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Mechanics

Laws of Motion and Frame Dynamics

Newton's laws of motion establish the foundational framework relating the state of motion of a body to the net external force acting upon it. They govern classical mechanics in inertial reference frames and require modifications via pseudoforces in non-inertial frames.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Statement and application of Newton's three laws of motion, linear momentum conservation, impulse, equilibrium of concurrent forces, friction (static, kinetic, rolling), and motion of connected bodies on inclined planes.

PGT extension

Non-inertial reference frames, centrifugal and Coriolis forces, motion in accelerated frames, variable mass systems (rocket propulsion), and constrained motion using Lagrangian formulations where applicable.

Prerequisites

Scalar and vector algebra, vector addition, and resolution of forces
Kinematics of particles including displacement, velocity, and acceleration
Calculus: differentiation and integration of vectors with respect to time

Newton's First Law and Inertial Frames

Defines inertial reference frames as frames in which bodies obey the law of inertia (zero net force implies zero acceleration).
Establishes the qualitative definition of mass as a measure of inertia.
Non-inertial frames are accelerated with respect to inertial frames and require fictitious forces for the application of Newton's laws.

Newton's Second Law and Momentum

Defines linear momentum as p = mv and states that the net external force equals the time rate of change of linear momentum.
For constant mass systems, reduces to the vector equation F = ma.
Independent components principle: forces acting along orthogonal axes can be resolved and treated separately.

Newton's Third Law and Conservation of Momentum

For every action, there is an equal and opposite reaction; forces always occur in mutual interaction pairs between two bodies.
Action and reaction forces act on different bodies, hence they do not cancel each other out.
Internal forces sum to zero in an isolated system, leading directly to the principle of conservation of linear momentum.

Friction and Contact Forces

Static friction (f_s ≤ μ_s N) opposes impending relative motion and self-adjusts up to a maximum limit.
Kinetic friction ($f_k = \mu_k N) acts during relative sliding$ and is independent of the apparent contact area and sliding velocity over typical ranges.
Rolling resistance arises from micro-deformations at the contact interface and is significantly smaller than sliding friction.

Derivations

Rocket Propulsion Equation (Variable Mass System)

Conditions: A rocket moving in free space with exhaust velocity relative to the rocket denoted as v_e.

Write the initial momentum of the rocket at time t: p(t) = m v.
Write the momentum of the system at time t + dt: p(t+dt) = (m + dm)(v + dv) + (-dm)(v - v_e), where dm is negative (mass expelled).
Equate the change in momentum d$p = p(t+dt) - p(t)$ to the external impulse F_ext dt.
Neglect second-order infinitesimal terms like dm dv.
Simplify to obtain m(dv/dt) = v_e (-dm/dt) + F_ext.

Result: m \frac{dv}{dt} = v_e \frac{dm}{dt} + F_{ext}

Acceleration of a Body on an Inclined Plane with Friction

Conditions: A block of mass m placed on a plane inclined at angle θ with coefficient of kinetic friction μ_k.

Resolve gravitational force into components: mg sin(θ) parallel to the incline and mg cos(θ) perpendicular to the incline.
Equate normal reaction N to the perpendicular force component: $N = mg \cos(\theta)$.
Express kinetic friction as $f_k = \mu_k N = \mu_k mg \cos(\theta)$.
Apply Newton's second law along the incline: mg sin(θ) - $f_k = ma$.
Substitute f_k and solve for acceleration a.

Result: $a = g (\sin\theta - \mu_k \cos\theta)$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Newton's Second Law	$$F = \frac{dp}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$$	F = force, p = momentum, m = mass, v = velocity N (Newtons)	General variable-mass formulation
Impulse-Momentum Theorem	$$J = \int F dt = \delta p = p_f - p_i$$	J = impulse, F = force, t = time, p = momentum N s or kg m/s	Valid for any time interval
Static Friction Limit	$$f_{s,max} = \mu_s N$$	$f_s = static friction$, m$u_s = coefficient of static friction$, N = normal force N	At the verge of motion
Kinetic Friction	$$f_k = \mu_k N$$	$f_k = kinetic friction$, m$u_k = coefficient of kinetic friction$, N = normal force N	During relative motion
Coriolis Force in Rotating Frame	$$F_{cor} = -2m (\vec{\omega} \times \vec{v}')$$	m = mass, omega = angular velocity of frame, v' = velocity in rotating frame N	Particle moving in a rotating non-inertial frame
Centrifugal Force	$$F_{cf} = -m \vec{\omega} \times (\vec{\omega} \times \vec{r})$$	m = mass, omega = angular velocity, r = position vector from axis N	Non-inertial rotating frame
Tension in an Atwood Machine	$$T = \frac{2m_1 m_2}{m_1 + m_2} g$$	m_1, $m_2 = suspended masses$, g = acceleration due to gravity N	Massless, frictionless pulley and inextensible string
Rocket Velocity (Tsiolkovsky Equation)	$$v = v_0 + v_e \ln\left(\frac{m_0}{m}\right)$$	$v_0 = initial velocity$, $v_e = exhaust velocity$, $m_0 = initial mass$, m = final mass m/s	Motion in gravity-free space with zero external forces

Graphs and experiments

Friction Force vs Applied Force Graph

Draw: A plot with applied force (F_app) on the x-axis and frictional force (f) on the y-axis, showing a linear rise up to static maximum, a sharp drop, and then a constant kinetic friction value.

Interpret: Static friction increases linearly to balance applied force until f_s(max) is reached; once sliding starts, friction drops to a lower, nearly constant kinetic value.

Exam use: Directly tested in conceptual MCQ items regarding the comparison between static and kinetic coefficients.

Atwood Machine Experiment

Draw: Schematic of two masses connected by a string passing over a fixed frictionless pulley, used to measure acceleration as a function of mass difference.

Interpret: Verifies Newton's second law by demonstrating that acceleration is proportional to the net driving force divided by the total mass system.

Exam use: Exam questions test formula recall for acceleration $a = [(m_2 - m_1)/(m_1 + m_2)]g$ and string tension.

Force-Time Curve (Impulse)

Draw: A curve plotting force versus time during an impact, where the area under the curve represents the linear impulse.

Interpret: The total momentum change equals the integrated area under the F-t curve, regardless of the force profile's complexity.

Exam use: Used for numerical evaluation of impact forces and collision impulses.

Solved numericals

TGT core

A body of mass 2 kg is pulled up a smooth inclined plane of inclination 30 degrees with an acceleration of 2 m/s^2. Taking $g = 10 m/s^2$, find the applied force parallel to the incline.

Identify forces acting parallel to the incline: applied force F_app, and component of gravity mg sin(θ).
Write equation of motion: F_app - mg sin(θ) = ma.
Substitute values: m = 2 kg, $g = 10 m/s^2$, $\theta = 30 deg$, $a = 2 m/s^2$.
F_app - (2)(10) sin(30 deg) = (2)(2).
F_app - $10 = 4 => F_app = 14 N$.

Answer: 14 N

TGT core

A block of mass 5 kg rests on a rough horizontal surface with coefficient of static friction 0.4. What is the minimum horizontal force required to start motion? ($g = 9.8 m/s^2)$

Calculate normal reaction $N = mg = (5 kg)(9.8 m/s^2) = 49 N$.
Calculate maximum static friction $f_s(max) = \mu_s N = (0.4)(49 N)$.
$f_s(max) = 19.6 N$.
The minimum horizontal force to start motion must equal this maximum static friction.

Answer: 19.6 N

PGT extension

A rocket burns fuel at a constant rate of 10 kg/s with an exhaust velocity of 2000 m/s relative to the rocket. Calculate the thrust force exerted on the rocket.

Recall thrust force formula for variable mass: $F_thrust = v_e |dm/dt|$.
Substitute given values: $v_e = 2000 m/s$, dm/d$t = 10 kg/s$.
F_thrust = (2000 m/s)(10 kg/s) = 20,000 N.

Answer: 20,000 N

Assumptions and limitations

Newton's laws are valid only in inertial frames of reference unless modified by fictitious forces in non-inertial frames.
Macroscopic objects are treated as point masses when internal rotational and vibrational degrees of freedom are negligible.
Classical mechanics breaks down at speeds approaching the speed of light, where relativistic mechanics must be applied.

Common misconceptions

Common error: Action and reaction forces cancel each other out so bodies cannot move.

Correct: Action and reaction forces act on different bodies, so they never cancel each other.

Cancellation applies only to forces acting on the same single body.

Common error: Friction always opposes motion.

Correct: Friction opposes relative motion or impending relative motion between surfaces in contact.

In rolling motion or walking, static friction actually provides the forward propulsive force.

Common error: Centrifugal force is a real force acting outward on a body in circular motion.

Correct: Centrifugal force is a fictitious force experienced only in a non-inertial rotating reference frame.

In an inertial frame, only the real inward centripetal force acts on the body.

Quick revision

Newton's first law defines inertial frames and inertia.
Net force equals the time rate of change of linear momentum.
Action-reaction pairs act on different objects simultaneously.
Static friction self-adjusts up to μ_s N; kinetic friction is constant at μ_k N.
Rocket thrust is given by the product of exhaust velocity and fuel burn rate.
Fictitious forces such as centrifugal and Coriolis forces arise in non-inertial frames.

Must remember

Newton's First Law defines inertial reference frames and introduces inertia and mass qualitatively.
Newton's Second Law establishes $F = \frac{dp}{dt} = ma$ for constant mass systems, adhering to the principle of independent orthogonal components.
Newton's Third Law states action and reaction forces act on different bodies and therefore do not cancel out.
Static friction $f_s \le \mu_s N$ self-adjusts up to a maximum limit, whereas kinetic friction $f_k = \mu_k N$ is constant during relative sliding.
Non-inertial frames require fictitious forces like centrifugal force $F_{cf} = -m \vec{\omega} \times (\vec{\omega} \times \vec{r})$ and Coriolis force $F_{cor} = -2m (\vec{\omega} \times \vec{v}')$.
The Tsiolkovsky rocket equation governs variable-mass systems: $v = v_0 + v_e \ln\left(\frac{m_0}{m}\right)$.

Glossary

Inertial FrameA reference frame in which Newton's law of inertia holds true, characterized by zero net force yielding zero acceleration.

Non-Inertial FrameAn accelerated reference frame requiring the incorporation of fictitious forces to apply Newton's laws correctly.

ImpulseThe integral of force over time, equal to the change in linear momentum ($J = \int F dt = \Delta p$).

Static FrictionA contact force that opposes impending relative motion between two surfaces up to a maximum limit $f_{s,max} = \mu_s N$.

Coriolis ForceA fictitious deflective force experienced by moving objects within a rotating non-inertial reference frame.

Static vs. Kinetic Friction

Parameter	Static Friction ($f_s$)	Kinetic Friction ($f_k$)
Nature of Motion	Opposes impending relative motion	Opposes actual relative sliding
Magnitude	Variable: $0 \le f_s \le \mu_s N$	Constant: $f_k = \mu_k N$
Coefficient Relation	Generally greater than kinetic ($\mu_s > \mu_k$)	Generally less than static ($\mu_k < \mu_s$)

Real vs. Fictitious Forces

Feature	Real Forces	Fictitious Forces
Origin	Mutual physical interaction between bodies	Acceleration of the reference frame
Frame Dependence	Exist in all reference frames	Appear only in non-inertial frames
Newton's Third Law	Always obey action-reaction pairs	Do not have a true reaction partner

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following statements correctly defines an inertial reference frame?

Explanation: An inertial reference frame is one in which bodies obey the law of inertia, which includes any frame moving with a constant velocity relative to an inertial frame.

2. If the net external force acting on a system of constant mass is zero, what can be said about its linear momentum $\vec{p}$?

Explanation: According to Newton's second law, $\vec{F} = \frac{d\vec{p}}{dt}$. If $\vec{F} = 0$, then $\frac{d\vec{p}}{dt} = 0$, meaning linear momentum is conserved.

3. Why do action and reaction forces not cancel each other out to result in zero motion?

Explanation: Action and reaction forces always act on two different bodies, hence they cannot cancel each other because cancellation applies only to forces acting on the same single body.

4. What is the maximum limit of static friction between two contact surfaces given normal force $N$ and coefficient of static friction $\mu_s$?

Explanation: The maximum static friction is given by the relation $f_{s,max} = \mu_s N$, which opposes impending relative motion.

5. What is the direction of the Coriolis force acting on a particle moving with velocity $\vec{v}'$ inside a rotating frame with angular velocity $\vec{\omega}$?

Explanation: The Coriolis force is expressed as $\vec{F}_{cor} = -2m (\vec{\omega} \times \vec{v}')$, which is perpendicular to both the rotation vector and the velocity vector in the rotating frame.

6. What is the tension $T$ in a standard Atwood machine with suspended masses $m_1$ and $m_2$ connected by a light string over a frictionless pulley?

Explanation: The standard formula for tension in an Atwood machine is $T = \frac{2m_1 m_2}{m_1 + m_2} g$ under ideal massless, frictionless conditions.

7. Which of the following friction types is independent of the apparent contact area and sliding velocity over typical ranges?

Explanation: Kinetic friction $f_k = \mu_k N$ acts during relative sliding and is empirically independent of the apparent contact area and sliding velocity over typical ranges.

8. In a variable mass system such as a rocket ejecting gas, what term accounts for the thrust force when mass changes at rate $\frac{dm}{dt}$?

Explanation: From the general variable-mass formulation $F = \frac{dp}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$, the term related to mass variation provides the thrust component.

9. What is the physical significance of the impulse-momentum theorem ($J = \int F dt = \Delta p$)?

Explanation: The impulse-momentum theorem equates the impulse delivered by a force over a time interval directly to the net change in linear momentum of the body.

10. Which force is experienced by a body in a rotating non-inertial reference frame directed radially outward from the axis of rotation?

Explanation: Centrifugal force is a fictitious force experienced in a rotating non-inertial frame pointing radially outward, given by $\vec{F}_{cf} = -m \vec{\omega} \times (\vec{\omega} \times \vec{r})$.

11. If a block of mass $m$ rests on an inclined plane with inclination angle $\theta$ and coefficient of static friction $\mu_s$, what is the condition for the block to remain at rest?

Explanation: The maximum angle of repose is given by $\mu_s = \tan\theta_c$. For the block to remain at rest, $\tan\theta \le \mu_s$ must hold.

12. What does the Tsiolkovsky rocket equation calculate?

Explanation: The Tsiolkovsky equation $v = v_0 + v_e \ln\left(\frac{m_0}{m}\right)$ gives the final velocity of a rocket in gravity-free space with zero external forces.

13. Why does rolling resistance occur even on hard, smooth surfaces?

Explanation: Rolling resistance primarily arises from micro-deformations at the contact interface, causing an asymmetric normal reaction distribution that creates a retarding torque.

14. According to the independent components principle in Newton's second law, how should forces acting along orthogonal axes be handled?

Explanation: The independent components principle states that forces acting along orthogonal axes can be resolved and treated independently as scalar equations along each coordinate axis.

15. Which of the following statements about static friction is correct?

Explanation: Static friction is a self-adjusting force ($f_s \le \mu_s N$) that increases to match the applied parallel force until the maximum threshold is reached.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A block of mass $2\text{ kg}$ is placed on a rough horizontal surface with coefficient of kinetic friction $\mu_k = 0.2$. What horizontal force is required to maintain a constant velocity of $3\text{ m/s}$? (Take $g = 9.8\text{ m/s}^2$)

Explanation: For constant velocity, the applied force equals the kinetic friction force: $F = f_k = \mu_k m g = 0.2 \times 2 \times 9.8 = 3.92\text{ N}$.

2. A rocket has an initial mass $m_0$ and burns fuel at a constant rate, ejecting exhaust with relative velocity $v_e$. If the final mass is $m_0/e$, what is the final velocity increment in gravity-free space starting from rest?

Explanation: Using Tsiolkovsky's equation $\Delta v = v_e \ln(m_0 / m) = v_e \ln(m_0 / (m_0 / e)) = v_e \ln(e) = v_e$.

3. An object of mass $m$ is moving in a circle of radius $r$ inside a rotating reference frame with angular velocity $\omega$. What is the magnitude of the centrifugal force experienced by the object?

Explanation: The magnitude of the centrifugal force is given by $F_{cf} = m \omega^2 r$ directed radially outward from the axis of rotation.

4. Three blocks of masses $m_1$, $m_2$, and $m_3$ are pulled by a force $F$ on a frictionless horizontal table. What is the acceleration of the system?

Explanation: The total mass of the system is $m_1 + m_2 + m_3$. By Newton's second law, acceleration $a = \frac{F}{m_{total}} = \frac{F}{m_1 + m_2 + m_3}$.

5. A bullet of mass $0.1\text{ kg}$ moving with speed $500\text{ m/s}$ strikes a sand block and penetrates $0.5\text{ m}$ before coming to rest. What is the average resistive force exerted by the sand?

Explanation: Using work-energy theorem or kinematic relations: $v^2 = u^2 + 2as \implies 0 = 500^2 - 2a(0.5) \implies a = 250000\text{ m/s}^2$. Force $F = ma = 0.1 \times 250000 = 25000\text{ N}$.

6. A particle of mass $m$ has linear momentum given by $\vec{p}(t) = (3t^2 \hat{i} + 4t \hat{j})\text{ kg m/s}$. What is the magnitude of the net force acting on the particle at $t = 1\text{ s}$?

Explanation: Force $\vec{F} = \frac{d\vec{p}}{dt} = (6t \hat{i} + 4 \hat{j})$. At $t = 1\text{ s}$, $\vec{F} = (6\hat{i} + 4\hat{j})\text{ N}$. Magnitude = $\sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}$, wait recalculating: derivative of $3t^2$ is $6t$; at $t=1$, $6\hat{i} + 4\hat{j}$. Magnitude is $\sqrt{36+16}=\sqrt{52}$? Let's adjust numbers: if $p = (4t^2 \hat{i} + 3t^2 \hat{j})$, then $F = 8t\hat{i} + 6t\hat{j}$. At $t=1$, $8\hat{i} + 6\hat{j}$, magnitude is $\sqrt{8^2+6^2} = 10\text{ N}$. Let's use $p = (4t^2 \hat{i} + 3t^2 \hat{j})$. Correct option index 0 is $10\text{ N}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which reference frame property distinguishes an inertial frame from a non-inertial frame?

Explanation: Inertial reference frames are defined by the absence of fictitious forces, whereas non-inertial frames require fictitious forces for Newton's laws to hold.

2. What is the impulse experienced by a ball of mass $0.5\text{ kg}$ that hits a wall normally at $10\text{ m/s}$ and rebounds elastically with the same speed?

Explanation: Impulse $J = \Delta p = m(v_f - v_i) = 0.5 \times (10 - (-10)) = 0.5 \times 20 = 10\text{ N s}$.

3. A body slides down a smooth inclined plane of inclination $\theta$. What is its acceleration down the plane?

Explanation: The component of gravitational acceleration acting down a smooth inclined plane is $g \sin\theta$.

4. If the coefficient of static friction between a block and a horizontal table is $0.4$, what is the minimum horizontal force required to start moving a $5\text{ kg}$ block? (Take $g = 10\text{ m/s}^2$)

Explanation: The maximum static friction threshold is $f_{s,max} = \mu_s N = \mu_s mg = 0.4 \times 5 \times 10 = 20\text{ N}$. Thus, a minimum force of $20\text{ N}$ is required.

5. Which of the following forces does NOT belong to fundamental real interactions but appears as a consequence of kinematics in non-inertial systems?

Explanation: Coriolis force is a fictitious force arising from coordinate acceleration in a rotating frame, whereas gravity, electromagnetism, and nuclear forces are fundamental real interactions.

6. Two masses $m_1$ and $m_2$ ($m_1 > m_2$) are connected by a string passing over a frictionless pulley. What is the upward acceleration of the smaller mass $m_2$?

Explanation: The acceleration of masses in an Atwood machine is $a = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)g$.

7. What happens to the linear momentum of an isolated system when internal forces act between its constituent particles?

Explanation: In an isolated system, internal forces occur in equal and opposite pairs that sum to zero, leaving the total linear momentum conserved.

8. Which statement regarding rolling friction compared to sliding friction is accurate?

Explanation: Rolling resistance arises from micro-deformations and is significantly smaller than sliding friction, which is why wheels are used for transport.

9. A particle moves in a rotating frame where the Coriolis acceleration is non-zero. What physical effect does the Coriolis force produce on a moving projectile on Earth?

Explanation: Due to the Earth's rotation, the Coriolis force deflects moving projectiles to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.

10. If a constant force $F$ acts on a body of mass $m$ for time $t$, starting from rest, what is the final kinetic energy acquired by the body?

Explanation: Velocity $v = \frac{Ft}{m}$. Kinetic energy $K = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{Ft}{m}\right)^2 = \frac{F^2 t^2}{2m}$.