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.
The frictional force exerted by air on a meter of mass 0.25 kg moving with acceleration 9.2 m/s² is:
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Q2.
Apparent weight of a person is more than real weight when lift goes:
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Q3.
There are two blocks, each of mass \(M\), resting on frictionless inclined planes as shown in the figure (45° for A and 60° for B). Then:
Image 1
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Q4.
Which one of the following is not an example of non-contact force?
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Q5.
If the net external force on a system is zero, which quantity is conserved?
📅BP 2011
Q6.
No net external force is required to maintain a body in:
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Q7.
Two blocks of masses \(6\,\mathrm{kg}\) and \(4\,\mathrm{kg}\) are connected over a frictionless pulley. Find their acceleration.
Q8.
A fireman slides down a vertical rope whose breaking strength equals \(\alpha\) times his weight (\(0<\alpha<1\)). What minimum downward acceleration prevents the rope from breaking?
Q9.
A \(2\,\mathrm{kg}\) block on a horizontal surface is pulled by \(10\,\mathrm{N}\) at \(30^\circ\) above the horizontal. Find the normal reaction (take \(g=10\,\mathrm{m/s^2}\)).
Q10.
A stone falls freely from a height of \(100\,\mathrm m\) and penetrates \(2\,\mathrm m\) into sand before stopping. Assuming uniform retardation in sand, find the penetration time.
Q11.
A balloon of total mass \(M\) descends with acceleration \(a\). When a mass \(m\) is dropped, the remaining balloon ascends with acceleration \(a\). Assuming constant upthrust, find \(\frac mM\).
Q12.
A balloon of mass \(M\) rises with acceleration \(a\). What fraction of its mass should be discarded to double the upward acceleration, with upthrust unchanged?
Q13.
A block \(B\) of mass \(0.5\,\mathrm{kg}\) rests on block \(A\) in a lift accelerating downward at \(2\,\mathrm{m/s^2}\). Find the contact force exerted by \(B\) on \(A\) (take \(g=10\,\mathrm{m/s^2}\)).
Q14.
A man of mass \(80\,\mathrm{kg}\) walks at \(1\,\mathrm{m/s}\) relative to a trolley of mass \(320\,\mathrm{kg}\) on frictionless rails. How far does he move relative to the ground in \(4\,\mathrm s\)?
Q15.
A rocket of mass \(600\,\mathrm{kg}\) ejects gas at \(100\,\mathrm{m/s}\) relative to the rocket. What mass of gas must be ejected per second to balance its weight?
Q16.
A bullet of mass \(10\,\mathrm g\) is fired from a gun of mass \(1\,\mathrm{kg}\) whose recoil speed is \(5\,\mathrm{m/s}\). Find the bullet speed relative to the ground.
Q17.
A uniform vertical rod of mass \(6\,\mathrm{kg}\) hangs at rest. Find its internal tension at a point \(\frac L4\) above its lower free end (take \(g=10\,\mathrm{m/s^2}\)).
Q18.
A block of mass \(M\) is pulled horizontally by a rope of mass \(m\) on a frictionless floor. A force \(F\) acts on the rope’s free end. What force does the rope exert on the block?
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Q19.
A boy hangs symmetrically by his two arms, which make an angle \(\theta\) with each other. At which listed angle is tension in each arm greatest?
Q20.
Gravel falls vertically onto a horizontal conveyor belt at \(0.5\,\mathrm{kg/s}\). What additional horizontal force keeps the belt moving at \(2\,\mathrm{m/s}\)?
Q21.
A satellite of instantaneous mass \(M\) moves through stationary dust with speed \(v\) and collects dust at the rate \(\frac{dM}{dt}=\alpha v\). Neglecting external forces, find its acceleration along the direction of motion.
Q22.
A water jet of cross-sectional area \(2\,\mathrm{cm^2}\) strikes a wall at \(60^\circ\) to the normal and rebounds elastically at the same speed \(10\,\mathrm{m/s}\). Find the force normal to the wall (water density \(1000\,\mathrm{kg/m^3}\)).
Q23.
A \(0.5\,\mathrm{kg}\) ball hits a wall normally at \(2\,\mathrm{m/s}\) and rebounds at the same speed. Contact time is \(1\,\mathrm{ms}\). Find the average force magnitude.
Q24.
A boy of mass \(60\,\mathrm{kg}\) holds a bag weighing \(40\,\mathrm N\) while standing still. Find the normal force from the floor (take \(g=10\,\mathrm{m/s^2}\)).
Q25.
A \(20\,\mathrm{kg}\) crate hangs from a rope with upward tension \(150\,\mathrm N\). Find its acceleration (take \(g=9.8\,\mathrm{m/s^2}\)).
Q26.
A scooter of mass \(120\,\mathrm{kg}\) travels at \(108\,\mathrm{km/h}\). What constant braking force stops it in \(10\,\mathrm s\)?
Q27.
A ball falls \(10\,\mathrm m\) to a floor and rebounds to \(2.5\,\mathrm m\). If contact lasts \(0.01\,\mathrm s\), find its average upward acceleration during impact (take \(g=9.8\,\mathrm{m/s^2}\)).
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Q28.
An engine pulls \(80\) identical railway wagons, each of mass \(5\times10^3\,\mathrm{kg}\), with force \(4\times10^5\,\mathrm N\) on frictionless tracks. Find tension between wagons 30 and 31.
Q29.
Two trains \(A\) and \(B\) travel in the same direction, with \(A\) faster than \(B\). They exchange equal-mass packets laterally without changing each packet’s forward velocity at release. What happens to the train speeds?
Q30.
A knife of mass \(M\) falls from height \(h\) and penetrates distance \(s\) into wood before stopping. Find the average upward resistance of the wood.
Q31.
Masses \(10\,\mathrm{kg}\) and \(20\,\mathrm{kg}\) are connected by a massless spring on a frictionless surface. A force of \(200\,\mathrm N\) pulls the \(20\,\mathrm{kg}\) block. At an instant the \(10\,\mathrm{kg}\) block accelerates at \(12\,\mathrm{m/s^2}\). Find the acceleration of the other block.
Q32.
A body at rest explodes into fragments with masses in the ratio \(2:2:1\). The two equal fragments move perpendicularly at \(15\,\mathrm{m/s}\) each. Find the lightest fragment’s speed.
Q33.
Two teams pull opposite ends of a rope with equal and opposite forces of \(10\,\mathrm{kN}\). What is rope tension?
Q34.
A frictionless truck of mass \(8000\,\mathrm{kg}\) moves at \(1.8\,\mathrm{m/s}\). It collects \(1000\,\mathrm{kg}\) of vertically falling rain. Find its final horizontal speed.
Q35.
A \(10\,\mathrm{kg}\) body travels east at \(2\,\mathrm{m/s}\). A constant \(20\,\mathrm N\) force acts north. Find the displacement magnitude after \(2\,\mathrm s\).
Q36.
Which set of three force magnitudes CAN keep a body in equilibrium when suitably directed?
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Q37.
The same force acts separately on \(5\,\mathrm{kg}\) and \(4\,\mathrm{kg}\) bodies. The heavier one accelerates at \(2\,\mathrm{m/s^2}\). Find the lighter body’s acceleration.
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Q38.
Two forces of \(6\,\mathrm N\) and \(2\,\mathrm N\) act on a \(2\,\mathrm{kg}\) body. What is the minimum possible acceleration magnitude?
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Q39.
A body encounters a constant resistive force of \(4\,\mathrm N\). What applied force maintains constant velocity?
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Q40.
A force \(P\) pulls the free end of a uniform rope of mass \(m\) attached to a block of mass \(M\) on a frictionless floor. Find the force on the block.
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Q41.
A \(8\,\mathrm{kg}\) body covers \(4,5,6\,\mathrm m\) in three successive one-second intervals. Find the net force.
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Q42.
A \(60\,\mathrm{kg}\) man pushes a \(40\,\mathrm{kg}\) man with \(60\,\mathrm N\). Find the reaction force magnitude.
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Q43.
A uniform rope of length \(L\) is pulled horizontally by constant force \(F\) at one end on a frictionless surface. Find tension a distance \(x\) from the pulled end.
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Q44.
A force \(F_1\) accelerates a particle from rest to speed \(v\). Another force \(F_2\) then brings it to rest. Which statement about their magnitudes is correct?
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Q45.
A motorcycle moves at constant speed over a convex circular bridge. As it ascends toward the top, what happens to the normal reaction?
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Q46.
A person holding a book jumps from a wall. Before reaching the ground, what is their apparent weight during free fall?
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Q47.
A \(10\,\mathrm{kg}\) person stands on a lift accelerating downward at \(1\,\mathrm{m/s^2}\). Find apparent weight (take \(g=9.8\,\mathrm{m/s^2}\)).
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Q48.
A body at rest breaks into three fragments. Two fragments of mass \(m\) each move at \(30\,\mathrm{m/s}\) in perpendicular directions. The third has mass \(3m\). Find its speed.
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