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Mechanics
Force and Acceleration in Classical Mechanics
Force is an external agency capable of changing the state of rest or uniform motion of a body, directly causing acceleration according to Newton's laws of motion. Acceleration quantifies the rate of change of velocity vector per unit time within an inertial or non-inertial reference frame.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Newton's laws of motion, linear acceleration, tension in strings, pulley systems, apparent weight in lifts, and basic conservation of linear momentum.
PGT extension
Motion in non-inertial frames, pseudoforces, central forces, variable mass systems (rocket propulsion), and Lagrangian formulation of equations of motion.
Prerequisites
Vector algebra (addition, dot product, cross product, and differentiation of vectors)
Kinematics of particles (displacement, velocity, and acceleration definitions)
Calculus (ordinary differential equations and integration)
Newton's First Law and Inertial Frames
Defines inertial reference frames as frames in which Newton's first law of motion holds true.
An object remains at rest or in uniform motion unless acted upon by an external net resultant force.
Provides the qualitative definition of force and establishes the concept of inertia as inherent mass.
Newton's Second Law and Momentum
The rate of change of linear momentum of a body is directly proportional to the applied external force and takes place in the direction in which the force acts.
Mathematically expressed as $F = dp/dt$, which reduces to F = ma for a system of constant rest mass m.
Establishes the fundamental equation of motion for a particle in classical mechanics under both constant and variable forces.
Newton's Third Law and Action-Reaction Pairs
To every action, there is always an equal and opposite reaction acting simultaneously on different bodies.
Internal forces within an isolated system cancel out vectorially, leading directly to the law of conservation of linear momentum.
Applies universally to contact forces, gravitational forces, and electromagnetic forces alike.
Non-Inertial Frames and Pseudoforces
Frames accelerating with respect to an inertial frame are non-inertial frames where Newton's laws do not directly apply without modification.
A fictitious or pseudoforce equal to -ma_0 must be introduced, where a_0 is the acceleration of the frame.
Coriolis and centrifugal forces arise as standard pseudoforces in rotating coordinate frames.
Derivations
Equation of Motion for Variable Mass Systems (Rocket Equation)
Conditions: A rocket moving in a gravity-free space expelling exhaust gas at a constant relative velocity.
Write the initial momentum of the rocket-gas system at time t: p(t) = m(v + dv).
Write the final momentum at time t + dt: p(t+dt) = (m - dm)(v + dv) + dm(v - v_rel), where v_rel is exhaust velocity relative to the rocket.
Equate the change in momentum dp to zero for an isolated system, leading to m d$v = v_rel dm$.
Integrate the resulting differential equation from initial mass m_0 to m and initial velocity v_0 to v.
Result: v - $v_0 = v_{rel} \ln\left(\frac{m_0}{m}\right)$
Acceleration of Two Bodies Connected by a String over a Fixed Pulley (Atwood Machine)
Conditions: Inextensible, massless string and frictionless, massless pulley.
Write equations of motion for mass m_1 moving upwards: T - m_1 $g = m_1 a$.
Write equation of motion for mass m_2 moving downwards: m_2 g - $T = m_2 a$.
Add the two equations to eliminate the internal tension force T: (m_2 - $m_1)g = (m_1 + m_2)a$.
Solve for acceleration a and substitute back to find tension T.
m = mass, omega = angular velocity, r = position vector from axis of rotation Newton (N)
Observed from a rotating non-inertial frame
Force in Central Force Field
$$F(r) = f(r) \hat{r}$$
f(r) = function of distance r from the force center, r-hat = unit position vector Newton (N)
Force is directed towards or away from a fixed center point
Graphs and experiments
Force versus Acceleration Graph
Draw: A straight line passing through the origin on a Cartesian plot with acceleration a on the x-axis and force F on the y-axis.
Interpret: The slope of the linear curve represents the inertial mass (m) of the experimental body, verifying F = ma.
Exam use: Direct conceptual questions regarding slope interpretation and proportionality.
Atwood Machine Acceleration vs Mass Difference
Draw: Plot of system acceleration a on the y-axis against mass difference (m_2 - m_1) on the x-axis for constant total mass.
Interpret: Yields a straight line whose gradient gives g / (m_1 + m_2), confirming Newton's second law dynamics.
Exam use: Used in numerical graph analysis problems for PGT examinations.
Spring Balance Reading in Accelerated Lift
Draw: Apparent weight W' plotted against upward acceleration a of the elevator.
Interpret: Linear positive slope starting from mg at a = 0, illustrating linear increase of apparent weight with upward acceleration.
Exam use: Problem-solving involving apparent weight variations in elevators.
Solved numericals
TGT core
A body of mass 5 kg is acted upon by a force $F = (3t^2 \hat{i} + 4t \hat{j}) N$. Find the magnitude of its acceleration at t = 2 s.
Calculate force at $t = 2 s: F = (3(2)^2 \hat{i} + 4(2) \hat{j}) = (12 \hat{i} + 8 \hat{j}) N$.
Use Newton's second law to find acceleration: $a = F / m = (12/5) \hat{i} + (8/5) \hat{j} = 2.4 \hat{i} + 1.6 \hat{j} m s^{-2}$.
Calculate magnitude of acceleration: |a| = \sqrt{(2.4)^2 + (1.6)^2} = \sqrt{5.76 + 2.56} = \sqrt{8.32} m s^{-2} \approx 2.88 m s^{-2}.
Answer: 2.88 m s^{-2}
PGT extension
A rocket burns fuel at a constant rate of 10 kg s^{-1} and ejects gas with a relative speed of 2000 m s^{-1}. Calculate the thrust force exerted on the rocket.
Identify given parameters: dm/d$t = 10 kg s^{-1}$, $v_rel = 2000 m s^{-1}$.
Apply the thrust force formula: $F_thrust = v_rel \left(\frac{dm}{dt}\right)$.
Newton's laws of motion are valid strictly in inertial reference frames unless modified by pseudoforces in non-inertial frames.
Classical mechanics assumes particle speeds are much smaller than the speed of light in vacuum (v << c), where relativistic mass-energy variations can be neglected.
Common misconceptions
Common error: Force is always required to keep an object moving at a constant velocity.
Correct: Force is required only to change the velocity (accelerate); a body in uniform motion requires zero net external force.
Friction and air resistance are opposing forces that must be balanced to maintain uniform motion, confusing everyday experience with idealized first-law conditions.
Common error: Action and reaction forces cancel each other out so net force is always zero.
Correct: Action and reaction forces act on two different bodies, hence they never cancel each other out.
Cancellation applies only to forces acting on the same single body when evaluating equilibrium.
Common error: Centrifugal force is a real inward pulling force.
Correct: Centrifugal force is a fictitious or pseudoforce experienced only in rotating non-inertial frames directed radially outwards.
Real forces in circular motion are purely centripetal, directed towards the center of curvature.
Quick revision
Force equals the time derivative of linear momentum: $F = dp/dt$.
Inertial frames obey Newton's laws directly without auxiliary pseudoforces.
Apparent weight in a lift moving upward with acceleration a is m(g + a).
Thrust on a variable mass rocket is given by v_rel (dm/dt).
Must remember
Newton's first law defines inertial reference frames and the qualitative concept of force and inertia.
Newton's second law is fundamentally expressed as $F = \frac{dp}{dt}$, which reduces to $F = ma$ for a system of constant rest mass.
Action and reaction forces act on two different bodies simultaneously and therefore do not cancel each other out.
In a non-inertial frame accelerating at $a_0$, a pseudoforce equal to $-ma_0$ must be incorporated to apply Newton's laws.
The Tsiolkovsky rocket equation is given by $\Delta v = v_{rel} \ln\left(\frac{m_0}{m_f}\right)$ in the absence of external forces.
Coriolis acceleration in a rotating frame is expressed vectorially as $a_{cor} = 2(\vec{\omega} \times \vec{v})$.
Glossary
Inertial FrameA reference frame in which Newton's first law of motion holds true and bodies maintain uniform velocity unless acted upon by a net force.
PseudoforceA fictitious force introduced in non-inertial reference frames to account for the acceleration of the coordinate system itself, such as $-ma_0$.
ImpulseThe integral of force over a time interval, equal to the change in linear momentum, given by $J = \int F dt = \Delta p$.
Coriolis ForceA standard pseudoforce arising in rotating coordinate systems when a particle possesses a velocity relative to that rotating frame.
Variable Mass SystemA mechanical system whose mass changes continuously due to accretion or ejection of matter, famously analyzed using rocket equations.
Inertial vs Non-Inertial Frames
Parameter
Inertial Frame
Non-Inertial Frame
Newton's Laws Validity
Hold strictly true without modification
Require incorporation of pseudoforces
Acceleration of Frame ($a_0$)
Zero ($a_0 = 0$)
Non-zero ($a_0 \neq 0$)
Examples
Fixed stars, unaccelerated lab
Accelerating lift, rotating merry-go-round
Centripetal vs Centrifugal Force
Parameter
Centripetal Force
Centrifugal Force
Nature
Real inward-pulling force
Fictitious/pseudoforce directed outwards
Reference Frame
Observed from inertial frames
Observed from rotating non-inertial frames
Magnitude
$F_c = \frac{mv^2}{r}$
$F_{cf} = m \omega^2 r$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following defines an inertial reference frame?
Explanation: An inertial frame is defined as a reference frame in which Newton's first law of motion is strictly valid without needing auxiliary pseudoforces.
2. The fundamental mathematical expression for Newton's second law for a general system is:
Explanation: Newton's second law states that the rate of change of linear momentum is directly proportional to the applied force, expressed as $F = \frac{dp}{dt}$.
3. Why do action and reaction forces not cancel each other out?
Explanation: Action and reaction forces act simultaneously on two different bodies. Forces can only cancel out when they act on the same single body.
4. What is the apparent weight of a body of mass $m$ in an elevator accelerating downwards with an acceleration $a$ ($a < g$)?
Explanation: When an elevator accelerates downwards, the apparent weight is given by $W' = m(g - a)$ due to the effective reduction in normal reaction.
5. The Tsiolkovsky rocket equation calculates velocity change as $\Delta v = v_{rel} \ln\left(\frac{m_0}{m_f}\right)$. What does $v_{rel}$ represent?
Explanation: In the Tsiolkovsky rocket equation, $v_{rel}$ represents the relative exhaust velocity of the expelled fuel with respect to the rocket body.
6. What is the magnitude of centripetal acceleration for a particle moving in a circle of radius $r$ with uniform speed $v$?
Explanation: Centripetal acceleration is given by the standard kinematic formula $a_c = \frac{v^2}{r} = \omega^2 r$.
7. When a coordinate frame rotates with an angular velocity vector $\vec{\omega}$, a particle moving inside it with velocity $\vec{v}$ experiences which acceleration?
Explanation: A particle moving in a rotating reference frame experiences a Coriolis acceleration given by $a_{cor} = 2(\vec{\omega} \times \vec{v})$.
8. Which of the following is a classic misconception regarding Newton's first law?
Explanation: A common misconception is that a continuous force is needed to maintain uniform motion, whereas force is only needed to change velocity (accelerate).
9. Impulse is mathematically equal to:
Explanation: By the Impulse-Momentum theorem, impulse $J = \int F dt = \Delta p$, which represents the total change in linear momentum.
10. In a non-inertial frame accelerating with acceleration $\vec{a}_0$, what is the magnitude and direction of the pseudoforce acting on a mass $m$?
Explanation: The pseudoforce introduced in an accelerating reference frame is given by $\vec{F}_{pseudo} = -m \vec{a}_0$, acting oppositely to the frame's acceleration.
11. Which force formula governs motion under a central force field directed towards a fixed center?
Explanation: A central force field is directed along the position vector towards or away from a center, represented as $F(r) = f(r)\hat{r}$.
12. What is the direction of centrifugal force experienced in a rotating frame?
Explanation: Centrifugal force is a pseudoforce experienced in rotating frames that acts radially outwards, away from the center of rotation.
13. If an object of constant mass $m$ has its velocity change from $v_1$ to $v_2$, what is the impulse imparted to it?
Explanation: Impulse equals the change in momentum: $J = \Delta p = m v_2 - m v_1 = m(v_2 - v_1)$.
14. Internal forces within an isolated system vectorially sum to zero. This leads directly to:
Explanation: Since internal forces occur in equal and opposite action-reaction pairs, their vector sum is zero, which ensures the conservation of total linear momentum of an isolated system.
15. Under what condition does the general equation $F = \frac{dp}{dt}$ simplify to $F = ma$?
Explanation: For a system of constant rest mass $m$ in an inertial frame, $F = \frac{d(mv)}{dt} = m \frac{dv}{dt} = ma$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. An elevator is moving upwards with a constant acceleration $g$. The apparent weight of a passenger of mass $m$ inside the elevator is:
Explanation: Apparent weight in an upward accelerating lift is given by $W' = m(g + a)$. Substituting $a = g$ gives $W' = m(g + g) = 2mg$.
2. A rocket burns fuel at a constant rate with a relative exhaust velocity $v_{rel}$. If gravity and drag are neglected, the thrust force acting on the rocket is given by:
Explanation: The thrust force on a variable-mass rocket system is given by $F_{thrust} = v_{rel} \left(-\frac{dm}{dt}\right)$, where $-\frac{dm}{dt}$ is the rate of mass ejection, magnitude-wise expressed as $v_{rel} \frac{dm}{dt}$.
3. Which of the following forces is a real force present in both inertial and non-inertial frames?
Explanation: Gravitational force is a fundamental real interaction between masses, whereas centrifugal, Coriolis, and Euler forces are pseudoforces dependent on the choice of coordinate frame.
4. A particle of mass $m$ moves in a circle of radius $r$ with angular speed $\omega$. The magnitude of the centrifugal force experienced by the particle in the rotating frame is:
Explanation: The centrifugal force in a rotating frame is given by $F_{cf} = m \omega^2 r$ directed radially outwards.
5. If a force is given by $F = 6t^2 \hat{i} + 4t \hat{j}$, what is the change in momentum of the particle from $t = 0$ to $t = 2$ seconds?
Explanation: Change in momentum $\Delta p = \int_0^2 F dt = \left[ \frac{6t^3}{3}\hat{i} + \frac{4t^2}{2}\hat{j} \right]_0^2 = [2(2^3)\hat{i} + 2(2^2)\hat{j}] = 16\hat{i} + 8\hat{j}$ Ns.
6. A block of mass $m$ rests on a rough inclined plane with angle of inclination $\theta$. If the coefficient of static friction is $\mu_s$, the minimum force parallel to the plane required to push the block up is:
Explanation: To push the block up the incline, the applied force must overcome both the downward component of gravity ($mg\sin\theta$) and the maximum static friction acting down the plane ($\mu_s mg\cos\theta$), giving $F = mg(\sin\theta + \mu_s \cos\theta)$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Inertial reference frames are characterized by which of the following properties?
Explanation: An inertial frame is defined purely by the validity of Newton's laws of motion without requiring any auxiliary pseudoforces.
2. A spacecraft accelerates in deep space by ejecting gas. If the initial mass is $m_0$, final mass is $m_f$, and exhaust speed is $v_{rel}$, the rocket velocity increment $\Delta v$ is:
Explanation: According to the Tsiolkovsky rocket equation, $\Delta v = v_{rel} \ln\left(\frac{m_0}{m_f}\right)$.
3. What is the Coriolis acceleration for a particle moving with a velocity $\vec{v}$ along the axis of rotation where $\vec{\omega} \parallel \vec{v}$?
Explanation: Coriolis acceleration is given by $a_{cor} = 2(\vec{\omega} \times \vec{v})$. If $\vec{\omega}$ and $\vec{v}$ are parallel, their cross product is zero.
4. A lift is falling freely under gravity. What is the apparent weight of a body of mass $m$ inside it?
5. Which of the following statements correctly describes Newton's third law?
Explanation: Newton's third law states that action and reaction forces are equal in magnitude and opposite in direction, acting simultaneously on two different bodies.
6. The central force acting on a particle is attractive and inversely proportional to the cube of the distance ($r^3$). The force law is expressed as:
Explanation: An inverse-cube attractive central force is represented by $F(r) = -\frac{k}{r^3}\hat{r}$, where the negative sign denotes attraction towards the center.
7. A particle of mass $2\text{ kg}$ has linear momentum given by $p = (4t \hat{i} + 3 \hat{j})\text{ kg m s}^{-1}$. The magnitude of the net force acting on the particle at $t = 1\text{ s}$ is:
Explanation: Force $F = \frac{dp}{dt} = \frac{d}{dt}(4t \hat{i} + 3 \hat{j}) = 4\hat{i}$ N. Its magnitude is $4\text{ N}$.
8. Which pseudoforce is responsible for the apparent deflection of winds (Coriolis effect) on the rotating Earth?
Explanation: The Coriolis force is the standard pseudoforce in a rotating frame (Earth) that deflects moving objects and winds sideways.
9. A bullet of mass $0.1\text{ kg}$ moving at $200\text{ m s}^{-1}$ strikes a sandbag and is brought to rest in $0.05\text{ s}$. The average force exerted by the sandbag is:
Explanation: Using impulse-momentum theorem, $F_{avg} \Delta t = \Delta p = m(0 - v) \implies F_{avg} (0.05) = 0.1 \times 200 = 20 \implies F_{avg} = \frac{20}{0.05} = 400\text{ N}$.
10. In classical mechanics, the concept of inertia of a body is quantitatively measured by its:
Explanation: Inertia is the inherent property of a body to resist changes in its state of motion, and it is quantitatively measured by its rest mass $m$.