot be created or destroyed.
It only changes from one form to another.
Total energy of an isolated system always remains constant.
Solid Angle
Three-dimensional angle.
Symbol: Ω.
SI Unit: Steradian (sr).
Formula: Ω = A/r².
Sphere at Centre
Surface area of sphere = 4πr².
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Solid angle = 4π steradian.
Conclusion
The idea of homogeneity of time tells us that nature behaves consistently, no matter when
an experiment is performed. Because the laws of physics never change with time, the total
energy of an isolated system always remains constant, giving rise to the law of
conservation of energy. Likewise, a solid angle helps us measure how large a three-
dimensional object appears from a point. Since a complete sphere surrounds its centre in
every direction, it subtends the maximum possible solid angle of steradians. These two
concepts are fundamental in physics because they explain why natural laws are reliable and
how we describe objects in three-dimensional space.
SECTION-B
3. State and prove Kepler's law of planetary motion.
Ans: Introduction
Kepler's Laws of Planetary Motion are three important laws that describe how planets move
around the Sun. These laws were discovered by the German astronomer Johannes Kepler in
the early 17th century after studying many years of observations made by Tycho Brahe.
Before Kepler, people believed that planets moved in perfect circles. However, Kepler
proved that planets actually move in elliptical paths.
Later, Sir Isaac Newton explained why these laws are true using his Law of Universal
Gravitation and the Laws of Motion.
Kepler's Three Laws
1. First Law (Law of Orbits)
Statement:
Every planet moves around the Sun in an elliptical orbit, with the Sun located at one of
the two foci of the ellipse.
Simple Explanation
Imagine stretching a rubber band around two pins on a board and drawing a curve. The
shape formed is called an ellipse.
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Unlike a circle, an ellipse has two fixed points called foci.
A planet does not travel in a perfect circle. Instead, it follows this oval-shaped path, and the
Sun lies at one focus, not at the center.
Diagram
Planet
●
.-------------.
.-' '-.
.' '.
/ \
| ☀ Sun (Focus F1) |
| |
\ Focus F2 /
'. .'
'-. .-'
'-------------'
2. Second Law (Law of Areas)
Statement:
The line joining the Sun and a planet sweeps out equal areas in equal intervals of time.
Simple Explanation
Suppose a planet moves around the Sun.
When the planet is close to the Sun, the Sun's gravitational pull is stronger.
Therefore, the planet moves faster.
When the planet is far from the Sun, gravity becomes weaker.
Therefore, the planet moves slower.
Even though the speed changes, the area covered by the line joining the Sun and the planet
during equal intervals of time remains exactly the same.
Diagram
P2
●
/ |
/ |
/ |
☀ Sun------● P1
Area A1
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-------------------------
● P4
/ |
/ |
/ |
☀ Sun------● P3
Area A2
A1 = A2 (Equal areas in equal time)
3. Third Law (Law of Periods)
Statement:
The square of the time period (T²) of a planet is directly proportional to the cube of the
semi-major axis (r³) of its orbit.
Mathematically,
or
Constant
Simple Explanation
T = Time taken by a planet to complete one revolution around the Sun.
r = Average distance (semi-major axis) from the Sun.
This law tells us:
Planets that are farther from the Sun take more time to complete one revolution.
Planets that are closer to the Sun complete their revolution much faster.
For example:
Mercury completes one revolution in about 88 days.
Earth takes 365 days.
Neptune takes about 165 years.
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Thus, greater distance means a longer orbital period.
Proof of Kepler's Third Law
Using Newton's Law of Gravitation.
The gravitational force between the Sun and a planet is
where
G = Universal gravitational constant
M = Mass of the Sun
m = Mass of the planet
r = Distance between the Sun and the planet
The centripetal force needed to keep the planet moving in its orbit is
Since gravitational force provides the centripetal force,
Cancelling m
The orbital speed is
Substituting,
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Simplifying,
Therefore,
or
Since G and M remain constant for all planets revolving around the same Sun,
Hence, Kepler's Third Law is proved.
Why Are Kepler's Laws Important?
Kepler's laws are extremely important because they help scientists understand the motion
of planets, moons, satellites, and even comets. These laws are also used to:
Calculate the orbital period of planets and artificial satellites.
Predict the position of celestial bodies.
Design satellite missions and space exploration programs.
Understand the gravitational relationship between the Sun and planets.
Form the foundation for Newton's theory of gravitation.
Key Points to Remember
First Law: Planets move in elliptical orbits, with the Sun at one focus.
Second Law: A planet moves faster when closer to the Sun and slower when farther
away, while sweeping equal areas in equal times.
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Third Law: The square of the orbital period is proportional to the cube of the
average orbital radius, i.e.,
Newton's Law of Gravitation provides the mathematical proof of the third law.
These laws are fundamental to astronomy, astrophysics, satellite technology, and
space science.
Conclusion:
Kepler's three laws transformed our understanding of the Solar System by showing that
planets move in elliptical orbits, change their speed depending on their distance from the
Sun, and follow a precise mathematical relationship between their orbital period and
distance. Newton later explained these observations through the force of gravity, making
Kepler's laws one of the cornerstones of modern physics and astronomy.
4. (a) Define a conservative force. Give one example. How a conservative force is related
to potential energy?
(b) Show that the total energy E is constant of motion under the central force field.
Ans: Introduction
In Physics, whenever an object moves, some force acts on it. Sometimes the force only
depends on the starting and ending positions of the object, while other times it depends on
the path taken. A conservative force is a very special type of force because it allows energy
to change from one form to another without any loss. This concept helps us understand why
planets keep revolving around the Sun, why pendulums keep swinging, and why objects fall
towards the Earth.
(a) What is a Conservative Force?
A conservative force is a force for which the work done depends only on the initial and
final positions of the object and not on the path followed.
In simple words, imagine you are climbing to the roof of a building.
You can use the stairs.
You can use a lift.
You can climb a ladder.
Although the paths are different, your final height is the same. Therefore, the work done
against gravity remains the same.
This is why gravity is called a conservative force.
Definition
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A conservative force is a force whose work done between two points is independent of
the path followed and depends only on the initial and final positions.
Examples of Conservative Forces
Some common examples are:
1. Gravitational Force (Earth attracting objects)
2. Spring Force (Elastic force)
3. Electrostatic Force (Force between electric charges)
Among these, gravitational force is the easiest and most common example.
Diagram of Conservative Force
B
●
/ \
/ \
/ \
A ●-------------● C
Different paths:
A → B
A → C
A → B → C
Work done by a conservative force
depends only on the starting and
ending points, not on the path.
Relationship Between Conservative Force and Potential Energy
One of the most important properties of a conservative force is that it has potential energy
associated with it.
Whenever a conservative force does work,
Potential Energy decreases.
Kinetic Energy increases.
Similarly,
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If Potential Energy increases,
Kinetic Energy decreases.
The mathematical relation is
where
W = Work done by conservative force
U = Potential Energy
ΔU = Change in Potential Energy
The negative sign means:
If the force does positive work, potential energy decreases.
If potential energy increases, the force does negative work.
Example
Suppose you hold a ball at a height.
At the top, the ball has maximum potential energy.
When released, gravity pulls it downward.
Potential energy decreases.
Kinetic energy increases.
Just before reaching the ground, almost all potential energy becomes kinetic energy.
Thus, energy is only converted from one form to another.
(b) Show that Total Energy Remains Constant Under a Central Force Field
What is a Central Force?
A central force is a force that always acts along the line joining the object and a fixed centre.
Its magnitude depends only on the distance from the centre.
Examples:
Gravitational force between the Earth and the Sun.
Electrostatic force between two charges.
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Diagram of Central Force
Planet
●
/
/
/
/
● Sun (Centre)
Force always acts
towards the centre.
Total Mechanical Energy
The total energy of a particle is
where
K = Kinetic Energy
U = Potential Energy
Since
therefore,
where U(r) is the potential energy depending only on the distance .
Proof
For a conservative (central) force,
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From the Work-Energy Theorem,
Therefore,
or
This becomes
Hence,
Constant
Therefore,
Constant
Thus, the total mechanical energy of a particle moving under a central force remains
constant. Although kinetic and potential energies may continuously change into one
another, their sum never changes as long as only the conservative central force acts on the
particle.
Real-Life Example
Think about a roller coaster.
At the highest point, it moves slowly but has maximum potential energy.
As it comes down, potential energy decreases.
Its speed increases, so kinetic energy increases.
When it climbs another hill, kinetic energy decreases while potential energy
increases again.
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Ignoring friction, the total energy remains constant throughout the ride. The same principle
applies to planets orbiting the Sun under the gravitational (central) force.
Key Points to Remember
A conservative force does work that depends only on the starting and ending
positions, not on the path taken.
Examples include gravitational force, spring force, and electrostatic force.
Conservative forces are associated with potential energy.
The relation between work and potential energy is:
A central force always acts toward or away from a fixed centre and depends only on
the distance from that centre.
Under a central (conservative) force, kinetic energy and potential energy keep
converting into each other, but their sum remains constant:
Constant
This is known as the Law of Conservation of Mechanical Energy, and it explains the
motion of falling objects, pendulums, satellites, and planets.
SECTION-C
5. (a) What are inertial and non-inertial frames of reference? Is earth an inertial frame of
reference?
(b) Show that the laws of conservation of linear momentum and energy are invariant
under Galilean transformations.
Ans: Introduction
Whenever we study the motion of an object, we need a frame of reference. A frame of
reference is simply the place or observer from which motion is measured. For example,
when you say that a bus is moving at 60 km/h, you are measuring its motion with respect to
the road. The road acts as the frame of reference.
There are two main types of frames of reference:
1. Inertial Frame of Reference
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2. Non-Inertial Frame of Reference
1. Inertial Frame of Reference
An inertial frame of reference is a frame that is either at rest or moving with a constant
velocity (constant speed in a straight line).
In such a frame, Newton's First Law of Motion is perfectly valid.
Newton's First Law states:
"A body remains at rest or continues to move with uniform velocity in a straight line unless
acted upon by an external force."
This means if no external force acts on an object, it will neither speed up nor slow down nor
change its direction.
Example
Imagine a train moving smoothly on a straight track at a constant speed.
------------------------------->
Train moving with constant speed
Passenger
O
/|\
/ \
A passenger sitting inside feels normal because there is no acceleration. If he throws a ball
upward, it comes back into his hand. Therefore, the train behaves like an inertial frame.
Characteristics of an Inertial Frame
It is either at rest or moves with constant velocity.
Newton's laws are valid.
No imaginary (pseudo) forces are required.
Objects obey the law of inertia.
2. Non-Inertial Frame of Reference
A non-inertial frame is a frame that is accelerating or rotating.
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Since the frame itself is changing its motion, Newton's laws do not work directly unless we
introduce an imaginary force called the pseudo force.
Example 1: Accelerating Bus
Suppose you are standing inside a bus.
Bus accelerates →
____________
| |
| O ← |
| /|\ |
| / \ |
|____________|
When the bus suddenly moves forward, you feel as if you are pushed backward.
Actually, nobody pushes you backward.
Your body wants to remain at rest due to inertia while the bus moves forward.
This backward effect is explained using a pseudo force.
Example 2: Turning Car
When a car suddenly turns left, passengers feel pushed towards the right.
Again, no real force pushes them.
The car is a non-inertial frame because it is changing direction.
Characteristics of a Non-Inertial Frame
It has acceleration or rotation.
Newton's laws do not directly apply.
Pseudo forces appear.
Motion appears different from reality.
Is Earth an Inertial Frame?
The answer is No, not exactly.
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Earth is continuously:
Rotating about its own axis.
Revolving around the Sun.
Both of these motions involve acceleration.
Therefore, Earth is technically a non-inertial frame.
However, the acceleration due to Earth's rotation and revolution is very small.
For most laboratory experiments and everyday problems, we ignore these effects.
Hence,
Practically, Earth is treated as an inertial frame of reference.
Conclusion
Theoretically: Earth is a non-inertial frame.
Practically: Earth is considered an inertial frame because its acceleration is very
small.
(b) Show that the Laws of Conservation of Linear Momentum and Energy are Invariant
under Galilean Transformations
What is Galilean Transformation?
Galilean transformation tells us how the position, velocity, and time of an object change
when observed from two different inertial frames moving with constant velocity relative to
each other.
Suppose there are two observers.
Frame S (Stationary)
---------------------------->
Frame S' moving with velocity V
---------------------------->
If
S = stationary frame
S = moving frame
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Relative velocity = V
Then,
x' = x − Vt
y' = y
z' = z
t' = t
The time remains the same for both observers.
Velocity transforms as:
u' = u − V
where:
= velocity in frame S
= velocity in frame S
= relative velocity between the frames
Conservation of Linear Momentum
What is Linear Momentum?
Linear momentum is the quantity of motion possessed by a body.
It is given by:
where:
= mass
= velocity
Statement of Law
The total linear momentum of an isolated system remains constant if no external force acts
on it.
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Suppose two bodies collide.
Before collision
Body A ----->
Body B <-----
After collision
Body A <----
Body B ----->
Total momentum before collision
After collision
Since momentum is conserved,
Applying Galilean Transformation
In the moving frame,
Similarly,
Substituting into the momentum equation:
Expanding,
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The common term
cancels from both sides.
We again obtain
Thus, the law of conservation of linear momentum has the same form in both inertial
frames. Therefore, it is invariant under Galilean transformation.
Conservation of Energy
What is Energy?
Energy is the ability of a body to do work.
The kinetic energy of a body is
Statement of Law
The total energy of an isolated system remains constant. Energy can neither be created nor
destroyed; it can only change from one form to another.
Under Galilean Transformation
Velocity changes according to
Hence,
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The value of kinetic energy changes because the measured velocity changes from one frame
to another. However, all observers apply the same law of conservation of energy: the total
energy before an interaction equals the total energy after it.
Therefore,
Total Energy Before Total Energy After
holds true in every inertial frame related by a Galilean transformation.
Final Conclusion
An inertial frame is at rest or moves with constant velocity, and Newton's laws are
directly valid.
A non-inertial frame accelerates or rotates, so pseudo forces must be introduced.
Earth is theoretically a non-inertial frame because it rotates and revolves around
the Sun, but for most practical calculations it is treated as an inertial frame.
A Galilean transformation relates two inertial frames moving with constant relative
velocity.
The law of conservation of linear momentum remains unchanged after applying a
Galilean transformation, proving that it is Galilean invariant.
The law of conservation of energy also retains the same conservation principle in all
inertial frames, even though the numerical value of kinetic energy depends on the
observer's frame. Thus, the conservation law itself is invariant under Galilean
transformations.
6. (a) What is Coriolis force? Name its two geographical consequences.
(b) Discuss the effect of Coriolis force on the free fall of a body from a height h above the
surface of earth.
Ans: Introduction
When we stand on Earth, it looks completely still. We feel as if the ground beneath us never
moves. But in reality, the Earth is rotating continuously from west to east. It completes one
full rotation in about 24 hours. Because of this rotation, moving objects such as winds,
ocean currents, airplanes, missiles, and even a freely falling object do not always move in a
perfectly straight path. Instead, they appear to bend from their original path. This apparent
bending is caused by a special force called the Coriolis Force.
The Coriolis force is not a real pushing or pulling force like gravity. It is an apparent (or
fictitious) force that appears only because we are observing motion from the rotating Earth.
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(a) What is Coriolis Force?
Definition:
Coriolis Force is the apparent force that acts on moving objects due to the rotation of the
Earth. It changes only the direction of motion, not the speed of the object.
A simple way to understand it is to imagine sitting on a rotating merry-go-round. If you
throw a ball straight ahead, another person standing on the ground will see the ball moving
straight. But to you, sitting on the rotating platform, the ball appears to curve. A similar
effect happens on Earth because our planet is constantly rotating.
Direction of Deflection
In the Northern Hemisphere, moving objects are deflected towards the right.
In the Southern Hemisphere, moving objects are deflected towards the left.
This deflection becomes stronger near the poles and becomes zero at the equator.
Diagram of Coriolis Force
North Pole
▲
|
Right Deflection
/
Moving Object
\
Left Deflection
|
▼
South Pole
Or simply remember:
Northern Hemisphere → Right Deflection
Southern Hemisphere → Left Deflection
Two Geographical Consequences of Coriolis Force
The Coriolis force greatly affects many natural phenomena on Earth.
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1. Deflection of Winds
Air always moves from high-pressure areas to low-pressure areas. Due to the Coriolis force,
these winds do not travel in straight lines.
In the Northern Hemisphere, winds bend to the right.
In the Southern Hemisphere, winds bend to the left.
Because of this bending, different wind systems such as trade winds, westerlies, and polar
winds are formed.
2. Rotation of Cyclones and Ocean Currents
The Coriolis force causes cyclones and ocean currents to rotate.
Cyclones rotate anticlockwise in the Northern Hemisphere.
Cyclones rotate clockwise in the Southern Hemisphere.
Similarly, major ocean currents also change their direction because of the Earth's rotation.
(b) Effect of Coriolis Force on the Free Fall of a Body
Now suppose a person drops a stone from the top of a very tall tower.
Without Earth's rotation, we would expect the stone to fall exactly at the foot of the tower.
However, because the Earth is rotating, something interesting happens.
Why Does the Stone Deviate?
The top of the tower is farther from Earth's centre than the ground.
Since the top has a slightly larger radius, it moves eastward slightly faster than the ground
due to Earth's rotation.
When the stone is released, it already has this slightly higher eastward speed.
As it falls downward, it keeps that extra eastward velocity.
Therefore, when it reaches the ground, it lands a little east of the point directly below
where it was dropped.
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Diagram
Tower
● Stone
|
|
|
|
|
-------------X---------------- Earth Surface
|
|
Actual Landing Point
\
\
● (Slightly East)
Earth rotates from West → East
Explanation in Simple Words
Imagine you are standing on the roof of a moving train and you drop a ball.
The ball already has the same forward speed as the train.
Even while falling, it continues moving forward with the train and lands slightly ahead rather
than straight down.
Similarly, the Earth is rotating continuously. The stone already possesses the Earth's
eastward rotational speed when released. Since the top of the tower moves slightly faster
than the bottom, the stone carries this extra speed throughout its fall and lands a little
towards the east.
This small deviation is caused by the Coriolis effect resulting from Earth's rotation.
Important Points to Remember
Earth rotates from west to east.
Coriolis force is an apparent force due to Earth's rotation.
It changes the direction of moving objects but not their speed.
Deflection is:
o Right in the Northern Hemisphere.
o Left in the Southern Hemisphere.
The effect is maximum at the poles and zero at the equator.
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A freely falling body lands slightly east of the vertical line because it retains the
higher eastward velocity it had at the top of the tower.
Conclusion
The Coriolis force is an important consequence of the Earth's rotation. Although it cannot be
felt directly, its effects are seen everywhere in nature. It controls the direction of winds,
ocean currents, and the rotation of cyclones. Even a freely falling object does not land
exactly beneath its starting point. Instead, because the upper part of the Earth moves
slightly faster than the lower part, the object lands a little towards the east. Understanding
the Coriolis force helps us explain many geographical and physical phenomena occurring on
our rotating planet.
SECTION-D
7. (a) Define elastic collision and in-elastic collision.
(b) Define centre of mass system. Prove that in centre of mass system, the magnitude of
the velocities of the particles remains unaltered in elastic collision.
Ans: (a) Elastic Collision and Inelastic Collision
In our daily life, we often see objects colliding with each other. For example, two cricket
balls hitting each other, a football bouncing from a wall, or two vehicles meeting in an
accident. Such events are called collisions. Depending on how energy behaves during the
collision, collisions are of two types: Elastic Collision and Inelastic Collision.
1. Elastic Collision
An elastic collision is a collision in which both momentum and kinetic energy remain
conserved.
This means:
Total momentum before collision = Total momentum after collision.
Total kinetic energy before collision = Total kinetic energy after collision.
No kinetic energy is lost as heat, sound, or deformation.
Examples
Collision between billiard balls.
Collision between gas molecules.
Collision of steel balls in Newton's cradle.
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Characteristics
Momentum is conserved.
Kinetic energy is conserved.
Objects usually bounce back after collision.
No permanent deformation occurs.
2. Inelastic Collision
An inelastic collision is a collision in which momentum is conserved but kinetic energy is
not conserved.
Some kinetic energy changes into:
Heat
Sound
Light
Deformation of objects
Examples
Car accidents.
A lump of clay striking a wall.
A bullet getting embedded in a wooden block.
Characteristics
Momentum is conserved.
Kinetic energy decreases.
Some energy changes into other forms.
Objects may stick together after collision (perfectly inelastic collision).
Difference Between Elastic and Inelastic Collision
Elastic Collision
Inelastic Collision
Momentum is conserved
Momentum is conserved
Kinetic energy is conserved
Kinetic energy is not conserved
No energy loss
Some energy changes into heat, sound, etc.
Objects rebound
Objects may stick together
(b) Centre of Mass (COM) System
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Before understanding the Centre of Mass System, imagine two students standing on a
moving skateboard. Instead of watching each student separately, suppose you imagine one
single point that represents the average position of both students. This imaginary point is
called the Centre of Mass (COM).
The Centre of Mass behaves as if the entire mass of the system is concentrated at one
point.
Definition
The Centre of Mass System is a reference frame in which the centre of mass of the system
remains at rest.
In this frame,
Total momentum of the system is zero.
One particle moves in one direction while the other moves in the opposite direction.
Diagram of Centre of Mass System
Before Collision (COM Frame)
Particle A COM Particle B
← u₁ ● u₂ →
Total Momentum = 0
After Collision
Particle A COM Particle B
v₁ → ● ← v₂
Total Momentum = 0
The particles simply reverse or change their directions, but the Centre of Mass remains
stationary.
Why Do We Study the COM System?
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The Centre of Mass System makes collision problems much easier because:
The total momentum is always zero.
Only the directions of motion change during an elastic collision.
The mathematical calculations become simple.
Proof: In the COM System, the Magnitude of Velocities Remains Unchanged in an Elastic
Collision
Consider two particles having masses m₁ and m₂.
Before collision:
Velocities = u₁ and u₂
After collision:
Velocities = v₁ and v₂
Step 1: Conservation of Momentum
Since the Centre of Mass is at rest,
Total momentum before collision = 0
Similarly,
Step 2: Conservation of Kinetic Energy
Because the collision is elastic,
Total kinetic energy before collision = Total kinetic energy after collision.
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Multiply both sides by 2,
Step 3: Using Momentum Relation
From momentum conservation,
and
Substituting these relations into the kinetic energy equation shows that the speed
(magnitude of velocity) of each particle remains the same before and after the collision.
Only the direction of motion changes.
Therefore,
and
Hence, the magnitudes of the velocities remain unchanged in the Centre of Mass System
during an elastic collision.
Simple Understanding of the Proof
Imagine two identical rubber balls moving toward each other with equal speed.
Before Collision
A -----> ● COM <----- B
Speed = 5 m/s Speed = 5 m/s
After collision:
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After Collision
A <----- ● COM -----> B
Speed = 5 m/s Speed = 5 m/s
Notice carefully:
Before collision, each ball moves at 5 m/s.
After collision, each ball still moves at 5 m/s.
Only the direction has changed.
This is exactly what happens in the Centre of Mass System during an elastic collision.
Key Points to Remember (Exam Revision)
Elastic Collision: Momentum and kinetic energy are both conserved.
Inelastic Collision: Momentum is conserved, but kinetic energy is not.
Centre of Mass (COM): A point representing the average position of the total mass
of the system.
COM Frame: The centre of mass remains at rest, and the total momentum is zero.
Elastic Collision in COM Frame: The magnitudes of the velocities of both particles
remain unchanged; only their directions change.
Reason: Since both momentum and kinetic energy are conserved in the COM frame,
the particles cannot gain or lose speedthey simply reverse or alter their directions.
This is why the Centre of Mass System is one of the most useful reference frames for
studying collisions in mechanics.
8. Derive Euler's equations of rotation of rigid body about a fixed point.
Ans: Introduction
Euler's equations describe how a rigid body rotates about a fixed point when external
forces or torques act on it. These equations were developed by the famous mathematician
Leonhard Euler and are very important in engineering, physics, robotics, satellites, aircraft,
and mechanical systems.
To understand Euler's equations, imagine a spinning top. When you spin it, it rotates around
a fixed point touching the ground. As it spins, gravity applies a torque, changing its motion.
Euler's equations explain exactly how this rotational motion changes with time.
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What is a Rigid Body?
A rigid body is an object whose shape and size never change, even when forces are applied.
Examples:
A cricket bat
A spinning wheel
A ceiling fan
A bicycle wheel
A satellite in space
The distance between any two particles inside a rigid body always remains constant.
What is a Fixed Point?
A fixed point is a point that does not move during rotation.
Example:
The center of a ceiling fan remains fixed while the blades rotate.
The bottom point of a spinning top touching the ground is approximately fixed.
Important Terms
1. Angular Velocity (ω)
Angular velocity tells how fast an object rotates.
It has three components:
ω₁ → Rotation about X-axis
ω₂ → Rotation about Y-axis
ω₃ → Rotation about Z-axis
2. Moment of Inertia (I)
Moment of inertia is the rotational equivalent of mass.
It tells how difficult it is to rotate an object.
For the three principal axes:
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I₁ = Moment of inertia about X-axis
I₂ = Moment of inertia about Y-axis
I₃ = Moment of inertia about Z-axis
A larger moment of inertia means the object is harder to rotate.
3. Torque (N)
Torque is the turning effect of a force.
Examples:
Opening a door.
Tightening a bolt with a wrench.
Pedaling a bicycle.
The torque components are:
N₁
N₂
N₃
Diagram
Z-axis (ω)
↑
|
|
● Fixed Point
/ | \
/ | \
/ | \
Y-axis ←──────┼──────→ X-axis
(ω) (ω)
Rigid body rotating about the fixed point
Derivation of Euler's Equations
For a rigid body rotating about a fixed point, Newton's Second Law for rotational motion
states:
Torque = Rate of change of Angular Momentum
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The angular momentum along the principal axes is:
L₁ = I₁ω₁
L₂ = I₂ω₂
L₃ = I₃ω₃
When the body rotates, the angular momentum changes because:
Angular velocity changes.
The direction of the rotating axes also changes.
Considering both effects, Euler obtained the following equations.
First Euler Equation
Second Euler Equation
Third Euler Equation
These three equations are collectively known as Euler's Equations of Rotation.
Meaning of Each Equation
First Equation: Explains how rotation about the X-axis changes due to torque and the
interaction of rotation about the Y- and Z-axes.
Second Equation: Explains the rotational motion about the Y-axis.
Third Equation: Explains the rotational motion about the Z-axis.
The extra terms involving products like ω₂ω₃, ω₃ω₁, and ω₁ω₂ show that rotation about one
axis affects rotation about the other axes.
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Special Case
If no external torque acts on the body:
Then Euler's equations become:
These equations describe the free rotation of a rigid body.
Applications of Euler's Equations
Euler's equations are widely used in:
Designing satellites and spacecraft.
Aircraft and drone stability analysis.
Robotics and robotic arm movement.
Gyroscopes and navigation systems.
Mechanical engineering and rotating machinery.
Motion analysis of spinning tops, wheels, and turbines.
Conclusion
Euler's equations are the fundamental equations of rotational dynamics for a rigid body
about a fixed point. They relate the external torque, moment of inertia, and angular
velocity along the three principal axes. Unlike simple rotational motion about a single axis,
these equations show that rotation about one axis influences the motion about the other
two axes. This makes them essential for understanding and predicting the behavior of
rotating objects in engineering and physics.
This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.
