Fundamental Physics Concepts: Questions and Solutions

Posted on Sep 18, 2025 in Physics

Natural Phenomena for Time Standards

Q.1 Name several repetitive phenomena in nature for time standard.

Earth-Sun System
Moon-Earth System
Human Pulse
Quartz Crystal Vibration

Pendulum as a Time Standard: Drawbacks

Q.2 Drawbacks of using a pendulum as a time standard.

Length and gravitational acceleration (g) vary.
Friction affects its motion.
Air resistance influences its period.

Kilogram vs. Mole: Understanding Mass and Quantity

Q.3 Why use both kilogram (kg) and mole?

Kilogram (kg): Represents the mass of a substance.
Mole: Represents the count of particles (e.g., atoms, molecules) in a substance.

Accurate Measurement: Understanding Least Count

Q.4 Which needle measurement is correct and why?

The measurement 0.214 m is correct because the least count of the scale is 0.001 m, indicating precision to three decimal places.

Accuracy in Measurement: The Weakest Link Principle

Q.5 Chain is as strong as weakest link – experimental data?

This principle implies that the accuracy of a measurement or calculation is limited by the least accurate data or component involved.

Common Errors in Pendulum Time Measurement

Q.6 Errors in measuring pendulum time with a stopwatch?

Zero error of the stopwatch.
Parallax error when reading the scale.
Human reaction time delay.
Air friction affecting the pendulum’s swing.

Dimensional Analysis: Deriving Relations, Not Constants

Q.7 Does dimensional analysis give constants?

No. Dimensional analysis only provides the relationship between physical quantities. Constants (like 2π) must be determined through experimentation.

Dimensional Analysis: Pressure and Density

Q.8 Write the dimensions of (i) Pressure, (ii) Density.

(i) Dimensions of Pressure:

Pressure = Force / Area
Dimensions of Force [F] = [M L T^-2] (mass × acceleration)
Dimensions of Area [A] = [L²]
Dimensions of Pressure [P] = [F] / [A] = [M L T^-2] / [L²] = [M L^-1 T^-2]

(ii) Dimensions of Density:

Density = Mass / Volume
Dimensions of Mass [M] = [M]
Dimensions of Volume [V] = [L³]
Dimensions of Density [D] = [M] / [V] = [M L^-3]

Dimensional Consistency: Wavelength, Speed, and Frequency

Q.9 The wavelength λ of a wave depends on the speed v of the wave and its frequency f. Knowing that [λ] = [L], [v] = [L T^-1] and [f] = [T^-1]. Decide which of the following is correct, f = v λ or f = v / λ.

Applying dimensional analysis:

Case 1: f = v λ

Dimensions of f = [T^-1] … (1)
Dimensions of vλ = [L T^-1] × [L] = [L² T^-1] … (2)

From equation (1) and (2): [T^-1] ≠ [L² T^-1]. So, this equation is dimensionally not correct.

Case 2: f = v / λ

Dimensions of f = [T^-1] … (3)
Dimensions of v / λ = [L T^-1] / [L] = [T^-1] … (4)

From equation (3) and (4): [T^-1] = [T^-1]. So, this equation is dimensionally correct.

Vector Fundamentals: Definitions

Q.1 Define the terms:

(i) Unit Vector

A vector whose magnitude is one, used to indicate the direction of any other vector. It is represented as Â = A / |A|.

(ii) Position Vector

A vector that describes the location of a point or particle with respect to the origin. It is commonly represented by r.

(iii) Components of a Vector

A component of a vector is its effective value in a given direction.

Vector Sum: Zero Resultant and Triangle Rule

Q.2 The vector sum of three vectors gives a zero resultant. What can be the orientation of the vectors?

If three vectors are drawn in such a way that they form a closed triangle (head-to-tail), then their vector sum will be zero (A + B + C = 0).

Vector Components: Negative Values in Quadrants

Q.3 Vector A lies in the xy plane. For what orientation will both of its rectangular components be negative?

When vector A lies in the 3rd quadrant, both its x and y components will be negative.
In the 2nd or 4th quadrant, the components have opposite signs (one positive, one negative).

Vector Magnitude: Non-Zero Components

Q.4 If one of the components of a vector is not zero, can its magnitude be zero? Explain.

No. Its magnitude cannot be zero.

Example: If A_x ≠ 0 and A_y = 0, then the magnitude |A| = √(A_x² + A_y²) = √(A_x²) = |A_x| ≠ 0.

Vector Components vs. Magnitude

Q.5 Can a vector have a component greater than the vector’s magnitude?

No. A vector cannot have a component greater than its magnitude.

A component is only a part of the vector in a specific direction.
A component may be equal to the vector’s magnitude if the other components are zero (i.e., the vector lies entirely along that axis).

Vector Magnitude: Always Positive

Q.6 Can the magnitude of a vector have a negative value?

No. The magnitude of a vector is always positive.

The magnitude is calculated as |A| = √(A_x² + A_y²). Even if A_x and A_y are negative, their squares make them positive, resulting in a positive magnitude.

Vector Sum of Zero: Component Relationship

Q.7 If A + B = 0, what can you say about the components of the two vectors?

If A + B = 0, then the corresponding components of both vectors are equal in magnitude and opposite in direction.

Example: If A = A_x î + A_y ĵ and B = B_x î + B_y ĵ, and A + B = 0, then:

A_x + B_x = 0 (meaning A_x = -B_x)
A_y + B_y = 0 (meaning A_y = -B_y)

Equal Vector Components: The 45-Degree Angle

Q.8 Under what circumstances would a vector have components that are equal in magnitude?

We know that A_x = A cosθ and A_y = A sinθ.

Since the sine and cosine functions have equal values for an angle of 45°, when a vector makes an angle of 45° with the X-axis (or Y-axis), its components will have equal magnitude.

Alternatively:

Suppose the magnitudes of both components are equal, then:

A_y = A_x
A sinθ = A cosθ
A sinθ / A cosθ = 1
tan θ = 1
θ = tan^-1(1) = 45°

Adding Vectors and Scalars: Fundamental Differences

Q.9 Is it possible to add a vector quantity to a scalar quantity? Explain.

No. It is not possible to add a vector quantity to a scalar quantity. Scalar quantities are added using simple algebraic rules, while vector quantities are added using specific vector addition rules (e.g., Head-to-Tail Rule or component methods). Therefore, scalars can only be added to scalars, and vectors can only be added to vectors.

Adding Zero to a Null Vector: Scalar vs. Vector

Q.10 Can you add zero to a null vector?

No. We cannot add the scalar quantity ‘zero’ to a null vector. As explained previously, scalar quantities are added algebraically, whereas vector quantities require vector addition methods. A scalar cannot be added to a vector, even if the scalar value is zero or the vector is a null vector.

Sum of Unequal Vectors: Can it be Zero?

Q.11 Two vectors have unequal magnitudes. Can their sum be zero? Explain.

No. The sum of two vectors with unequal magnitudes cannot be zero. For the sum of two vectors to be zero, they must have equal magnitudes and opposite directions.

Perpendicular Vectors: Sum and Difference Properties

Q.12 Show that the sum and difference of two perpendicular vectors of equal lengths are also perpendicular and of the same length.

Consider two vectors A and B of equal length, perpendicular to each other. Their resultant sum (A + B) and resultant difference (A – B) will each make an angle of 45° with the original vectors (if placed appropriately). This implies that the sum and difference vectors are perpendicular to each other.

Furthermore, the magnitude of the sum |A + B| = √(A² + B²) and the magnitude of the difference |A – B| = √(A² + (-B)²) = √(A² + B²). Since |A| = |B|, both magnitudes will be equal, meaning the sum and difference vectors have the same length.

Resultant Vector: Equal Magnitude Orientation

Q.13 How would two vectors of the same magnitude have to be oriented if they were to be combined to give a resultant equal to a vector of the same magnitude?

When the angle between two vectors of the same magnitude is 120°, the magnitude of their resultant will be the same as the magnitude of each individual vector. In this configuration, the three vectors (the two original vectors and their resultant) form an equilateral triangle.

Vector Addition: Range of Resultant Magnitudes

Q.14 The two vectors to be combined have magnitudes 60 N and 35 N. Pick the correct answer from those given and tell why it is the only one of the three that is correct.
i) 100 N ii) 70 N iii) 20 N

Given magnitudes: A₁ = 60 N and A₂ = 35 N.

The range of possible resultant magnitudes is between the minimum (difference) and maximum (sum) values:

Maximum possible magnitude: 60 N + 35 N = 95 N
Minimum possible magnitude: |60 N – 35 N| = 25 N

Therefore, the resultant magnitude must be between 25 N and 95 N (inclusive).

(i) 100 N is incorrect because 100 N > 95 N.
(ii) 70 N is correct because 25 N ≤ 70 N ≤ 95 N.
(iii) 20 N is incorrect because 20 N < 25 N.

Thus, answer (ii) 70 N is the only correct option.

Closed Polygon of Vectors: Zero Sum

Q.15 Suppose the sides of a closed polygon represent vectors arranged head to tail. What is the sum of these vectors?

The vector sum will be zero, because the tail of the first vector meets the head of the last vector, indicating no net displacement. Hence: A + B + C + D + E = 0.

Vector Applications: Multiple Choice Questions

Q.16 Identify the correct answer:

i) Relative Motion of Ships

Two ships X and Y are travelling in different directions at equal speeds. The actual direction of motion of X is due north, but to an observer on Y, the apparent direction of motion of X is north-east. The actual direction of motion of Y as observed from the shore will be:

(A) East (B) West (C) South-East (D) South-West

Ans: (B) West

ii) Resultant Force on an Inclined Plane

A horizontal force F is applied to a small object P of mass M at rest on a smooth plane inclined at an angle θ to the horizontal. The magnitude of the resultant force acting up and along the surface of the plane, on the object is:

(A) F cos θ – mg sin θ
(B) F sin θ – mg cos θ
(C) F cos θ + mg cos θ
(D) F sin θ + mg sin θ
(E) mg tan θ

Ans: (A) F cos θ – mg sin θ

Vector Cross Product: Effect of Component Reversal

Q.17 If all the components of the vectors, A₁ and A₂ were reversed, how would this alter A₁ × A₂?

If A₁ becomes -A₁ and A₂ becomes -A₂, then the new cross product is:

(-A₁) × (-A₂) = (-1)(-1)(A₁ × A₂) = A₁ × A₂

So, there would be no effect on the cross product; it remains unchanged.

Conditions for Zero Vector Cross Product

Q.18 Name the three different conditions that could make A₁ × A₂ = 0.

The cross product A₁ × A₂ = 0 if:

A₁ is a null vector: 0 × A₂ = 0
A₂ is a null vector: A₁ × 0 = 0
A₁ and A₂ are parallel or anti-parallel: This means the angle θ between them is 0° or 180°. Since A₁ × A₂ = A₁A₂ sin(θ), and sin(0°) = 0 and sin(180°) = 0, the cross product will be zero.

Equilibrium Concepts: True or False Statements

Q.19 Identify true or false statements and explain the reason.

a) A body in equilibrium implies that it is not moving nor rotating.

False. A body in equilibrium means its net force and net torque are zero. This implies it is either at rest or moving/rotating with uniform velocity (i.e., no acceleration, no angular acceleration).

b) If coplanar forces acting on a body form a closed polygon, then the body is said to be in equilibrium.

True. A closed polygon of forces indicates that the net force acting on the body is zero, which is a primary condition for translational equilibrium.

Minimum Tension in Suspended Objects

Q.20 A picture is suspended from a wall by two strings. Show by diagram the configuration of the strings for which the tension in the strings will be minimum.

The tension in the strings will be minimum when the strings are as horizontal as possible, meaning the angle θ they make with the horizontal is close to 90° (or the angle with the vertical is close to 0°). The tension T in each string for a picture of weight W, suspended symmetrically, is given by:

T = W / (2 sin θ)

where θ is the angle each string makes with the horizontal. To minimize T, sin θ must be maximized. The maximum value of sin θ is 1, which occurs when θ = 90°.

So, minimum tension occurs when the strings are horizontal (θ = 90°), effectively supporting the weight directly upwards with minimal horizontal component.

Center of Gravity and Rotation: Weight’s Effect

Q.21 Can a body rotate about its center of gravity under the action of its weight?

No. A body cannot rotate about its center of gravity under the action of its own weight.

Weight acts through the center of gravity.
The moment arm (perpendicular distance from the pivot to the line of action of force) for the weight about the center of gravity is zero.
Therefore, the torque (τ = force × moment arm) due to weight about the center of gravity is zero.

Hence, no torque means no rotation caused by its own weight about its center of gravity.

Velocity and Acceleration: Definitions and Units

Q.1 What is the difference between uniform and variable velocity? From the explanation of variable velocity, define acceleration. Give SI units of velocity and acceleration.

Uniform Velocity: An object has uniform velocity if it covers equal distances in equal intervals of time in a specific direction. Both speed and direction remain constant.
Variable Velocity: An object has variable velocity if it covers unequal distances in equal intervals of time, or if its direction of motion changes, or both.

Acceleration: From the concept of variable velocity, acceleration is defined as the rate of change of velocity.

SI Units:

Velocity: meters per second (m/s)
Acceleration: meters per second squared (m/s²)

Vertical Motion: Acceleration and Velocity Signs

Q.2 An object is thrown vertically upward. Discuss the sign of acceleration due to gravity, relative to velocity, while the object is in air.

Upward Motion: The velocity is directed upward, while the acceleration due to gravity (g) is always directed downward. Therefore, the acceleration has a negative sign relative to the upward velocity.
Downward Motion: Both the velocity and acceleration due to gravity are directed downward. Therefore, the acceleration has a positive sign relative to the downward velocity.

Constant Acceleration and Velocity Reversal

Q.3 Can the velocity of an object reverse direction when acceleration is constant? If so, give an example.

Yes. The velocity of an object can reverse direction even when acceleration is constant.

Example: When a ball is thrown vertically upward, its acceleration due to gravity is constant and directed downward throughout its flight. As the ball moves upward, its velocity decreases, momentarily becomes zero at the highest point, and then reverses direction as the ball falls downward. The acceleration remains constant (g, downward) throughout this process.

Kinematics Concepts: Identifying Correct Statements

Q.4 Specify the correct statement:
a) An object can have a constant velocity even its speed is changing.
b) An object can have a constant speed even its velocity is changing.
c) An object can have a zero velocity even its acceleration is not zero.
d) An object subjected to a constant acceleration can reverse its velocity.

Correct Statements:

(a) False: Constant velocity implies constant speed and constant direction.
(b) True: An object in uniform circular motion has constant speed but its direction (and thus velocity) is continuously changing.
(c) True: At the peak of its trajectory when thrown vertically upward, an object momentarily has zero velocity but is still under the constant acceleration of gravity (g).
(d) True: As seen in Q.3, an object thrown upward under constant gravitational acceleration reverses its velocity.

Projectile Motion: Impact Speed from a Tower

Q.5 A man standing on the top of a tower throws a ball straight up with initial velocity v_i and at the same time throws a second ball straight downward with the same speed. Which ball will have larger speed when it strikes the ground? Ignore air friction.

Both balls will strike the ground with the same speed.

This is due to the conservation of energy (or symmetry of motion under gravity). The ball thrown upward will return to the height of the tower with the same speed v_i, but in the downward direction. From that point, both balls effectively start moving downward from the same height with the same initial speed v_i. Therefore, they will reach the ground with identical final speeds.

Velocity and Acceleration: Relative Orientations

Q.6 Explain the circumstances in which the velocity v and acceleration a of a car are:

(i) Parallel

When the car is speeding up (accelerating) in a straight line, its velocity and acceleration are in the same direction, hence parallel.

(ii) Anti-parallel

When the car is slowing down (decelerating or braking) in a straight line, its velocity and acceleration are in opposite directions, hence anti-parallel.

(iii) Perpendicular to one another

When the car is moving in a circular path at a constant speed (uniform circular motion), its velocity is tangential to the path, while its acceleration (centripetal acceleration) is directed towards the center of the circle. Thus, velocity and acceleration are perpendicular.

(iv) v is zero but a is not

When a car momentarily stops at the peak of an upward incline before rolling back down, or when it is momentarily at rest during braking, its velocity is zero, but there is still an acceleration acting on it (e.g., gravity, braking force).

(v) a is zero but v is not zero

When the car is moving with uniform velocity (constant speed in a straight line), its acceleration is zero, but its velocity is non-zero.

Constant Velocity as a Special Case of Acceleration

Q.7 Motion with constant velocity is a special case of motion with constant acceleration. Is this statement true? Discuss.

Yes, this statement is true.

Constant velocity implies that the acceleration is zero. Since zero is a constant value, motion with constant velocity can be considered a special case of motion with constant acceleration, where the constant acceleration happens to be zero.

Newton’s Second Law: Momentum Form

Q.8 Find the change in momentum for an object subjected to a given force for a given time and state law of motion in terms of momentum.

Newton’s Second Law of Motion (Momentum Form):

The rate of change of linear momentum of a body is directly proportional to the applied external force and takes place in the direction of the force.

Mathematically, if a force F acts on a mass m, changing its velocity from v_i to v_f in time t, then:

Change in momentum (Δp) = mv_f – mv_i
According to Newton’s Second Law: F = Δp / t

Impulse and Momentum Relationship

Q.9 Impulse and Its Relation to Momentum

Impulse (I): Impulse is defined as the product of the force applied and the time interval over which it acts.
I = F × Δt
Units of Impulse: Newton-second (Ns)
From Newton’s Second Law (F = Δp / Δt), we can rearrange to get: F Δt = Δp
Therefore, Impulse equals the change in linear momentum (I = Δp).

Law of Conservation of Linear Momentum

Q.10 Law of Conservation of Linear Momentum

Statement: For an isolated system (where no external forces act), the total linear momentum remains constant.

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

This law approximately holds true even when external forces are present but negligible compared to internal forces (e.g., gas pressure in a container, ignoring gravity).

Elastic and Inelastic Collisions

Q.11 Elastic vs Inelastic Collisions

Elastic Collision: In an elastic collision, both linear momentum and kinetic energy of the system are conserved.
Inelastic Collision: In an inelastic collision, linear momentum is conserved, but kinetic energy is not conserved. Some kinetic energy is converted into other forms of energy.
Bouncing Ball Example:
- If a ball rebounds to its original height, the collision is considered elastic (ideal case).
- If a ball does not rebound fully (e.g., to a lower height), the collision is inelastic; kinetic energy is lost.
Reason for Kinetic Energy Loss: In inelastic collisions, kinetic energy is typically converted into other forms such as heat, sound, and energy used for deformation of the colliding objects.

Projectile Motion Fundamentals

Q.12 Projectile Motion

Projectile motion describes the path of an object thrown into the air, subject only to gravity.
Key parameters include time of flight, maximum height, and range. Detailed derivations for these can be found in standard physics textbooks.
The range is maximum when the launch angle is 45° (assuming a level surface and no air resistance).

Projectile Speed: Minimum and Maximum Points

Q.13 Projectile Speed

Minimum Speed: The minimum speed of a projectile occurs at its highest point. At this point, the vertical component of velocity is zero, and only the horizontal component of velocity remains.
Maximum Speed: The maximum speed of a projectile occurs at its launch point and its landing point (assuming it lands at the same height from which it was launched). At these points, both vertical and horizontal components of velocity are present, contributing to the highest overall speed.

Dynamics: Multiple Choice Questions

Q.14 Multiple Choice Questions

Ballistic trajectory means:
(a) The path followed by an un-powered and unguided projectile.
In an elastic collision:
(b) The momentum of the system does not change. (Note: Kinetic energy also does not change in an elastic collision, but momentum conservation is a fundamental aspect of all collisions, elastic or inelastic, in an isolated system.)