4Newton's Laws of Motion

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NEWTON'S LAWS OF MOTION
▢ Newton's Laws:

Table 1: Three Laws of Motion

Law
Also called
Gives
Statement
Formula / Point
Newton's 1st law
Law of inertia
Qualitative definition of force
A body continues in its state of rest or uniform motion in straight line unless acted upon by external force
If \\(\vec F=0\), then \(\vec a=0\)
Newton's 2nd law
Real law of motion
Quantitative definition / measurement of force
Rate of change of momentum is directly proportional to applied force and occurs in direction of force
\(\vec F=\frac{d\vec p}{dt}=m\frac{d\vec v}{dt}+\frac{dm}{dt}\vec v\)
Newton's 3rd law
Action-reaction law
Force exists in pairs
To every action, there is equal and opposite reaction
\(\vec F*{AB}=-\vec F*{BA}\)
▢ First Law:
❖ Inertia:

Table 1: Inertia

Term
Meaning
Inertia
Property of body to continue in initial state in absence of external force
Measure of inertia
Mass
Greater mass
Greater inertia
❖ Momentum:

Table 1: Momentum

Term
Formula / Meaning
Momentum
Quantity of motion contained in a body
Linear momentum
\(\vec p=m\vec v\)
Change of momentum
By changing mass, velocity or both
❖ Conservation of Linear Momentum:
◉ Statement: If external force on a system is zero, total linear momentum remains constant
◉ Formula: If \(\vec F*{ext}=0\), then \(\vec p=\sum m\vec v=constant\)
◉ Centre of Mass: Centre of mass remains in initial state if net external force is zero
◉ Recoiling of Gun:

Table 1: Gun Recoil

Quantity
Formula / Point
Conservation equation
\(m_1v_1+m_2v_2=0\)
Gun recoil velocity
\(v_2=-\frac{m_1v_1}{m_2}\)
Negative sign
Gun recoils opposite to bullet direction
▢ Second Law:
❖ General Formula: \(\vec F=\frac{d\vec p}{dt}=m\frac{d\vec v}{dt}+\frac{dm}{dt}\vec v\)
❖ Special Cases:

Table 1: Newton's Second Law Special Cases

Condition
Formula
Mass constant
\(\frac{dm}{dt}=0 \Rightarrow \vec F=m\vec a\)
Velocity constant
\(\frac{d\vec v}{dt}=0 \Rightarrow \vec F=\frac{dm}{dt}\vec v\)
❖ Applications:

Table 1: Variable Mass Applications

Case
Formula / Point
\(n\) bullets per second, each mass \(m\), fired with velocity \(v\)
Force required to hold gun = \(F=nmv\)
Sand dropped on conveyor belt at rate \(\frac{dm}{dt}\), belt speed \(v\)
\(F=\frac{dm}{dt}v\)
Water of density \(\rho\), velocity \(v\), pipe area \(A\), strikes wall at angle \(\theta\)
Elastic collision: \(F=2\rho Av^2\cos\theta\)
Water strikes wall inelastically
\(F=\rho Av^2\cos\theta\)
▢ Third Law:
❖ Key Points:
    ◈ **type: bullet
  1. For every action, equal and opposite reaction
  2. Action and reaction act on different bodies
  3. They do not cancel each other
  4. Forces are mutual
❖ Formula: \(\vec F*{AB}=-\vec F*{BA}\)
❖ Rocket Propulsion:
◉ Principle: Conservation of linear momentum / Newton's third law

Table 1: Rocket Formulae

Quantity
Formula / Meaning
\(M_0\)
Initial mass of rocket
\(\frac{\Delta M}{\Delta t}\)
Rate of ejection of fuel
\(M\)
Mass of rocket at any instant
\(\vec v\)
Relative velocity of ejected gases with respect to rocket
Thrust without gravity
\(F=\frac{\Delta M}{\Delta t}\vec v\)
Thrust with gravity
\(F=\frac{\Delta M}{\Delta t}\vec v-Mg\)
Acceleration with gravity
\(\vec a=\frac{1}{M}\frac{\Delta M}{\Delta t}\vec v-g\)
▢ Impulse:

Table 1: Impulse

Term
Formula / Point
Impulse
Product of force and time
Formula
\(\vec I=\int \vec F\,dt\)
Using momentum
\(\vec I=\int \frac{d\vec p}{dt}dt=\int d\vec p=\vec p_f-\vec p_i\)
Impulse
Change in momentum
SI unit
N s
Dimensional formula
Same as linear momentum
Force-time graph
Impulse = area under force-time graph
▢ Motion of Connected Bodies:

Table 1: Connected Bodies Formulae

Case
Acceleration
Tension
Thrust on pulley
Hanging mass \(m_1\) pulls mass \(m_2\) on smooth horizontal surface
\(a=\frac{m_1}{m_1+m_2}g\)
\(T=\frac{m_1m_2}{m_1+m_2}g\)
\(T'=\sqrt2T\)
Hanging mass \(m_1\) pulls mass \(m_2\) on rough horizontal surface
\(a=\frac{m_1-\mu m_2}{m_1+m_2}g\)
\(T=(1+\mu)\frac{m_1m_2}{m_1+m_2}g\)
\(T'=\sqrt2T\)
Atwood machine
\(a=\frac{m_1-m_2}{m_1+m_2}g\)
\(T=2\frac{m_1m_2}{m_1+m_2}g\)
\(T'=2T\)
Two masses on smooth inclined planes \(\alpha\), \(\beta\)
\(a=\frac{m_1\sin\alpha-m_2\sin\beta}{m_1+m_2}g\)
\(T=(\sin\alpha+\sin\beta)\frac{m_1m_2}{m_1+m_2}g\)
\(T'=2T\cos\left(\frac{180^\circ-(\alpha+\beta)}{2}\right)\)
▢ Tension in System of Masses:

Table 1: Bodies Tied by Strings

Case
Acceleration
Tension Rule
Several masses pulled by force \(F\) on smooth surface
\(a=\frac{F}{m_1+m_2+m_3}\)
Tension in any part = \(\frac{Sum\ of\ masses\ behind\ it}{Total\ mass}\times F\)
Constant force on smooth surface with two masses
\(a=\frac{F}{m_1+m_2}\)
Tension in any part = \(\frac{Sum\ of\ masses\ ahead\ of\ it}{Total\ mass}\times F\)
Pulling a horizontal chain / rope
\(a=\frac{F}{m_1}\)
Tension in any part = \(\frac{Length\ before\ it}{Total\ length}\times F\)
❖ Example for Three Masses:
  • •
    \(T_1=\frac{m_2+m_3}{m_1+m_2+m_3}F\)
  • •
    \(T_2=\frac{m_3}{m_1+m_2+m_3}F\)
▢ Read and Digest:

Table 1: Special Points

Case
Result / Explanation
Man on weighing balance takes quick step
Weight first decreases, then increases
Constant force on body
Uniform acceleration
Force always perpendicular to velocity
Path becomes circular
Athlete runs before long jump
To gain greater inertia of motion
Iron-loaded truck vs cotton-loaded truck
Cotton truck overturns more easily due to higher centre of gravity
Action-reaction
Mutual forces; do not cancel because they act on different bodies
Spring balance in lift
Measures apparent weight
Light rifle vs heavy rifle
Light rifle recoils more; causes more shoulder injury
Motorcycle and car same velocity, same retardation
Both stop at same distance
Person on frictionless horizontal plane
Can move by throwing object / spitting / sneezing opposite direction
Balloons moving upward with velocities \(v,2v,3v\); bombs released at same height
Bomb from \(B_1\) reaches earth first
Coin tossed in uniformly moving train
Falls back in hand
Coin tossed in accelerating train
Falls behind passenger
Coin tossed in decelerating train
Falls in front of passenger
Oil drum in accelerating truck
Oil surface rises backward
Two 5 kg weights attached to spring scale
Reading = 5 kg wt
▢ Wedge and Accelerating Systems:
❖ Body on Frictionless Wedge:

Table 1: Horizontal Acceleration of Wedge

Condition
Required acceleration
Body remains at rest relative to wedge
\(a=g\tan\theta\)
Body falls freely
\(a=g\cot\theta\)
❖ Body Suspended in Accelerating Carriage:

Table 1: Pendulum in Accelerating Carriage

Quantity
Formula
Vertical balance
\(T\cos\theta=mg\)
Horizontal balance
\(T\sin\theta=ma\)
Acceleration
\(a=g\tan\theta\)
Tension
\(T=m\sqrt{g^2+a^2}\)
Time period
\(T_p=2\pi\sqrt{\frac{l}{\sqrt{a^2+g^2}}}\)
▢ False Balance:

Table 1: False Balance Formulae

Condition
True weight
False balance with equal arms
\(W=\frac{W_1+W_2}{2}\)
False balance with unequal arms
\(W=\sqrt{W_1W_2}\)
▢ Smooth Inclined Plane:

Table 1: Body Sliding on Smooth Inclined Plane

Quantity
Formula
Acceleration down plane
\(a=g\sin\theta\)
Velocity at bottom
\(v=\sqrt{2gh}=\sqrt{2gl\sin\theta}\)
Time to reach bottom
\(t=\sqrt{\frac{2l}{g\sin\theta}}\)
Same height, different angle
\(t\propto \csc\theta\)
Equal length, different angle
\(t\propto \frac{1}{\sqrt{\sin\theta}}\)
❖ Ratios:
  • •
    Same height → \(\frac{t_1}{t_2}=\frac{\sin\theta_2}{\sin\theta_1}\)
  • •
    Equal length → \(\frac{t_1}{t_2}=\sqrt{\frac{\sin\theta_2}{\sin\theta_1}}\)
▢ High-Yield Recall:

Table 1: Newton's Laws One-Liners

Fact
Answer
Newton's 1st law
Law of inertia
Newton's 1st law gives
Qualitative definition of force
Newton's 2nd law gives
Quantitative measurement of force
Real law of motion
Newton's 2nd law
Newton's 3rd law gives
Action-reaction pair property
Inertia measured by
Mass
Momentum
\(\vec p=m\vec v\)
Force
\(\vec F=\frac{d\vec p}{dt}\)
Constant mass force
\(\vec F=m\vec a\)
Variable mass, constant velocity
\(F=\frac{dm}{dt}v\)
Action-reaction
\(\vec F*{AB}=-\vec F*{BA}\)
Action-reaction cancel?
No; they act on different bodies
Gun recoil principle
Conservation of linear momentum
Rocket propulsion principle
Newton's 3rd law / conservation of momentum
Rocket thrust without gravity
\(F=\frac{\Delta M}{\Delta t}v\)
Impulse
Change in momentum
Impulse unit
N s
Impulse from F-t graph
Area under graph
Force perpendicular to velocity
Circular path
Lift spring balance reading
Apparent weight
Acceleration of suspended body system
\(a=g\tan\theta\)
Tension in accelerating carriage
\(T=m\sqrt{g^2+a^2}\)
False balance unequal arms
\(W=\sqrt{W_1W_2}\)
Smooth inclined plane acceleration
\(g\sin\theta\)
Smooth inclined plane bottom velocity
\(\sqrt{2gh}\)
Q1.
If the external force applied is zero, then\nwhich of the following is conserved?\n[BP 2011]
📅BP 2011
Q2.
\nNo force is required for a body moving\nwith\n[MOE 2014]
📅MOE 2014
Q3.
\nTwo blocks of masses 6kg and 4kg tied to\nends of a light inextensible string passing\n\nover a frictionless pulley are released.\nThen acceleration of the system will be
Q4.
\nWhat is the minimum acceleration of a\nfireman sliding down a fixed vertical rope\nfor which breaking strength is a times his\nweight
Q5.
\nA block of 2kg on a horizontal surface is\npulled at 30" with horizontal by a force of\n\n10N. Then normal reaction on the block is
Q6.
. A 5kg stone falls from a height of 100m\nand penetrates 2m is a layer of sand. The\ntime of penetration is
Q7.
A balloon is descending at a constant\nacceleration a. The mass of the balloon is\nM. When a mass m is released from the\nmass of the balloon it starts rising with the\n\nsame acceleration a. Assuming that the\nvolume doesn't change when the mass is\nreleased, what is the value of m/M ?
Q8.
A balloon of mass M is rising up with\nacceleration a, then to double the\n\nacceleration, the fraction of weight of\nballoon to be detached is
Q9.
What is the force exerted by the block B on\nblock A if lift accelerates\ndownward at acceleration\n2m/s' if mass of block B is\n0.5kg
Q10.
A man weighing 80kg is standing on a\ntrolley weighing 320kg. The trolley is\n\nresting on frictionless horizontal rails. .If\nthe man starts walking on the trolley along\nthe rails at speed 1m/s, then after 4sec, his\ndisplacement relative to ground will be
Q11.
A 600kg rocket is set for a vertical firing.\nIf the exhaust speed is 100ms , the mass of\nthe gas ejected per second to supply the\nthrust needed to overcome the weight of\nrocket is
Q12.
A bullet of mass 10g is fired from a gun of\nmass 1kg with recoil velocity of gun =\n5m/s. The muzzle velocity will be
Q13.
A uniform rod of mass 6kg and length is\nsuspended from . a rigid support. The\ntension at a distance - from the free end is
Q14.
A block of mass M is pulled along a\nhorizontal frictionless surface by a rope of\nmass m. If a force F is applied at one end\nof the rope, the force which the rope exert\non the block is\n[BPKIHS]
📅BPKIHS
Q15.
A boy of mass 40kg is hanging from the\nhorizontal branch of a tree. The tension is\narms is maximum when angle between the\narms is :
Q16.
Gravel is dropped onto a conveyer belt at\nthe rate of 0.5kg/s. The extra force\nrequired to keep the belt moving at 2m/s is
Q17.
A satellite in force free space sweeps\nstationary interplanetary dust at the rate\nof at = a v. The acceleration of the\nsatellite is
Q18.
A jet of water with area of cross-section\n2cm strikes a wall at an angle 60 to th\nnormal and rebounds elastically from the\nwall with the same speed. If the speed of\nwater in the jet is 10m/s, then the force\nacting on the wall is
Q19.
A ball of mass 0.5kg moving with a velocity\nof 2m/s strikes a wall normally and\nbounces back with the same speed. If the\n\ntime of contact between the ball and the\nwall is 1 millisecond, then average force\nexerted by the wall on the ball is
Q20.
A boy having a mass of 60kg holds i\nhands a school bag of weight 40N. Wit\nwhat force the floor will push up on his\nfeet? (g = 10m/s-)
Q21.
A 20 kg crate hangs at the end of a long\nrope. Find its acceleration when\ntension in the rope is 150N.
Q22.
A scooter of mass 120kg is moving with a\nuniform velocity of 108 km/hr. The force\nrequired to stop the velocity in 10sec is
Q23.
A ball is dropped onto a floor from a\n\nheight of 10m. It rebounds to a height of\n2.5m. If the ball is in contact with the floor\nfor 0.01 sec, then upward average\nacceleration at the time of contact is:\n[BPKIHS]
📅BPKIHS
Q24.
80 railway wagons all of same 5x10'kg ar\npulled by an engine with a force of 4x10'N\nThe tension in the coupling between 30th\nand 31st wagon from the engine is
Q25.
Two trains A and B are running in th\nsame direction on parallel tracks such that\nA is faster than B. If packets of equal\nweights are exchanged between the two\nthen
Q26.
An open knife edge of mass M is dropped\n\nfrom a height h on a wooden floor. If the\nblade penetrates S into the wood, th\naverage resistance offered by the wood to\nthe blade is
Q27.
The surfaces are frictionless. The ratio of\nT, to T2 is\n12kg 15kg30
Q28.
Two masses of 10kg and 20kg respectively\nare tied together by a massless spring. A\nforce of 200N is applied on a 20kg mass. At\nacceleration\n10kg mass is 12ms", the acceleration of 20\nkg mass is
Q29.
A body of mass 5kg at rest explodes into 3\nsegments having masses in the ratio 2:2:1.\nThe fragments with equal masses fly in\nmutually perpendicular directions with\nspeed 15ms". What will be the velocity of\nthe lighter segment?
Q30.
In a tug of war two opposite teams are\npulling the rope with an equal and\n\nopposite force of 10KN at each end of the\nrope so that condition of equilibrium\nexists. What is the tension in the rope ?
Q31.
A truck weighing 8000kg is moving along a\n\ntrack with negligible friction at 1.8ms'\nwith the engine turn off when it begins t\nrain hard. The rain drops fall vertically\nwith respect to the ground. The speed of\nthe truck, when it. has collect 1000kg of\nrain water is
Q32.
A body of mass 10kg is moving eastward\nwith a uniform speed of 2m/s. A force of\n20N is applied to it towards north. What is\nthe magnitude of displacement after\n\n2second?
Q33.
Which one of the following group of three\nforces will not produce accelerat\nbody acted by the forces?\n[MOE 2009]
📅MOE 2009
Q34.
Two bodies of masses 4 Kg and 5Kg are\nacted upon one after the other by the same\nforce. If the acceleration of the heavier\n2ms', the acceleration of the lighter\nbody is\n[MOE 20651
📅MOE 2065
Q35.
Whatever may be the direction of the two\nforces 6N and 2N acting on a body of mass\n2kg, the minimum acceleration of the body\ncannot be less than.\n[MOE 2058]
📅MOE 2058
Q36.
A body of mass 2 kg moving with a certain\nvelocity is acted upon by an opposing force\nof 4N. It stops in 2s. For the same body to\ncontinue motion with the same velocity, we\nshould apply:\n[BPKIHS 01]
📅BPKIHS 01
Q37.
A force of P magnitude is acting on the\nfree end of a rope of mass m attached with\na block of mass M. What is the force on the\nblock?\n[BPKIHS 02]
📅BPKIHS 02
Q38.
. A body having a mass of 8kg travels\ndistances of 4, 5 and 6 m respectively in\nsuccessive seconds. The force acting on it is:\n[BPKIHS 05]
📅BPKIHS 05
Q39.
A 60 Kg man pushes a 40 Kg man by a\nforce of 60 N. The 40 Kg man has pushed\nthe other man with a force of [BPKIHS-95]
📅BPKIHS-95
Q40.
A rope of length / is pulled by a constant\nforce F. What is the tension in the rope at a\ndistance x from the end where force is\napplied\n[BPKIHS 98]
📅BPKIHS 98
Q41.
A force F, acts on a particle so as to\naccelerate it from rest to velocity v. The\nforce F, is replaced by a force F, which\n[KU 2015)\n\ndecelerates it to rest. Then
📅KU 2015
Q42.
A thief stole the book weighing 'w' then\njump vertically down the wall. What is the\nresultant wt of the body before he reach\nthe ground?\n[KU 2016]
📅KU 2016
Q43.
When a man weighing 10kg in lift is\naccelerated downward\nwith\nthe\nacceleration of 1m/s' then apparent wt is:\n[KU 2016
📅KU 2016
Q44.
A rest substance is broken into three pieces.\nFirst two pieces have equal mass with the\n\nvelocity of 30 m/s move in perpendicular\ndirection, 3" piece has its mass 3 times the\nmass of each equal piece. Find the velocity of\nthird piece.\n[TOM 2016)
📅TOM 2016