Mathematical Methods for Physics I: Master the Fundamentals of Physics

Question 1

In rigid body dynamics, Newton's second law can be expressed using angular quantities. Which equation accurately represents this law for rotational motion?

Accepted Answer

τ = Iα

Answer

F = ma

Answer

W = Fd

Answer

p = mv

Question 2

Which of the following is NOT a fundamental principle of statics?

Accepted Answer

The law of conservation of energy

Answer

The law of equilibrium

Answer

The principle of moments

Answer

The principle of virtual work

Question 3

Which of the following concepts is not a core component of engineering mechanics?

Accepted Answer

Thermodynamics

Answer

Statics

Answer

Dynamics

Answer

Strength of materials

Question 4

Which of the following is NOT a fundamental principle of engineering mechanics?

Accepted Answer

Thermodynamics

Answer

Dynamics

Answer

Statics

Answer

Strength of Materials

Question 5

In engineering mechanics, which of the following forces is NOT considered fundamental?

Accepted Answer

Frictional force

Answer

Gravitational force

Answer

Electromagnetic force

Answer

Strong nuclear force

Question 6

Which of the following is not a fundamental concept in statics?

Accepted Answer

Deformation

Answer

Equilibrium

Answer

Moment

Answer

Couple

Question 7

A rigid body is in equilibrium under the influence of three concurrent forces. What are the necessary conditions for equilibrium?

Accepted Answer

The vector sum of the forces acting on the body is zero.

Accepted Answer

The vector sum of the moments of the forces about any point is zero.

Answer

The body is not moving.

Answer

The body is accelerating.

Question 8

Which of the following is NOT a fundamental concept within statics?

Accepted Answer

Motion

Answer

Equilibrium

Answer

Moment

Answer

Force

Question 9

What is the unit for moment of inertia, a quantity that measures an object's resistance to rotation?

Accepted Answer

kg.m^2

Answer

kg.m

Answer

N.m

Question 10

Which force causes an object to stretch or elongate lengthwise in engineering mechanics?

Accepted Answer

Tensile force

Answer

Shear force

Answer

Compressive force

Question 11

Which of the following physical quantities is a scalar, meaning it has only magnitude and no direction?

Accepted Answer

Mass

Answer

Force

Answer

Displacement

Question 12

A force of 100 Newtons acts on a 10-kilogram object. Calculate the object's acceleration using Newton's second law of motion, F = m * a.

Accepted Answer

10 m/s²

Answer

1 m/s²

Answer

100 m/s²

Answer

1000 m/s²

Question 13

In engineering mechanics, how does the magnitude of the moment generated by a force applied to a rigid body change as the distance from the point of application to the axis of rotation increases?

Accepted Answer

The moment generated increases as the distance increases.

Answer

The moment generated remains constant.

Answer

The moment generated decreases as the distance increases.

Answer

The moment generated is independent of the distance.

Question 14

Newton's second law of motion states that the force acting on an object is directly proportional to its mass and acceleration. Which equation represents this relationship?

Accepted Answer

F = m * a

Answer

F = m/a

Answer

F = a/m

Question 15

According to D'Alembert's principle, what is the resultant force acting on a rigid body in dynamic equilibrium?

Accepted Answer

Zero, indicating a balanced state with no net force

Answer

Non-zero and opposing the body's acceleration

Answer

Equal to the net external force acting on the body

Answer

Equal to the frictional force acting on the body

Question 16

For a rigid body in equilibrium under the influence of three coplanar forces, what is the relationship between the lines of action of these forces?

Accepted Answer

They intersect at a single point.

Answer

They are parallel to each other.

Answer

Two are parallel and the third is perpendicular to them.

Question 17

In engineering design, what causes stress concentration?

Accepted Answer

Sudden changes in shape, causing stresses to redistribute.

Answer

Using brittle materials in the design.

Answer

External forces stronger than the material's yield strength.

Answer

Materials with low fatigue strength.

Question 18

Calculate the dot product of vectors A = (2, 3) and B = (-1, 4).

Accepted Answer

10

Answer

5

Answer

15

Answer

20

Question 19

What is the significance of the Wronskian determinant in systems of differential equations?

Accepted Answer

It determines the linear independence of the solutions

Answer

It provides a measure of the stability of the system

Answer

It can be used to find the general solution

Answer

It has no physical significance

Question 20

Identify the invertible matrix among the following options:

Accepted Answer

[[1, 2], [3, 4]]

Answer

[[1, 0], [0, 0]]

Answer

[[1, 2], [2, 4]]

Answer

[[0, 1], [0, 0]]

Question 21

During an impulsive collision, which physical quantity remains unchanged?

Accepted Answer

Momentum

Answer

Kinetic energy

Answer

Angular momentum

Answer

Total mechanical energy

Question 22

Which of the following is NOT a fundamental principle of fluid mechanics?

Accepted Answer

Entropy always increases

Answer

Mass conservation

Answer

Momentum conservation

Answer

Energy conservation

Question 23

Find the general solution to the differential equation y'' + 4y = 0.

Accepted Answer

y = C1cos(2x) + C2sin(2x)

Answer

y = C1e^(2x) + C2e^(-2x)

Answer

y = C1x + C2

Answer

y = C1x^2 + C2x + C3

Question 24

Find the solution to the differential equation dy/dx = y(1 - y), given the initial condition y(0) = 0.5.

Accepted Answer

y = C / (1 - Ce^x)

Answer

y = 1 - Ce^x

Answer

y = Ce^x

Answer

y = 1 + Ce^x

Question 25

Which quantity is NOT a characteristic of a vector?

Accepted Answer

Speed

Answer

Magnitude

Answer

Direction

Question 26

What mathematical operator is known as the Laplacian?

Accepted Answer

∇^2

Answer

∇ x

Answer

∇ .

Answer

∇

Question 27

Find the general solution to the differential equation: dy/dx = x^2.

Accepted Answer

y = x^3/3 + C

Answer

y = x^2/2 + C

Answer

y = 2x + C

Answer

y = x + C

Question 28

Which property is characteristic of orthogonal matrices?

Accepted Answer

Transpose is equal to inverse

Answer

Determinant is 1

Answer

Eigenvalues are all real

Answer

Trace is zero

Question 29

In mechanical systems, what phenomenon causes a significant increase in amplitude when the driving frequency aligns with the natural frequency?

Accepted Answer

Resonance

Answer

Damping

Answer

Forced vibration

Answer

Harmonic motion

Question 30

A particle moves in a circular path at a constant speed. Which direction does its acceleration vector point at any instant?

Accepted Answer

Toward the center of the circle

Answer

Away from the center of the circle

Answer

In the direction of motion

Answer

Zero

Question 31

Which theorem in vector calculus is used to calculate the circulation of a vector field around a surface?

Accepted Answer

Stokes' theorem

Answer

Green's theorem

Answer

Gauss's theorem

Answer

The divergence theorem

Question 32

What is the Boolean expression for an XOR (exclusive OR) gate?

Accepted Answer

A XOR B

Answer

A AND B

Answer

A OR B

Answer

NOT (A AND B)

Question 33

Consider a scalar field Φ(x,y,z) representing a potential energy function. Which of the following expressions gives the gradient of Φ?

Accepted Answer

∇Φ = (∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)

Answer

∇Φ = (∂Φ/∂y, ∂Φ/∂z, ∂Φ/∂x)

Answer

∇Φ = (Φ/∂x, Φ/∂y, Φ/∂z)

Question 34

In a horizontal pipe with varying cross-sectional area, how does the fluid velocity change? Explain the reason for this change.

Accepted Answer

The fluid velocity increases as the cross-sectional area decreases to maintain a constant flow rate.

Answer

The fluid velocity remains constant throughout the pipe regardless of the cross-sectional area.

Answer

The fluid velocity decreases as the cross-sectional area decreases due to increased friction.

Question 35

The dot product of two vectors A and B is a:

Accepted Answer

Scalar quantity

Answer

Vector quantity

Answer

Matrix quantity

Answer

Tensor quantity

Question 36

The gradient of a scalar field f(x, y, z) is a:

Accepted Answer

Vector field

Answer

Tensor field

Answer

Matrix field

Answer

Scalar field

Question 37

Which of the following matrices is orthogonal?

Accepted Answer

A matrix whose inverse is its transpose

Answer

A matrix whose determinant is 1

Answer

A matrix whose eigenvalues are all real

Answer

A matrix whose trace is 0

Question 38

The eigenvalue equation for a matrix A is:

Accepted Answer

Av = λv

Answer

A²v = λv

Answer

Av² = λv

Answer

A⁻¹v = λv

Question 39

Which of the following is a fundamental equation in electromagnetism?

Accepted Answer

Maxwell's equations

Answer

Schrödinger's equation

Answer

Newton's laws of motion

Answer

Einstein's field equations

Question 40

Which of the following is the correct expression for the dot product of two vectors A = (A1, A2, A3) and B = (B1, B2, B3)?

Accepted Answer

A·B = A1*B1 + A2*B2 + A3*B3

Answer

A·B = |A| * |B| * cos(θ)

Answer

A·B = A1*B2 - A2*B1

Answer

A·B = A1*B1 - A2*B2 + A3*B3

Question 41

What is the divergence of the vector field F = (x^2, y^2, z^2)?

Accepted Answer

2(x + y + z)

Answer

1

Answer

0

Answer

x^2 + y^2 + z^2

Question 42

Which of the following is the solution to the system of equations: x + y = 5, x - y = 1?

Accepted Answer

x = 3, y = 2

Answer

x = 1, y = 4

Answer

x = 2, y = 3

Question 43

What is the Laplace transform of the function f(t) = e^(at)?

Accepted Answer

1/(s - a)

Answer

s/(s - a)

Answer

(s - a)/s

Answer

1 - a/s

Question 44

Which of the following is the correct form of the chain rule for derivatives?

Accepted Answer

d/dx(f(g(x))) = f'(g(x)) * g'(x)

Answer

d/dx(f(g(x))) = f(g'(x)) * g(x)

Answer

d/dx(f(g(x))) = f'(g(x)) + g'(x)

Question 45

Identify the scalar quantity among the following:

Accepted Answer

Mass

Answer

Electric field

Answer

Velocity

Answer

Force

Question 46

What is the result when taking the dot product of two vectors?

Accepted Answer

A scalar

Answer

A tensor

Answer

A matrix

Answer

A vector

Question 47

Classify the following as a first-order differential equation or not: dy/dx = y

Accepted Answer

First-order differential equation

Answer

Not a differential equation

Answer

Second-order differential equation

Answer

Third-order differential equation

Question 48

Identify the following among the given options: Electric field

Accepted Answer

Vector field

Answer

Tensor field

Answer

Matrix field

Answer

Scalar field

Question 49

What is the divergence of a vector field?

Accepted Answer

A scalar

Answer

A tensor

Answer

A matrix

Answer

A vector

Question 50

What is the curl of a vector field?

Accepted Answer

A vector

Answer

A tensor

Answer

A matrix

Answer

A scalar

Question 51

Which of the following represents the Laplacian operator?

Accepted Answer

∇^2

Answer

∇

Answer

∇.

Answer

∇x

Question 52

What is the solution to the Laplace equation ∇^2u = 0 called?

Accepted Answer

Harmonic function

Answer

Exponential function

Answer

Polynomial function

Question 53

Find the curl of the vector field **F(x, y, z) = (x^2, y^2, z^2)**.

Accepted Answer

(0, 0, 0)

Answer

(-x, -y, -z)

Answer

(x, y, z)

Answer

(2x, 2y, 2z)

Question 54

Solve the system of linear equations using Cramer's rule:
      2x - 3y = 1
      x + 2y = 5

Accepted Answer

x = 7/5, y = 1/5

Answer

x = 1/5, y = 7/5

Answer

x = 5/7, y = 1/7

Answer

x = -7/5, y = -1/5

Question 55

Which of the following is the Fourier series representation of the function **f(x) = x** on **[-π, π]**?

Accepted Answer

f(x) = Σ((-1)^n-1 / n) sin(nx), n = 1, 2, 3, ...

Answer

f(x) = Σ((-1)^n / n) sin(nx), n = 0, 1, 2, ...

Answer

f(x) = Σ(1 / n) sin(nx), n = 1, 2, 3, ...

Answer

f(x) = Σ((-1)^n / n) cos(nx), n = 1, 2, 3, ...

Question 56

Find the inverse Laplace transform of the function **F(s) = 1/(s^2 + 4)**.

Accepted Answer

(1/2) sin(2t)

Answer

sin(2t)

Answer

(1/2) cos(2t)

Question 57

Which of the following vector fields is NOT conservative?

Accepted Answer

(y^2, x, z)

Answer

(x, y, z)

Answer

(2x, 2y, 2z)

Answer

(yz, xz, xy)

Question 58

Find the determinant of the matrix A = [[2, -1], [4, 3]]

Accepted Answer

10

Answer

2

Answer

-2

Question 59

Calculate the work done by the force F = (2x + y)i + xj in moving a particle along the straight-line path from (0, 0) to (1, 2).

Accepted Answer

3 J

Answer

2 J

Answer

4 J

Question 60

Which of the following is true about the dot product of two vectors?

Accepted Answer

It is a scalar quantity.

Answer

It is a tensor quantity.

Answer

It is a vector quantity.

Answer

It is a matrix quantity.

Question 61

Which of the following is true about the cross product of two vectors?

Accepted Answer

It is a vector quantity perpendicular to both input vectors.

Answer

It is a matrix quantity.

Answer

It is a tensor quantity.

Answer

It is a scalar quantity.

Question 62

Which of the following is true about the gradient of a scalar field?

Accepted Answer

It is a vector quantity that points in the direction of the greatest rate of change of the scalar field.

Answer

It is a tensor quantity.

Answer

It is a scalar quantity.

Answer

It is a matrix quantity.

Question 63

Which of the following is true about the divergence of a vector field?

Accepted Answer

It is a scalar quantity that measures the outward flux of the vector field from a point.

Answer

It is a tensor quantity.

Answer

It is a matrix quantity.

Answer

It is a vector quantity.

Question 64

Which of the following is true about the curl of a vector field?

Accepted Answer

It is a vector quantity that measures the circulation of the vector field around a point.

Answer

It is a tensor quantity.

Answer

It is a matrix quantity.

Answer

It is a scalar quantity.