Mechanics

It is a core branch of physics that studies the motion of objects and the forces that cause it. It includes Kinematics, which describes motion without forces, and Newton’s Laws of Motion, explaining how forces affect motion. It also covers Work, Power, and Energy, exploring force-related work and energy transfer, and System of Particles & Centre of Mass, which analyzes the behavior of multiple particles. Rotational Motion focuses on rotating objects, with concepts like angular velocity and torque. Lastly, Gravitation explains the influence of gravity on motion. Mechanics is essential for understanding the physical world and is crucial in fields like engineering, astronomy.

1.0Introduction to Mechanics

• Mechanics is a foundational branch of physics that deals with motion and forces. It helps us understand how and why objects move—whether it's a ball rolling down a hill or a satellite orbiting Earth. Mechanics is broadly divided into kinematics (study of motion) and dynamics (study of the causes of motion).

Motion in 1D

• This involves movement along a straight line. The three key variables are displacement (s), velocity (v), and acceleration (a).

Equations of Motion

$$\begin{gathered} v=u+at \\ s=ut+\frac{1}{2}a{t}^2 \\ {v}^2=u^2+2as \\ \text{where }u=\text{initial velocity},a=\text{acceleration},t=\text{time},s=\text{displacement} \end{gathered}$$

Motion in 2D

• This includes motion in a plane, like an object moving on both horizontal and vertical axes.

Projectile Motion

• An object projected into the air under gravity follows a parabolic path.

Important Results

$$\begin{gathered} 1.\text{Time of Flight:}T=\frac{2u\sin \theta }{g} \\ 2.\text{Maximum Height:}H=\frac{u^2{\sin }^2\theta }{2g} \\ 3.\text{Horizontal Range:}R=\frac{u^2\sin 2\theta }{g} \end{gathered}$$

Relative Motion

• Used when observing motion from different reference frames

${\vec{v}}_{AB}={\vec{v}}_A-{\vec{v}}_B$

• This concept is essential in river-boat problems, moving trains, and aircraft navigation.

2.0Graphical Analysis

Displacement-Time Graph:	Velocity-Time Graph:	Acceleration-Time Graph:
Slope = velocity	Slope = acceleration  Area = displacement	Area = change in velocity

3.0Laws of Motion

Newton's First Law (Inertia)

• An object remains at rest or continues in uniform motion unless a net force acts on it.

Second Law (F = ma)

• Net force is directly proportional to the rate of change of momentum: F=ma

Third Law

• Every action has an equal and opposite reaction.

Friction

• It is a force that resists the relative motion (or the tendency to move) between two surfaces in contact.

Types of Friction

4.0Pulleys and Connected Motion

• Pulley: A simple machine to change direction of force.
• Ideal pulley system: Massless, frictionless pulley; inextensible, light string.
• Connected motion: Masses connected by string move with related accelerations.
  For systems of masses  $m_1$ and  $m_2$.

If 

Net acceleration (ideal pulley): $a=\left(\frac{m_1-m_2}{m_1+m_2}\right)g$

Tension: $T=\left(\frac{2m_1{m}_2}{m_1+m_2}\right)g$  and 

Rotational Dynamics

Key Concepts

• Deals with rotation of rigid bodies about a fixed axis.
• Analogous to linear motion, but with angular quantities.

Rotational analogue of Newton’s second law

• The net external torque acted on a rigid body is equal to the product of its moment of inertia and angular acceleration.

$\tau =I\alpha$

5.0Torque and Angular Momentum

Torque

• It is the physical agency which is responsible for change in state of rotation. Torque is essential for producing turning or toppling phenomena.
• For producing torque the force is required & it is product of force and perpendicular distance of line of action of force (lever arm from axis).

 $$\begin{gathered} \tau =Fr\sin \theta \\ \text{In vector form: }\vec{\tau }=\vec{r}\times \vec{F} \end{gathered}$$

Angular Momentum

• The angular momentum of a body about a specific axis is the product of its linear momentum and the orthogonal distance from the axis of rotation to the line of action of the linear momentum.

$$\begin{gathered} L=mv\cdot r\sin \theta \\ \vec{L}=\vec{r}\times \vec{p} \end{gathered}$$

6.0Moment of Inertia

 It is the measure of an object's resistance to change in its rotational motion about an axis.

$I=m{r}^2$

7.0(MOI) for Continuous Mass Distribution

$I_{AB}=\int r^2\mathrm{d}m$

Rotational Kinetic Energy

• The energy possessed due to rotational motion of a body is known as rotational kinetic energy.

$K.E_r=\frac{1}{2}I{\omega }^2$

Rolling Motion

• When a body performs translatory motion as well as rotatory motion combinedly then it is said to undergo rolling motion.
• If the relative velocity of the point of contact of the rolling body with the surface is zero then it is known as pure rolling, $v_{CM}=R\omega$
• If a body is performing rolling then the velocity of any point of the body with respect to the surface is given by $\vec{v}={\vec{v}}_{CM}+\vec{\omega }\times \vec{R}$

Conservation of Angular Momentum

• If the net external torque exerted on a system is zero then the total angular momentum of the system remains constant.

${\vec{\tau }}_{\text{ext }}=0\Rightarrow \frac{\overrightarrow{dL}}{dt}=0\Rightarrow \vec{L}\text{ is constant }\Rightarrow L_i=L_f$

• If a system is isolated from its surroundings any internal interaction between its different parts cannot alter its total angular momentum.

Inclined Planes

• It is a flat surface tilted at an angle to the horizontal, facilitating the raising or lowering of objects with less effort by spreading the required force over a longer distance.

Downward sliding on rough inclined plane

• If angle of inclination is greater than the angle of repose, then the body accelerates down the incline.

$a=g\left[\sin \theta -{\mu }_k\cos \theta \right](a<g)$

Upward sliding on rough inclined Plane

 $a=\left[g\sin \theta +{\mu }_{k}g\cos \theta \right]$

8.0Work, Power, and Energy

Work

• If a constant force F is applied on a body and displaces it through displacement s then work done by F on the body is given by a scalar product of F and s  .

$W=\vec{F}\cdot \vec{s}=Fs\cos \theta$

• SI unit of work:Joule(J)

Kinetic Energy

• It is the inherent ability of an object to perform work due to its motion.

$$\begin{gathered} K.E=\frac{1}{2}m{v}^2 \\ K.E.=\frac{p^2}{2m} \end{gathered}$$

Potential Energy

• It is the energy due to its position or arrangement within a conservative force field.

$U_f-U_i=mgh$

Work-Energy Theorem 

• It states that work done by all the forces (internal and external) on a particle equals to change in its kinetic energy.

$$\begin{gathered} W_{\text{all}}=\Delta K.E \\ {W}_{\text{all}}=K_f-K_i=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2 \end{gathered}$$

Power

• It is the ratio of work done(W) to the time taken (t) to do that work. Therefore, it is the rate of energy transfer or work done concerning time, and the symbol (P) denotes it.

$P=\frac{W}{T}$

Newton’s Law of Gravitation

• This law states that every particle attracts others with a force proportional to their masses and inversely proportional to the square of separation between them.

$F_g=G\frac{m_1{m}_2}{r^2}$

• Value of G in $\mathrm{SI}=6.67\times 10^{-11}{\mathrm{Nm}}^2{\mathrm{kg}}^{-2}or\text{ CG.S }=6.67\times 10^{-8}\text{ dyne }{\mathrm{cm}}^2{\mathrm{g}}^{-2}$

9.0Acceleration Due to Gravity

• If a body is positioned inside a gravitational field having intensity I, gravitational force exerted on it is,

$F=ml$

• The Earth exerts a force on every object, pulling it towards its center.

$F=mg$

On relating both the equations we get $g=|\vec{I}|$

Where g = acceleration due to gravity I is intensity of gravitational field due to Earth.

$g_s=\frac{GM}{R^2}\Rightarrow 9.81{\text{m/s}}^2$

10.0Orbital Motion and Satellites

Orbital Motion

• It is the movement of an object along a curved path around a central body, such as a planet orbiting a star or a satellite orbiting a planet. This motion is governed by gravitational forces, which provide the necessary centripetal force to maintain the orbit.

Orbital Velocity (V_o):

$V_0=\sqrt{\frac{G{M}_e}{R_e}}\approx 8.1\text{km/s}$

Time period of orbital motion:

$T=2\pi \sqrt{\frac{R_e^3}{G{M}_e}}\approx 84\text{min}$

Satellites

• Satellites are artificial objects placed into orbit around celestial bodies, primarily Earth, to perform various functions such as communication, navigation, weather monitoring, and scientific research.

Essential conditions for satellite motion –

• The center of a satellite’s orbit aligns with the center of the Earth.
• The satellite's orbital plane passes through the Earth's center.
• A satellite can have two stable types of orbits: equatorial orbit and polar orbit.

1.Kinetic energy of satellite

$KE=\frac{GMm}{2r}$

2.Potential energy of satellite

$PE=-GMemr$

Total Energy = Kinetic Energy + Potential Energy

$$\begin{gathered} PE=-\frac{GMm}{r} \\ \text{Total Energy = Kinetic Energy + Potential Energy} \\ TE=-\frac{GMm}{2r} \\ KE=-TE=-\frac{PE}{2}=\frac{G{M}_{e}m}{2r} \end{gathered}$$

• Since a satellite's total energy is negative, it is considered a bound system.

Escape Velocity

• The minimum velocity required for an object to escape a planet's gravitational pull from its surface.

$$\begin{gathered} V_e=\sqrt{\frac{2GM}{R_e}} \\ {V}_e=\sqrt{2g_s{R}_e}=11.2\text{km/s},R_e=6400\text{km} \end{gathered}$$

Pressure in a Fluid

• Pressure in a fluid is the force exerted per unit area by the fluid particles on the walls of the container or on objects submerged in the fluid. The pressure at any point in a fluid at rest is given by 

$P=\frac{F}{A}$

In fluids, pressure increases with depth due to the weight of the fluid above

$P=P_0+\rho gh$

Pascal’s Law

• Pascal’s Law states that any change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. This principle is the basis for hydraulic systems.

11.0Buoyancy And Archimedes Principle

Buoyancy

• The tendency for an immersed body to be lifted up in a fluid ,due to an upward force that acts opposite to the action of gravity.
• It is an upward force that is exerted by a fluid on any object immersed partly or wholly in fluid.

Archimedes Principle

• Any object completely or partially submerged in fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the liquid.

$$\begin{gathered} \text{Buoyant Force}=\text{Weight of fluid displaced}=m_{l}g \\ {m}_l=\text{mass of liquid displaced},v_l=\text{volume of liquid displaced} \\ {m}_l={\rho }_l\times v_l \\ {F}_B={\rho }_l\times v_l\times g \\ \text{Apparent Weight }{W}^{\prime }=\text{Weight of body in air}-\text{Buoyant Force} \end{gathered}$$

12.0Surface Energy and Surface Tension

Surface Energy

• It  is the energy required to increase the surface area of a material by one unit. It is a measure of the intermolecular forces at the surface of a solid.

$S.E.=\frac{W}{A}={\text{J/m}}^2$

Surface Tension

• It  is the force per unit length acting along the surface of a liquid, causing it to minimize its surface area. 

$\sigma =\frac{F}{l},\text{unit: N/m}$

Viscosity

• It is the property of a fluid (liquid or gas) by virtue of which it opposes the relative motion between its adjacent layers. It is the fluid friction or internal friction.

$F=\eta A\frac{\Delta v_x}{\Delta y}$

13.0Modulus of Rigidity and Bulk Modulus

 Modulus of Rigidity

• Within the elastic limit it is the ratio of shearing stress to shearing strain.

$$\begin{gathered} \eta =\frac{\text{shearing stress}}{\text{shearing strain}}=\frac{F_{\text{tangential}}/A}{\varphi }=\frac{F_{\text{tangential}}}{A\varphi } \\ \eta =\frac{F}{bc\varphi } \end{gathered}$$

Bulk Modulus

•  It is defined as the ratio of the volume stress to the volume strain

$B=\frac{\text{Pressure}}{\text{Volumetric Strain}}=-\frac{p}{\Delta V/V}=-\frac{pV}{\Delta V}$

Negative sign shows that increase in pressure (p) causes decrease in volume (V).

$B=-V\frac{\Delta p}{\Delta V}=-V\frac{dp}{dV}$

14.0Stoke’s Law and Terminal Velocity

Stoke’s Law

• Stoke showed that if a small sphere of radius r is moving with a velocity v through a homogeneous stationary medium (liquid or gas), of viscosity  then the viscous force acting on the sphere is,

$F=6\pi \eta rv$

Terminal Velocity

• When a solid sphere falls in a liquid, its increasing velocity is controlled by the viscous force due to liquid and hence it attains a constant velocity which is known as the terminal velocity.

$v_T=\frac{2}{9}\cdot \frac{r^2(\rho -\sigma )}{\eta }g$

15.0Types of Waves

Wave:It is a disturbance that transfers energy from one point to another without the transfer of matter.

16.0Superposition and Interference

Superposition

• When two or more waves meet at a point, the resultant displacement is the algebraic sum of the individual displacements.

$y=y_1+y_2$

Interference

• It is the phenomenon of superposition of two or more coherent waves resulting in increase (constructive) or decrease (destructive) in amplitude.

$$\begin{gathered} I_R=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi \\ 2\sqrt{I_1{I}_2}\cos \varphi =\text{Interference Factor} \end{gathered}$$

Vibration of Strings and Air Columns

• A string fixed at both ends vibrates in standing wave patterns at specific frequencies, determined by:

$f_n=\frac{n}{2l}\sqrt{\frac{T}{\mu }}$

Resonance and Beats

• Resonance appears when a system is driven at its natural frequency, producing large vibrations. Beats occur when two close frequencies interfere, causing periodic variations in sound intensity.

$\text{Beat Frequency}=\left|f_1-f_2\right|$

17.0Speed of Sound and the Doppler Effect

Speed of Sound

• Solid medium, $v=\sqrt{\frac{k+\frac{4}{3}\eta }{\rho }},kk==Bulkmodulus,\eta ==Modulusofrigidity,\rho =Density$
• Solid is in the form of long bar, Y== Young's modulus of rod material of rod.$v=\sqrt{\frac{Y}{\rho }}$
• Velocity of sound waves in a fluid medium (liquid or gas), $V=\sqrt{\frac{B}{\rho }}whereB=-V\frac{dP}{dV}$

Doppler Effect

• When a sound or light source and an observer move relative to each other along the line of sight, the observed frequency differs from the source frequency. This is known as the Doppler Effect.

$f^{\prime }=f\left(\frac{v+v_o}{v+v_s}\right)$

18.0Solved Examples On Mechanics

Q-1.For a particle moving along x-axis, acceleration is given as a = v. a = v Find the position as a function of time.Given that at t = 0, x = 0 v = 1.t = 0, x = 0, v = 1

Solution:

$$\begin{gathered} a=v\Rightarrow \frac{dv}{dt}=v\Rightarrow \frac{dv}{v}=dt \\ lnv=t+c\Rightarrow 0=0+c(\text{at }t=0,v=1) \\ v=e^t\Rightarrow \frac{dx}{dt}=e^t\Rightarrow dx=\int e^{t}dt \\ x=e^t+c\Rightarrow 0=1+c(\text{at }t=0,x=0) \\ x=e^t-1 \end{gathered}$$

Q-2.For a particle moving along the x-axis , acceleration is given as. $a=2v^2$  If the speed of the particle is vo at x = 0 ,find speed as a function of x.

Solution:

$$\begin{gathered} a=2v^2\Rightarrow \frac{dv}{dt}=2v^2\Rightarrow \frac{dv}{dx}\cdot \frac{dx}{dt}=2v^2 \\ \Rightarrow v\frac{dv}{dx}=2v^2\Rightarrow \frac{dv}{v}=2dx \\ {\int }_{v_0}^v\frac{dv}{v}={\int }_0^{x}2dx\Rightarrow [\ln v{]}_{v_0}^v=[2x{]}_0^x \\ \Rightarrow \ln \frac{v}{v_0}=2x\Rightarrow v=v_0{e}^{2x} \end{gathered}$$

Q-3.Two particles are projected at the same time with identical speeds v in the same vertical plane. Their angles of projection are θ and 2θ, whereθ<45°. Determine the time at which their velocities become parallel.

Solution:

Velocity of particle projected at angle '$\theta$' after time t 

${\vec{v}}_1=\left(v\cos \theta \hat{i}+v\sin \theta \hat{j}\right)-(gt\hat{j})$

Velocity of particle projected at angle '$2\theta$' after time t 

${\vec{v}}_2=\left(v\cos 2\theta \hat{i}+v\sin 2\theta \hat{j}\right)-(gt\hat{j})$

Since velocities are parallel so $\frac{v_x}{v_y}=\frac{v\cos \theta }{v\sin \theta -gt}=\frac{v\cos 2\theta }{v\sin 2\theta -gt}$

On solving we get $\frac{v}{g}\left(\frac{\cos \theta }{{\cos }^2\theta -\cos 2\theta }\right)$

Q-4.Two balls made of the same material having masses m and 8m respectively. Then find the ratio of their terminal velocity in the same liquid.

Solution:

$$\begin{gathered} m\propto r^3{v}_T\propto r^2 \\ \Rightarrow \frac{m_1}{m_2}={\left(\frac{r_1}{r_2}\right)}^3\Rightarrow \frac{v_{T1}}{v_{T2}}={\left(\frac{r_1}{r_2}\right)}^2 \\ \Rightarrow \frac{r_1}{r_2}={\left(\frac{m}{8m}\right)}^{1/3}=\frac{1}{2} \\ \Rightarrow \frac{v_{T1}}{v_{T2}}={\left(\frac{1}{2}\right)}^2=\frac{1}{4} \end{gathered}$$

Q-5.A 60 gm tennis ball thrown vertically up at 24 m/s rises to a maximum height of 26 m. What was the work done by resistive forces?

Solution:

$$\begin{gathered} W_g+W_{res}=\left(0-\frac{1}{2}m{v}^2\right) \\ \Rightarrow W_{res}=mgh-\frac{1}{2}m{v}^2 \\ {W}_{res}=\left(\frac{60}{1000}\times 10\times 26\right)-\frac{1}{2}\times \frac{60}{1000}\times (24{)}^2=-1.68\text{J} \end{gathered}$$

Q-6.A drop of water of radius is falling in the air. The co-efficient of viscosity of air is $1.8\times 10^{-5}\text{kg/m-s}$.What will be the terminal velocity of the drop? The density of air can be neglected.

Solution:

$v_T=\frac{2r^2(\rho -\sigma )g}{9\eta }=\frac{2\times {\left(\frac{15\times 10^{-5}}{1000}\right)}^2\times 10^3\times 9.8}{9\times 1.8\times 10^{-5}}=2.72\times 10^{-4}\text{m/s}$