JEE Important Chapter - Mechanics

Name of the Concept

Key Points of Concept

1. Kinematic equations in one-dimensional motion

• Equations of motions for uniformly accelerated motion is given by,

• $v=u+at$

• $s=ut+\dfrac{1}{2}at^{2}$

• $v^2=u^2+2as$

Where u is the initial velocity, v is the final velocity, a is the acceleration and t is the time taken.

• The instantaneous velocity and instantaneous acceleration of a body is given by the formula,

$v=\dfrac{dx}{dt}$

$a=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$

• For motion under gravity, take acceleration as 9.8 m/s2. 

• While applying equations of motion in motion under gravity, take one direction positive and the opposite direction. For eg: If you take upward direction as positive, then take upward velocity as positive and acceleration due to gravity in the downward direction negative.

2. Graphs for one-dimensional motion

• Displacement-time graph

The slope of a displacement time graph gives velocity.

• Velocity-time graph

The slope of a velocity-time graph gives the displacement covered by the body.

• Acceleration-time graph

The area under the curve of the acceleration time graph gives the change in velocity.

3. Motion along two-dimension

• When the motion of a body is along a plane, it is called two-dimensional motion. Examples are  Projectile motion and circular motion.

• In a projectile motion, the body moves under the influence of gravity.

• The range of the projectile motion is given by,

$R=\dfrac{u^2\sin (2\theta)}{g}$

• Time of flight is the total time taken by the body to reach back to the ground.

$T=\dfrac{2u\sin (\theta)}{g}$

• Maximum height of the projectile motion is given by,

$R=\dfrac{u^2\sin^2 (\theta)}{2g}$

Where u is the velocity at which the body is projected, θ is the angle of projection and g is the acceleration due to gravity. 

• In a uniform circular motion, speed remains constant.

• The relation between linear velocity (v) and angular velocity (⍵) of a particle moving in a circle of radius r is,

$v=r\omega$

• Angular acceleration is the rate of change of angular velocity with respect to time.

$\alpha=\dfrac{d\omega}{dt}=r\omega^2$

• The relationship between linear acceleration (a) and angular acceleration (α) is

$a=r\alpha$

4. Laws of Motion

• Newton’s first laws of motion state that a body continues to be in a state of rest or state of uniform motion along a straight line until and unless an external unbalanced force is acting on it.

• According to newton’s second law, the net force acting on a body is equal to the rate of change of momentum.

$\vec F=\dfrac{d\vec P}{dt}=m\vec a$

• Newton’s third law of motion states that every action has an equal and opposite reaction.

• While dealing with problems related to laws of motion, always draw the free body diagram and write equations using the laws of motion of the free body mechanism. 

• Law of Conservation of momentum: The total momentum of a system is conserved if the external force acting on the system is zero

• Friction: It is the force that opposes the relative motion between two surface in contact.

• Friction is independent of area of contact and depends on the normal force acting between the two surface.

• The formula  for static friction and kinetic friction is given below

$f_{s}=\mu_sN$

$f_{k}=\mu_kN$

• Static friction is a self adjusting force

5. Mass Pulley system

• For a mass pulley system as shown below, the magnitude acceleration for both blocks will be same.

• Tension acting on the string is given by,

$T=\dfrac{2m_1m_2g}{m_1+m_2}$

• The acceleration of each block is given by,

$a=\dfrac{(m_1-m_2)g}{m_1+m_2}$

6. Work done by a constant force and variable force

• Work is a scalar quantity and the work done by a constant force is given by,

$W=\vec F\cdot \vec d$

$W= Fd\cos\theta$

• Work done by a variable force is given by,

$W=\int Fds$

• The area under the curve of force-displacement graph also gives the work done by the force.

• Work- Energy theorem: It states that work done by net force on a body is equal to change in kinetic energy.

7. Rotational motion

• Parallel axis theorem: The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to an axis passing through the centre of mass is equal to the sum of the moment of inertia of body  about an axis passing through centre of mass and product of mass and square of the distance between the two axes.

$I=I_{COM}+Mr^2$

• Perpendicular axis theorem: This theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.

$I_z=I_{x}+I_y$

• The torque required to rotate a body having moment of inertia(I) with an angular acceleration(⍺) is given by,

$\tau=I\alpha$

• The anglular momentum of a rotating body is given by, 

$L=I\omega$

• The condition for a body to be in pure rolling motion is given by,

$v_{cm}=R\omega$

8. Gravitation

• The gravitational force acting between two bodies m_1 and m_2 is given by

$F=\dfrac{Gm_1m_2}{r^2}$

• Acceleration due to Gravity of Earth below the Earth’s Surface is given by

$g'=g\left(\dfrac{R_E-d}{R_E}\right)$

• Acceleration due to Gravity of Earth above the Earth’s Surface is given by,

$g'=g\left(1-\dfrac{2h}{R_E}\right)$

• The gravitational potential energy between two masses is given by,

$U=-\dfrac{Gm_1m_2}{r}$

• Escape velocity is the minimum velocity required for a body to escape from the gravitational field of the earth.

$v=\sqrt{2gR_E}$

9. Kepler’s Law of planetary motion

• Kepler's Law consists of three laws.

• Law of Orbits: All planets move in elliptical orbits with the Sun situated at one of the foci of the ellipse.  

• Law of Areas: The line that joins any planet to the sun sweeps equal areas in equal intervals of time.

• Law of Periods: The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet.

10. Stress-strain curve

• The behaviour of a material can be studied using stress-strain curve.

• The curve obeys Hooke’s law in the region from O to A. According to Hookes law. Stress is proportional to strain.

• In the stress-strain curve given above, the point A corresponds to proportional limit.

•  The yield point or elastic limit is the point upto which the wire shows elastic behaviour and the point B corresponds to the yield point.

• The stress corresponding to the yield point is called yield strength.

• Beyond point B, it loses its elastic behaviour and does not regain its original length even after deforming force is removed.

• The stress corresponding to point  D is called ultimate strength.

11. Archimedes principle

• According to Archimedes principle, whenever a body is fully or partially submerged in liquid, it experiences a net upward force which is equal to the weight of the liquid displaced.

• The formula to calculate the buoyancy force is given by,

$F_B=\rho_l Vg$

Where ⍴l is the density of the liquid, V is the volume of the body immersed in the liquid and g is the acceleration due to gravity.

• A body will float in a liquid if the buoyancy force acting on the body is equal to the weight  of the body.

12. Bernoulli’s theorem

• According to Bernoulli’s theorem, the total energy per unit volume of an incompressible fluid always remains constant at any point of the fluid.

$P+\rho gh+\dfrac{1}{2}\rho v^2=\text{constant}$

$P_1+\rho gh_1+\dfrac{1}{2}\rho v_1^2=P_2+\rho gh_2+\dfrac{1}{2}\rho v_2^2$

13. Laws of thermodynamics

• It states that heat given to a system(ΔQ) is equal to the change in internal energy(ΔU) and work done by the system(ΔW).

$\Delta Q=\Delta U+\Delta W$

• Second law of thermodynamics:

• Clausius statement: It is not possible to construct a device that operates on a cycle and produces no other effect than transferring heat from a colder body to a hotter body 

• Kelvin- Plancks statement: It is impossible to design an engine operating in cycles that absorb heat and deliver the equivalent amount of work.

14. Carnot engine and refrigerator

• Carnot engine is an ideal heat engine designed by french engineer Carnot.

• The working substance in the Carnot engine undergoes a cyclic process called carnot cycle.

• Area of the P-V graph of  carnot cycle gives the work done during one cycle.

• The efficiency of the carnot cycle is given by,

$\eta=\dfrac{T_1-T_2}{T_1}$

• A refrigerator is a device that transfers heat energy from colder body to a hotter body by doing some external work on it.

• The coefficient of performance of a refrigerator (C.O.P) is given by,

$\beta=\dfrac{Q_2}{W}$

$\beta=\dfrac{Q_2}{Q_1-Q_2}$

Si. No

Name of the Concept

Formula

1.

Addition and subtraction of two vectors

$|\vec A  +\vec B|=\sqrt{A^2+B^2+2AB\cos\theta}$

$|\vec A  -\vec B|=\sqrt{A^2+B^2-2AB\cos\theta}$

Where A and B are the magnitude of the two vectors at an angle θ.

2.

The motion of a body projected from the top of a tower

The trajectory of the body is given as

$y=\dfrac{gx^2}{2u^2}$

Where y is the vertical displacement and x is the horizontal displacement. 

3.

The average velocity of the body in a journey that covers two equal distances with two different speeds

$v_{avg}=\dfrac{2uv}{u+v}$

Where u and v are the two different speeds by which the body covers two equal distances.

4. 

The displacement covered in nth second of a uniformly accelerated body

$S_{n}=u+\dfrac{1}{2}a(2n-1)$

Where u is the initial velocity, a is the acceleration and Sn is the displacement covered by the body in the nth second.

5. 

Maximum safe speed for a a car moving in a circular track

$v_{max}=\sqrt{\mu Rg}$

6. 

Potential energy stored in a stretched string

$U=\dfrac{1}{2}kx^2$

7.

Moment of inertia of a disk

$I=\dfrac{MR^2}{2}$

8.

Moment of inertia of a ring

$I=MR^2$

9.

Moment of inertia of a solid sphere

$I=\dfrac{2}{3}MR^2$

10. 

Work done in stretching a strain

Work done in stretching a wire having young’s modulus(Y) under a force (F) is given by the formula,

$W=\dfrac{1}{2}\dfrac{YA\Delta l^2}{l}$

11.

Thermal stress on a rod fixed between two rigid supports.

$\text{Thermal stress }=Y\alpha\Delta T$

Where ⍺ is the coefficient of linear expansion and ΔT is the change in temperature

12.

Equation of continuity

$A_1V_1=A_2V_2$

Where A1 and A2 are the area of cross section of the pipe at the two ends and V1 and V2 are the corresponding velocity of the fluid.

13.

Velocity of efflux

Velcoity of water flowing out from the container is given by,

$v=\sqrt{2gh}$

14.

The height of capillary rise inside a tube

$h=\dfrac{2S\cos\theta}{\rho rg}$