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GNDU Question Paper-2022
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. (a) Find the expression for the displacement and velocity of a particle moving in space
both in Cartesian and Spherical polar coordinates.
(b) The plane polar coordinates of a particle at any instant are r = 3e2t and 0=2t. Find the
radial and transverse components of velocity and acceleration.
2. (a) Define solid angle and write its Sl unit. Find the solid angle subtended by the surface
of a sphere at its centre.
(b) Discuss the properties of space and show that the homogeneity of space leads to the
law of conservation of linear momentum.
SECTION-B
3. (a) Why we reduce a two body problem into one body problem by introducing the
concept of reduced mass? Derive the equation of motion of an equivalent one body
problem. Give physical meaning of reduced mass.
(b) Using the differential equation of the orbit under the central force, find the law of
force, if the orbit is rebo, where the symbols have their usual meaning.
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(c) Show that when a body moves under the action of a central force, its motion is
confined to a plane.
4. (a) Determine the turning points in the trajectory of a particle moving under inverse
square force field. Show that the shape of the trajectory depends upon the total energy.
(b) If the average distance of Mars from the Sun is 1.52 times than that of the Earth from
the Sun, find the period of revolution of Mars around Sun.
SECTION-C
5. (a) Show that the law of conservation of linear momentum and energy are invariant to
Galilean transformations.
(b) Derive an expression showing the effect of rotation of earth on acceleration due to
gravity. Where the value is maximum?
6. (a) Discuss the effect of coriolis force on a particle moving on the surface of earth, also
mention the geographical consequences of this force.
(b) Determine the latitude at which the plane of vibration of the Focault's pendulum does
not rotate at all..
(c) What does Focault's pendulum demonstrate ?
SECTION-D
7. (a) Discuss the elastic scattering in centre of mass system and show that magnitudes of
velocities of the particles remain unaltered after the collision.
(b) Two particles of masses 4 kg and 6 kg are moving with velocities 21 m/s and 3 j m/s
respectively in a laboratory frame. Find the total kinetic energy of the system relative to
the centre of mass frame.
8. (a) Derive the Euler's equations for rotation of a rigid body about a fixed point.
(b) A bicycle wheel of mass 3 kg and radius 0.5 m is rolling on a road at 8 m/s. What is the
torque required to be applied on the handle to turn it through half a radian in 0.1s ?
Assume that mass of the wheel is concentrated at the rim.
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GNDU Answer Paper-2022
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. (a) Find the expression for the displacement and velocity of a particle moving in space
both in Cartesian and Spherical polar coordinates.
(b) The plane polar coordinates of a particle at any instant are r = 3e2t and 0=2t. Find the
radial and transverse components of velocity and acceleration.
Ans: Simple Explanation
When we study the motion of an object, the first thing we need to know is where the object
is and how fast it is moving. These are described by displacement and velocity.
Imagine a drone flying in the sky. We can describe its position in two different coordinate
systems:
1. Cartesian Coordinates (x, y, z) Measures movement along three perpendicular
directions.
2. Spherical Polar Coordinates (r, θ, φ) Measures movement using distance and
angles.
Diagram of Cartesian Coordinates
Z-axis
|
|
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| P(x,y,z)
| *
| /
| /
| /
| /
---------------O---------------- X-axis
/
/
/
Y-axis
Here,
x = Distance along X-axis
y = Distance along Y-axis
z = Distance along Z-axis
The point P(x, y, z) represents the position of the particle.
Displacement in Cartesian Coordinates
The displacement vector is
where
= Unit vector along x-axis
ĵ = Unit vector along y-axis
k
= Unit vector along z-axis
This equation simply tells us the exact location of the particle.
Velocity in Cartesian Coordinates
Velocity is the rate of change of displacement with time.
Therefore,
which becomes
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Here,
= Velocity along x-axis
= Velocity along y-axis
= Velocity along z-axis
So, the total velocity is simply the combination of motion in all three directions.
Spherical Polar Coordinates
Instead of using x, y, and z, we use
r = Distance from the origin
θ = Polar angle
φ = Azimuth angle
Diagram
Z-axis
|
|\
| \
| \ r
| \
| θ *
| /
| /
-------------------O---------------- X-axis
/
φ/
Y-axis
The particle's position depends on one distance and two angles.
Displacement
In spherical coordinates,
where
r = Distance from origin
êr = Unit vector pointing outward from the origin
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Velocity
Velocity consists of three components:
Radial velocity (along r)
Polar velocity (along θ)
Azimuth velocity (along φ)
The expression is
This means the particle may move:
Away from the origin,
Up or down,
Around the origin.
Thus, spherical coordinates are very useful for describing the motion of planets, satellites,
and projectiles.
(b) The plane polar coordinates of a particle are
Find the radial and transverse components of velocity and acceleration.
Understanding the Question
This particle moves in a plane using polar coordinates.
Here,
r changes with time.
θ also changes with time.
Since both distance and direction change continuously, the particle has two types of motion:
1. Radial Motion → Moving toward or away from the origin.
2. Transverse Motion → Moving around the origin.
Diagram
P
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*
↑
Radial | Transverse
/
/
O
The radial direction points directly outward from the origin, while the transverse direction is
perpendicular to it.
Step 1: Differentiate r
Given
Differentiate:
Hence,
Radial velocity
Step 2: Transverse Velocity
Given
Differentiate:
Therefore,
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Substituting
,
Step 3: Radial Acceleration
The formula is
Differentiate again:
Also,
Hence,
So,
Radial acceleration = 0
Step 4: Transverse Acceleration
The formula is
Since
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we get
Final Answers
Radial Velocity
Transverse Velocity
Radial Acceleration
Transverse Acceleration
Key Concepts to Remember
Displacement tells us the position of a particle relative to the origin.
Velocity is the rate at which the position changes with time.
Cartesian coordinates use three perpendicular axes (x, y, z) and are convenient for
straight-line and general three-dimensional motion.
Spherical polar coordinates describe position using one distance (r) and two angles
(θ, φ), making them ideal for circular, orbital, and rotational motion.
In plane polar coordinates, velocity and acceleration are divided into:
o Radial component: motion toward or away from the origin.
o Transverse component: motion perpendicular to the radial direction,
representing rotational motion.
In this problem, the particle moves outward exponentially while also rotating with a
constant angular speed. Because the outward acceleration exactly balances the
centripetal requirement, the radial acceleration becomes zero, while the transverse
acceleration remains positive due to the combined effect of increasing radial speed
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and constant angular motion. This demonstrates how a particle can simultaneously
move away from the origin and rotate around it.
2. (a) Define solid angle and write its Sl unit. Find the solid angle subtended by the surface
of a sphere at its centre.
(b) Discuss the properties of space and show that the homogeneity of space leads to the
law of conservation of linear momentum.
Ans: Imagine you are standing in the middle of a room holding a torch. The light spreads in
all directions and covers a part of the wall. The amount of space covered by the light, as
seen from your position, is similar to the idea of a solid angle.
A plane angle (measured in radians) tells us how much of a circle is covered in a flat surface.
In the same way, a solid angle tells us how much of a three-dimensional space is covered by
an object when viewed from a point.
Definition of Solid Angle
A solid angle is the three-dimensional angle subtended by a surface at a point. It measures
how large an object appears from that point.
The solid angle is represented by the Greek letter Ω (Omega).
SI Unit
The SI unit of solid angle is the steradian (sr).
Formula
The solid angle subtended by a surface is
where:
Ω = Solid angle
A = Surface area
r = Distance from the point to the surface
Solid Angle Subtended by a Sphere
Consider a sphere of radius r.
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The total surface area of a sphere is
Putting this into the formula,
steradian
Hence,
The entire surface of a sphere subtends a solid angle of steradians at its centre.
Diagram
* * * * * * *
* *
* *
* ↑ *
* | r *
* O (Centre) *
* *
* *
* *
* * * * *
Entire surface of sphere
subtends a solid angle = 4π sr
Important Points
Plane angle → 2D → Unit = Radian
Solid angle → 3D → Unit = Steradian
Complete sphere at centre → 4π steradians
2. (b) Discuss the properties of space and show that the homogeneity of space leads to the
law of conservation of linear momentum.
Simple Explanation (More than 200 Words)
To understand this topic, first imagine playing football in different places. Whether you kick
the ball in your school playground, a nearby park, or another open field, the laws of physics
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remain the same. The ball moves according to the same physical rules everywhere. This idea
introduces us to the properties of space.
Properties of Space
Physicists describe space using three important properties.
1. Homogeneity of Space
Homogeneity means all points in space are identical. The laws of physics do not change
from one place to another.
For example:
A cricket ball falls in Amritsar according to the same laws as it falls in Delhi.
A pendulum behaves in the same way whether it is placed in one classroom or
another.
This means nature does not give preference to any particular location.
2. Isotropy of Space
Isotropy means space is the same in every direction.
For example:
Throw a ball towards the north, south, east, or west with the same force.
Ignoring air resistance, it behaves according to the same physical laws.
No direction is special.
3. Continuity of Space
Space has no gaps or breaks. An object can move smoothly from one position to another
without suddenly disappearing or jumping.
How Homogeneity Leads to Conservation of Linear Momentum
Suppose two ice skaters stand on smooth ice facing each other.
Before Push
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A B
( ) ( )
| |
---------------------------
After Push
← A B →
When they push each other:
One moves to the left.
The other moves to the right.
Even though they move in opposite directions, the total linear momentum of the system
remains constant because there is no external force acting on them.
Since space is homogeneous, the laws of physics are the same everywhere. If momentum
were different at different places, the same experiment would produce different results
simply because it was performed somewhere else. That would violate the idea of
homogeneity.
Therefore, because space is identical at every point, linear momentum must remain
conserved in an isolated system.
Mathematically,
Total Linear Momentum Before Collision Total Linear Momentum After Collision
or
where:
= masses of the objects
= initial velocities
= final velocities
Conclusion
The properties of spacehomogeneity, isotropy, and continuityare fundamental
assumptions of physics. Among them, homogeneity of space states that all locations in
space are physically equivalent. This directly leads to one of the most important
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conservation laws in mechanics: the law of conservation of linear momentum, which states
that the total linear momentum of an isolated system remains constant when no external
force acts on it. This principle is widely used to study collisions, rocket motion, explosions,
and many other physical phenomena.
SECTION-B
3. (a) Why we reduce a two body problem into one body problem by introducing the
concept of reduced mass? Derive the equation of motion of an equivalent one body
problem. Give physical meaning of reduced mass.
(b) Using the differential equation of the orbit under the central force, find the law of
force, if the orbit is rebo, where the symbols have their usual meaning.
(c) Show that when a body moves under the action of a central force, its motion is
confined to a plane.
Ans: Imagine two friends, A and B, are holding opposite ends of a rope and pulling each
other. Both of them move because each exerts an equal and opposite force on the other. If
we try to study the motion of both people separately, the calculations become difficult
because both are moving at the same time.
Now imagine replacing this complicated situation with a single imaginary person whose
mass represents both A and B together. This makes the calculations much easier while
giving exactly the same result. This is the idea behind reducing a two-body problem into a
one-body problem.
Why is it done?
In nature, many systems contain two bodies attracting each other, such as:
Earth and Moon
Sun and Earth
Electron and Proton
Since both bodies move due to mutual attraction, solving two equations of motion is
difficult. Therefore, physicists replace the system by one equivalent particle moving under
the same force. This simplifies the mathematical calculations without changing the actual
physics.
Reduced Mass
The equivalent particle has a special mass called the Reduced Mass (μ).
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The reduced mass is
where
= mass of first body
= mass of second body
Equation of Motion
Suppose the distance between the two bodies is r.
The force between them is F(r).
The equivalent one-body equation becomes
This equation tells us that instead of studying two moving bodies separately, we only study
the motion of one particle having reduced mass μ.
Physical Meaning of Reduced Mass
Reduced mass is not a real physical object. It is an imaginary mass that represents the
combined motion of two interacting bodies.
Its importance is:
Simplifies difficult calculations.
Gives the same result as the original two-body system.
Used in astronomy, atomic physics and orbital mechanics.
For example,
Earth revolves around the Sun.
Actually both Earth and Sun revolve around their common centre of mass.
Using reduced mass, we study the motion as if only one particle is moving.
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Diagram
Original Two-Body System
m ←── F ──→ m
●--------------●
distance r
Both bodies move.
Equivalent One-Body System
μ
●
|
| r
|
Force F(r)
Only one equivalent particle is considered.
(b) Using the differential equation of orbit under a central force, find the law of force.
Simple Explanation
A central force always acts along the line joining the particle and the centre.
Examples are:
Gravitational force
Electrostatic force
Suppose the orbit of a particle is already known. Using the differential equation of the
orbit, we can determine the force responsible for producing that orbit.
The orbit equation is substituted into the standard differential equation of motion.
After differentiation and simplification, the force law is obtained.
For planetary motion (elliptical orbit), the resulting force is
This means the force follows the Inverse Square Law.
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Hence,
where
G = Universal gravitational constant
M = mass of central body
m = mass of moving body
The negative sign indicates that the force is attractive.
Meaning
This result proves that planets revolve around the Sun because the gravitational force
decreases as the square of the distance increases.
(c) Show that when a body moves under a central force, its motion is confined to a plane.
Simple Explanation
A central force always acts directly toward or away from the centre.
Examples include:
Earth's gravity
Sun's gravitational attraction
Electric force between charged particles
Because the force always acts along the radius, it cannot twist the motion sideways.
The turning effect of a force is called torque.
For a central force,
because the force and radius are in the same direction.
Since torque is zero,
Angular momentum remains constant.
The direction of angular momentum never changes.
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When the direction of angular momentum remains fixed, the particle cannot leave its
original plane of motion.
Therefore, the entire motion remains confined to one fixed plane.
Diagram
Motion of Particle
●
.-' \
.-' \
.-' \
O-------------→
O = Centre of force
Force always points toward O.
Since there is no sideways torque,
the particle always stays in one plane.
Conclusion
The concept of reduced mass helps convert a difficult two-body problem into a much
simpler one-body problem while preserving the same physical behavior. The equivalent
particle follows the equation
, making orbital calculations much easier. By using
the differential equation of an orbit, the nature of the central force can be determined; for
planetary motion, this leads to the inverse-square law of gravitation. Finally, because a
central force always acts along the line joining the particle to the centre, it produces zero
torque, which keeps angular momentum constant. As a result, the particle's path is always
confined to a single fixed plane. These concepts form the foundation of celestial mechanics
and explain the motion of planets, satellites, and many atomic systems.
4. (a) Determine the turning points in the trajectory of a particle moving under inverse
square force field. Show that the shape of the trajectory depends upon the total energy.
(b) If the average distance of Mars from the Sun is 1.52 times than that of the Earth from
the Sun, find the period of revolution of Mars around Sun.
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Ans: Simple Explanation
Imagine you are spinning a ball tied to a string. The string always pulls the ball toward your
hand. In a similar way, the Sun pulls planets toward itself through gravity. This gravitational
force is called an inverse square force because its strength decreases as the square of the
distance increases.
Mathematically,
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where:
G = Universal Gravitational Constant
M = Mass of the Sun (or attracting body)
m = Mass of the particle
r = Distance between them
The force is always directed toward the center.
What are Turning Points?
A turning point is a point where the particle changes its direction of motion. At this point,
the particle is neither moving toward the center nor away from it. In other words, its radial
velocity becomes zero.
There are generally two turning points:
1. Nearest point (Perihelion or Perigee) The particle comes closest to the attracting
body.
2. Farthest point (Aphelion or Apogee) The particle reaches its maximum distance
before returning.
At these points,
This means the particle stops moving inward or outward for an instant and then reverses its
radial motion.
Why Does the Shape of the Orbit Depend on Total Energy?
The total energy (E) of the particle is the sum of:
Kinetic Energy (K.E.)
Potential Energy (P.E.)
The value of this total energy decides the type of path followed by the particle.
Case 1: Total Energy is Negative (E < 0)
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When the particle has less energy, it cannot escape the gravitational pull. It remains bound
to the attracting body and moves in a closed orbit, usually an ellipse.
Example: Earth and Mars revolve around the Sun in elliptical orbits.
Case 2: Total Energy is Zero (E = 0)
If the particle has exactly the energy needed to escape, it follows a parabolic path. It
escapes the gravitational field but slows down continuously.
Case 3: Total Energy is Positive (E > 0)
If the particle has more than enough energy, it escapes completely and follows a hyperbolic
path.
Conclusion
Thus, the shape of the trajectory depends entirely on the total energy.
Total Energy
Shape of Trajectory
Motion
Negative
Ellipse
Bound orbit
Zero
Parabola
Just escapes
Positive
Hyperbola
Escapes completely
Therefore, by knowing the total energy of a particle moving under an inverse square force,
we can predict whether it will remain in orbit or escape forever.
4(b) If the average distance of Mars from the Sun is 1.52 times that of the Earth, find the
period of revolution of Mars around the Sun.
Simple Explanation
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This question is based on Kepler's Third Law of Planetary Motion.
The law states:
The square of the time period of a planet is directly proportional to the cube of its average
distance from the Sun.
Instead of memorizing it, think of it like this:
A planet that is farther from the Sun has a larger orbit.
Since it has to travel a longer path, it takes more time to complete one revolution.
The mathematical relation is:
Where:
= Time period of Earth = 1 year
= Time period of Mars
= Average distance of Earth
= Average distance of Mars = 1.52
Substitute the values:
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Rearranging,
years
Final Answer
The period of revolution of Mars around the Sun is approximately 1.88 years (about 687
Earth days).
Why is Mars slower than Earth?
Mars is farther from the Sun, so:
It travels in a much larger orbit.
The Sun's gravitational pull on Mars is weaker because gravity decreases with the
square of the distance.
As a result, Mars moves more slowly and takes about 1.88 Earth years to complete
one revolution.
Key Points for Exams
Inverse square force:
Turning points: Positions where radial velocity becomes zero; they represent the
minimum and maximum distances from the attracting body.
Total energy determines orbit:
o : Elliptical orbit
o : Parabolic path
o : Hyperbolic path
Kepler's Third Law:
Mars' period of revolution: 1.88 years (≈687 days).
SECTION-C
5. (a) Show that the law of conservation of linear momentum and energy are invariant to
Galilean transformations.
(b) Derive an expression showing the effect of rotation of earth on acceleration due to
gravity. Where the value is maximum?
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Ans: Imagine two friends are watching a cricket match. One friend is standing on the
ground, while the other is sitting in a train moving at a constant speed. Both are watching a
player throw a ball. Although the ball appears to move differently to each observer because
one is moving and the other is not, the laws of physics remain exactly the same. This idea is
explained by Galilean Transformation.
A Galilean Transformation is a mathematical relation that connects the measurements
made by two observers moving with a constant velocity relative to each other.
The transformation equations are:
Here:
= relative velocity between two observers.
Time remains the same for both observers.
Conservation of Linear Momentum
Linear momentum is defined as:
where:
= mass
= velocity
Suppose two particles collide.
Before collision:
After collision:
According to the law of conservation of momentum,
Now consider another observer moving with speed .
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The observed velocities become:
Substituting these into the momentum equation, the extra terms cancel because the total
mass remains the same. Therefore,
This proves that linear momentum is conserved in every inertial frame. Hence, the law is
invariant under Galilean transformation.
Conservation of Energy
The kinetic energy of a particle is
For another moving observer,
The numerical value of kinetic energy changes because velocity changes.
However, during any collision or physical process, the total energy before and after the
event changes by the same amount for both observers. Thus, if energy is conserved in one
inertial frame, it is also conserved in every other inertial frame moving with constant
velocity.
Therefore, the law of conservation of energy is also invariant under Galilean
transformation.
Diagram
Ground Observer (S)
A -----> Ball -----> B
Velocity = u
-----------------------------
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Moving Observer (S')
Train moving with velocity V
A -----> Ball -----> B
Observed velocity = u − V
Although velocities differ,
Momentum and Energy remain conserved.
Key Points
Galilean transformation relates two inertial reference frames.
Time remains unchanged.
Velocity changes by subtracting the frame velocity.
Linear momentum remains conserved in every inertial frame.
Total energy is also conserved in every inertial frame.
Therefore, both conservation laws are Galilean invariant.
5. (b) Derive an expression showing the effect of rotation of Earth on acceleration due to
gravity. Where is its value maximum?
Simple Explanation (More than 200 words)
When we say the acceleration due to gravity is
, we usually think it is the same
everywhere. In reality, it is slightly different at different places on Earth because the Earth
is continuously rotating about its own axis.
Imagine tying a stone to a string and rotating it in a circle. The stone experiences an outward
effect called centrifugal force. Similarly, because the Earth rotates, every object on its
surface experiences a small outward centrifugal effect.
This centrifugal effect acts opposite to gravity, reducing the effective value of gravity.
Why does this happen?
The Earth's angular velocity is
At latitude , the distance from the Earth's axis is
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where:
= Earth's radius
= latitude
The centrifugal acceleration is
Only its vertical component opposes gravity.
Hence the effective acceleration due to gravity becomes
where:
= actual gravitational acceleration
= effective acceleration due to Earth's rotation
= angular velocity of Earth
= radius of Earth
= latitude
Special Cases
At the Equator
Latitude,
Since
The centrifugal effect is maximum, so gravity is minimum.
At the Poles
Latitude,
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Since
There is no centrifugal effect, so gravity is maximum.
Diagram
North Pole
●
|
|
Rotation Axis
|
|
----------------------------- Earth
/ \
/ \
| Equator |
|<----- Rotation -----> |
\ /
\_____________________________/
At Equator:
Maximum centrifugal effect
Gravity is minimum
At Poles:
No centrifugal effect
Gravity is maximum
Final Result
The effective acceleration due to gravity is
Where is the value maximum?
Maximum gravity: At the North Pole and South Pole (
)
Minimum gravity: At the Equator (
)
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6. (a) Discuss the effect of coriolis force on a particle moving on the surface of earth, also
mention the geographical consequences of this force.
(b) Determine the latitude at which the plane of vibration of the Focault's pendulum does
not rotate at all..
(c) What does Focault's pendulum demonstrate ?
Ans: 6. (a) Effect of Coriolis Force on a Particle Moving on the Surface of the Earth and its
Geographical Consequences
The Coriolis Force is an imaginary or apparent force that appears because the Earth is
continuously rotating from west to east. It does not actually push objects, but it makes
moving objects appear to change their direction when viewed from the Earth's surface.
Imagine you are sitting on a rotating merry-go-round. If you throw a ball straight to your
friend, the ball appears to curve instead of moving in a straight line. The same thing happens
on Earth because our planet is constantly spinning.
How Coriolis Force Affects a Moving Particle
Suppose a particle, airplane, missile, river water, or wind starts moving on the Earth's
surface.
In the Northern Hemisphere, the moving object is deflected towards the right of its
direction of motion.
In the Southern Hemisphere, the moving object is deflected towards the left.
The amount of deflection depends on:
The speed of the moving object.
The Earth's rotation.
The latitude (distance from the Equator).
At the Equator, the Coriolis force is zero, so there is no deflection.
At the North Pole and South Pole, the Coriolis force is maximum, so the deflection is
greatest.
Simple Diagram of Coriolis Force
North Pole
▲
|
Right Deflection (Northern Hemisphere)
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Object Moving ------>
|
------------------ Equator ------------------
No Coriolis Force (Zero)
|
Object Moving ------>
Left Deflection (Southern Hemisphere)
|
South Pole
Geographical Consequences of Coriolis Force
The Coriolis force has many important effects on the Earth.
1. Deflection of Winds
Instead of blowing in straight lines, winds bend due to Earth's rotation.
Northern Hemisphere → Right
Southern Hemisphere → Left
This creates the world's major wind systems.
2. Formation of Cyclones and Anticyclones
Cyclones rotate because of Coriolis force.
Northern Hemisphere → Counter-clockwise
Southern Hemisphere → Clockwise
Without Coriolis force, cyclones would not spin.
3. Ocean Currents
Large ocean currents also bend because of Coriolis force.
Example:
Gulf Stream
Kuroshio Current
These currents influence climate around the world.
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4. River Flow
Long rivers flowing from north to south experience slight deflection.
This causes one river bank to erode more than the other.
5. Aircraft and Missile Navigation
Pilots and scientists must consider Coriolis force while planning long-distance flights and
missile paths.
Ignoring it could make them miss their target.
(b) Latitude at Which the Plane of Vibration of Foucault's Pendulum Does Not Rotate
A Foucault's Pendulum is a long pendulum that swings freely in one plane. It was invented
by the French scientist Léon Foucault to demonstrate that the Earth rotates.
As the Earth rotates beneath the pendulum, the plane in which the pendulum swings
appears to rotate relative to the Earth's surface.
The rate of this apparent rotation depends on the latitude.
North Pole (90°): The plane completes one full rotation in about 24 hours.
Equator (0°): The plane does not rotate at all.
Between the Equator and the Poles, the rotation increases gradually.
Therefore,
The plane of vibration of the Foucault's pendulum does not rotate at the Equator (Latitude
= 0°).
Simple Diagram
North Pole (90°)
Plane rotates fully
|
|
|
45°
Plane rotates partially
|
|
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Equator (0°)
No rotation
(c) What Does Foucault's Pendulum Demonstrate?
The Foucault's Pendulum provides one of the simplest and most convincing proofs that the
Earth rotates on its own axis.
When the pendulum is set in motion:
It continues to swing in almost the same plane due to inertia.
Meanwhile, the Earth rotates beneath it.
To an observer standing on Earth, the swinging plane appears to rotate.
This apparent rotation is not because the pendulum changes direction, but because the
Earth itself is turning.
What It Demonstrates
The Earth rotates on its axis.
The rate of apparent rotation depends on latitude.
There is no rotation of the pendulum's plane at the Equator.
The maximum apparent rotation occurs at the Poles.
It provides experimental evidence of the Earth's daily rotation.
Conclusion
The Coriolis force is an apparent force caused by the Earth's rotation. It changes the
direction of moving objects, causing winds, ocean currents, rivers, aircraft, and cyclones to
follow curved paths instead of straight ones. This force is zero at the Equator and maximum
at the Poles, making it one of the most important forces in physical geography and
geophysics.
The Foucault's pendulum is a famous scientific experiment that clearly proves the Earth
rotates. Its plane of vibration rotates fastest at the poles, more slowly at intermediate
latitudes, and does not rotate at the Equator (0° latitude). Together, the Coriolis force and
Foucault's pendulum help us understand how the Earth's rotation influences natural
phenomena such as weather, ocean circulation, navigation, and planetary motion.
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SECTION-D
7. (a) Discuss the elastic scattering in centre of mass system and show that magnitudes of
velocities of the particles remain unaltered after the collision.
(b) Two particles of masses 4 kg and 6 kg are moving with velocities 21 m/s and 3 j m/s
respectively in a laboratory frame. Find the total kinetic energy of the system relative to
the centre of mass frame.
Ans: What is Elastic Scattering?
Elastic scattering is a type of collision in which both momentum and kinetic energy remain
conserved. This means that after the collision:
Total momentum before collision = Total momentum after collision.
Total kinetic energy before collision = Total kinetic energy after collision.
No energy is lost as heat, sound, or deformation. The particles simply change their
directions of motion.
What is the Centre of Mass (C.M.) System?
The centre of mass is an imaginary point where the entire mass of the system appears to be
concentrated.
In the Centre of Mass frame, we observe the motion of particles while moving along with
the centre of mass. This makes the collision much easier to understand because:
The total momentum in the C.M. frame is always zero.
Before collision, the particles move toward each other with equal and opposite
momenta.
After collision, they move away with equal and opposite momenta.
Simple Explanation
Imagine two friends are standing on roller skates and push each other. Since they are on
smooth ground, one moves left while the other moves right with opposite momenta. If they
bounce perfectly without losing energy, only their directions change, not their speeds.
The same thing happens in an elastic collision observed from the centre of mass.
Diagram
Before Collision (C.M. Frame)
Particle A -----> ● <----- Particle B
v1 v2
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Collision
After Collision
Particle A
\
●
/
Particle B
The particles scatter in different directions after collision.
Why do the magnitudes of velocities remain unchanged?
In the Centre of Mass frame:
Total momentum is zero before and after collision.
Total kinetic energy is also conserved.
Since kinetic energy depends on the square of the speed, and the total kinetic energy
remains constant, the speed (magnitude of velocity) of each particle remains the same.
Only the direction of the velocity changes.
Therefore,
and
where
and
are the velocities before collision, and
and
are the velocities after
collision.
Important Points for Exams
Elastic collision conserves momentum and kinetic energy.
In the C.M. frame, total momentum is always zero.
The particles move with equal and opposite momenta.
After collision, only the direction of motion changes.
Magnitude of velocity remains unchanged, making the Centre of Mass frame very
useful for studying scattering.
7(b) Numerical
Given
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Mass of particle 1,
kg
Mass of particle 2,
kg
Velocity of particle 1,
m/s
Velocity of particle 2,
m/s
Step 1: Velocity of Centre of Mass
m/s
Step 2: Velocities Relative to the C.M.
For particle 1:
m/s
For particle 2:
m/s
Step 3: Kinetic Energy in the C.M. Frame
For particle 1:
J
For particle 2:
J
Total Kinetic Energy
J
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Final Answer
(a) In an elastic scattering observed from the Centre of Mass frame, the total momentum is
zero and kinetic energy is conserved. Therefore, the particles only change their direction of
motion, while the magnitudes of their velocities remain unchanged.
(b) Total kinetic energy of the system relative to the Centre of Mass frame = J .
8. (a) Derive the Euler's equations for rotation of a rigid body about a fixed point.
(b) A bicycle wheel of mass 3 kg and radius 0.5 m is rolling on a road at 8 m/s. What is the
torque required to be applied on the handle to turn it through half a radian in 0.1s ?
Assume that mass of the wheel is concentrated at the rim.
Ans: Introduction
Imagine you are spinning a top, a bicycle wheel, or a ceiling fan. These objects rotate about
a fixed point or fixed axis. While rotating, different forces and torques act on them. To
understand how the speed and direction of rotation change, scientists use Euler's equations
of motion.
Euler's equations are simply Newton's Second Law for rotational motion. Just as Newton's
law tells us that force changes linear motion, Euler's equations tell us that torque changes
rotational motion.
Important Concepts
1. Rigid Body
A rigid body is an object whose shape and size remain unchanged while it moves or rotates.
Examples:
Bicycle wheel
Cricket bat
Ceiling fan
Spinning top
2. Fixed Point
A fixed point is a point that does not move, although the body can rotate around it.
Example:
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The axle of a bicycle wheel remains fixed while the wheel rotates.
3. Moment of Inertia (I)
Moment of inertia is the rotational equivalent of mass.
Mass resists linear motion.
Moment of inertia resists rotational motion.
It depends on:
Mass of the body
Distribution of mass from the axis
Greater the distance of mass from the axis, greater is the moment of inertia.
4. Torque (τ)
Torque is the turning effect of a force.
where
τ = Torque
I = Moment of Inertia
α = Angular acceleration
Euler's Equations
For a rigid body rotating about its principal axes,
where
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= Principal moments of inertia
= Angular velocities
= Components of external torque
These equations explain how the rotational speed changes when torque acts on a rigid
body.
Simple Diagram
T (Torque)
↑
|
----------------
/ \
| Bicycle Wheel |
\ /
----------------
●
Fixed Axle
Torque acts on the wheel, causing changes in its rotational motion around the fixed axle.
(b) Numerical Problem
Given
Mass of wheel
Radius
Speed
Angle turned
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Time
Mass is concentrated at the rim.
Hence,
Step 1: Calculate Moment of Inertia
Step 2: Angular Velocity of Wheel
Step 3: Angular Velocity of Turning
The wheel is turned through
in
Therefore,
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Step 4: Gyroscopic Torque
For a spinning wheel,
Substitute the values,
Final Answer
Torque required =
Easy Explanation
Think about riding a bicycle. When the bicycle is moving fast, you may notice that turning
the handle becomes slightly harder. This happens because the wheels are spinning rapidly
and naturally try to keep their direction unchanged. This effect is known as the gyroscopic
effect.
In this question, the bicycle wheel has a mass of 3 kg and a radius of 0.5 m. Since all the
mass is assumed to be concentrated at the rim, the wheel behaves like a thin ring. First, we
calculate its moment of inertia, which tells us how difficult it is to change its rotational
motion. Next, using the bicycle's speed, we find the wheel's angular velocity. When the
handle is turned through half a radian in 0.1 second, the axis of the spinning wheel changes
direction. Because the wheel is already rotating rapidly, an additional turning effect called
gyroscopic torque is required.
The required torque depends on three important quantities:
The wheel's moment of inertia (how resistant it is to changes in rotation),
Its spinning speed (angular velocity),
The rate at which its axis is turned.
By multiplying these values using the gyroscopic torque formula, we obtain a torque of 60
N·m.
This problem combines several important concepts from rotational mechanics: rigid bodies,
fixed-point rotation, moment of inertia, angular velocity, torque, and gyroscopic motion.
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Understanding how these ideas work together helps explain not only bicycle steering but
also the operation of motorcycles, aircraft, ships, and even spinning satellites.
This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.
