Mechanics Formula Sheet and Core Concepts - NDA Physics

Mastering Classical Mechanics is vital for cracking the Physics section of the National Defence Academy (NDA) General Ability Test (GAT). Mechanics accounts for a significant portion of the NDA physics weightage. This comprehensive, LibreTexts-style chapter guide synthesizes key formulas, conceptual breakdowns, and examination trends based on official Union Public Service Commission (UPSC) NDA patterns.

Chapter 1: Kinematics (Rectilinear & Projectile Motion)

Kinematics deals with the motion of points, bodies, and systems without considering the forces that cause the motion.

1.1 Rectilinear Motion with Constant Acceleration

For an object moving in a straight line with uniform acceleration \(a\), initial velocity \(v_0\), and final velocity \(v\) over displacement \(\Delta x\) in time \(t\):

\[v = v_0 + at\] \[\Delta x = v_0 t + \frac{1}{2}at^2\] \[v^2 = v_0^2 + 2a\Delta x\]

1.2 Projectile Motion

When an object is launched at an angle \(\theta\) relative to the horizontal with an initial velocity \(v_0\), its parametric position equations under constant gravitational acceleration \(g\) are:

\[x(t) = v_0 \cos(\theta) t\] \[y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2\]

Chapter 2: Dynamics & Newton's Laws of Motion

Dynamics links forces with the motion of physical bodies.

2.1 Newton's Second Law

The net external force \(\mathbf{F}_{net}\) acting on a body equals the rate of change of linear momentum \(\mathbf{p}\):

\[\mathbf{F}_{net} = \frac{d\mathbf{p}}{dt} = m\mathbf{a}\]

2.2 Friction Force

Frictional force opposes relative motion between contacting surfaces. Static friction \(f_s\) self-adjusts up to a maximum limit, while kinetic friction \(f_k\) acts during relative motion:

\[f_s \le \mu_s N, \quad f_k = \mu_k N\]

where \(\mu_s\) and \(\mu_k\) are the static and kinetic coefficients of friction, and \(N\) is the normal reaction force.

2.3 Centripetal Force

An object moving in a circular path of radius \(r\) with constant tangential speed \(v\) or angular speed \(\omega\) experiences a inward force directed toward the center:

\[F_c = \frac{m v^2}{r} = m \omega^2 r\]

Chapter 3: Work, Energy, and Power

3.1 Work Done and the Work-Energy Theorem

Work \(W\) performed by a force field along path \(\mathbf{r}\) equals the change in kinetic energy \(\Delta K\):

\[W = \int \mathbf{F} \cdot d\mathbf{r} = \Delta K\]

3.2 Potential Energy & Mechanical Energy Conservation

For conservative forces, force is the negative gradient of potential energy \(U\):

\[\mathbf{F} = - abla U\]

In the absence of non-conservative forces (like friction), total mechanical energy \(E\) is conserved:

\[E = K + U = \text{constant}\]

3.3 Instantaneous Power

Power \(P\) is the time rate at which work is performed:

\[P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v}\]

Chapter 4: Linear Momentum and Collisions

4.1 Linear Momentum and Impulse

Linear momentum is given by \(\mathbf{p} = m\mathbf{v}\). An impulse \(\mathbf{J}\) delivered by a force over a time interval equals the net change in momentum:

\[\mathbf{J} = \int \mathbf{F} dt = \Delta \mathbf{p}\]

4.2 One-Dimensional Elastic Collisions

For two colliding masses \(m_1\) and \(m_2\) with initial velocities \(v_1\) and \(v_2\), the final velocity \(v_1'\) of mass \(m_1\) after an elastic collision is:

\[v_1' = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2\]

Chapter 5: Rotational Dynamics

5.1 Rotational Kinematics & Moment of Inertia

Analogous to linear motion, angular position \(\theta\) and angular velocity \(\omega\) under constant angular acceleration \(\alpha\) obey:

\[\theta = \omega_0 t + \frac{1}{2}\alpha t^2, \quad \omega = \omega_0 + \alpha t\]

The moment of inertia \(I\) quantifies rotational inertia about an axis:

\[I = \int r^2 dm\]

5.2 Parallel Axis Theorem

The moment of inertia \(I\) about any parallel axis at distance \(d\) from the center-of-mass axis is:

\[I = I_{cm} + Md^2\]

5.3 Torque and Angular Momentum

Torque \(\boldsymbol{\tau}\) and angular momentum \(\mathbf{L}\) are defined as:

\[\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = I\boldsymbol{\alpha}, \quad \mathbf{L} = I\boldsymbol{\omega} = \mathbf{r} \times \mathbf{p}\]

Chapter 6: Gravitation & Orbital Motion

6.1 Newton's Law of Universal Gravitation

The attractive force between two point masses \(m_1\) and \(m_2\) separated by distance \(r\) is:

\[F = -G \frac{m_1 m_2}{r^2} \hat{\mathbf{r}}\]

The associated gravitational potential energy \(U\) is defined as:

\[U = -G \frac{m_1 m_2}{r}\]

6.2 Kepler's Third Law of Planetary Motion

The square of the orbital period \(T\) of a satellite or planet is proportional to the cube of its orbital radius \(r\):

\[T^2 = \bigl( \frac{4\pi^2}{G(m_1 + m_2)} \bigr) r^3\]

Chapter 7: Oscillations (Simple Harmonic Motion)

7.1 Simple Harmonic Motion (SHM) Governing Equation

Systems undergoing SHM obey the linear second-order differential equation:

\[\frac{d^2x}{dt^2} + \omega^2 x = 0\]

7.2 Angular Frequency Expressions

For a mass-spring system with spring constant \(k\) and a simple pendulum of length \(L\):

\[\omega = \sqrt{\frac{k}{m}} \quad \text{(Spring)}, \quad \omega = \sqrt{\frac{g}{L}} \quad \text{(Pendulum)}\]

7.3 Total Mechanical Energy in SHM

The total energy remains constant throughout the oscillation amplitude \(A\):

\[E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2\]

UPSC NDA Exam Weightage & Pattern Analysis

In the NDA GAT Physics syllabus, Mechanics contributes approximately 35% to 40% of the total Physics questions. The NTA/UPSC paper pattern emphasizes conceptual clarity, standard direct formula applications, and unit conversions.

• High Yield Topics: Work-Power-Energy conservation, Newton's laws with friction, simple pendulum time period, and projectile motion parameters.
• Common Question Styles: Statement-based reasoning, matching physical quantities with dimensions, and direct numerical evaluation based on ratio changes (e.g., doubling radius or mass).

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