When solving pulley problems, why is tension the same throughout an ideal string?

Strings are assumed light (massless) and inextensible, pulleys frictionless. A massless string cannot support tension differences; a frictionless pulley cannot change tension across it. Under these idealizations tension is uniform.

Newton's Laws Of Motion

Q: What is the difference between Newton's First and Second Laws?

The First Law is qualitative: it defines inertia and identifies when no net force acts. The Second Law is quantitative: it tells us how much a body accelerates for a given net force via F = ma. The First Law is a special case of the Second when F = 0.

Q: Why do action and reaction not cancel each other?

Action and reaction act on different bodies. When you push a wall, you exert force on the wall; the wall exerts equal opposite force on you. Since the two forces act on different objects, they never appear in the same free-body diagram and cannot cancel.

Q: What is a pseudo force and when do I need it?

A pseudo force is F = -m a_0, introduced only when solving inside a non-inertial (accelerating) frame. It has no physical origin and no reaction pair. Once added, Newton's Second Law works inside the frame as if it were inertial.

Q: Is centripetal force a real force?

Centripetal force is a label for the net inward radial force required for circular motion. The actual providers are real forces like tension, friction, gravity, or normal reaction. It is not a separate fundamental force.

Q: How does F = ma break down for a rocket?

F = ma assumes mass is constant. A rocket ejects fuel continuously, so its mass changes. The general form F = dp/dt must be used, and the extra term v(dm/dt) produces rocket thrust.

Q: What is impulse and how is it related to momentum?

Impulse J is the product of average force and time: J = F_avg * dt. By Newton's Second Law this equals change in momentum: J = dp. Athletes follow through to extend contact time and increase momentum transfer.

Q: What is the difference between mass and weight?

Mass is amount of matter (kg), a scalar and measure of inertia. Weight is gravitational force W = mg, a vector in newtons. Mass is invariant; weight varies with location such as on the Moon.

PhysicsLaws of MotionFor JEE aspirants

Newton's Laws of Motion are three foundational principles that describe how forces produce and change motion. The First Law defines inertia: a body stays at rest or moves uniformly in a straight line unless acted on by a net force. The Second Law quantifies this: net force equals rate of change of momentum, or $F = m a$ for constant mass. The Third Law states that every action has an equal and opposite reaction. Together with spring force, pseudo forces in non-inertial frames, and centripetal force in circular motion, these laws form the basis of classical mechanics for JEE and NEET.

Key Formulas — Quick Reference

Newton's Second Law (constant mass): $F_{n e t} = m a$
Momentum: $p = m v$ , with units kg·m/s
Second Law (general form): $F = \frac{d p}{d t}$
Impulse: $J = F_{a v g} \cdot Δ t = Δ p$
Spring force (Hooke's law): $F = - k x$
Pseudo force in non-inertial frame accelerating with $a_{0}$ : $F_{p se u d o} = - m a_{0}$
Centripetal force: $F_{c} = \frac{m v ^{2}}{r} = m ω^{2} r$
Rocket thrust: $F = u \frac{d M}{d t}$ , where $u$ is exhaust speed

1. What is a Force?

A force is a push or pull acting on a body. It is a vector quantity, having both magnitude and direction.

SI unit: newton (N). CGS unit: dyne. $1 N = 1 0^{5}$ dyne.
Dimensions: $[M L T^{- 2}]$

Contact vs Field (Non-contact) Forces

Type	Examples	Nature
Contact forces	Tension, normal reaction, friction	Act between bodies in physical contact
Field (non-contact) forces	Weight (gravity), electrostatic, magnetic	Act between bodies separated by a distance

Tension (T): When a string, rope, or spring is held taut, the ends pull on the bodies attached to them along the direction of the string. This pulling force is called tension.

Normal reaction (N): When two surfaces are in contact, they exert equal and opposite forces on each other, perpendicular to the surfaces.

Friction (f): A contact force that opposes relative motion or the tendency of relative motion between two surfaces (covered in detail in the Friction concept).

2. Newton's First Law of Motion (Law of Inertia)

Statement: Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by an external force.

Inertia is the property of a body by virtue of which it resists any change in its state of rest or uniform motion. Mass is a measure of inertia — heavier bodies have greater inertia.

Linear Momentum

Linear momentum is the product of a body's mass and its velocity, and it points in the direction of the velocity.

p = m v

SI unit: kg·m/s. CGS unit: g·cm/s.
Dimensions: $[M L T^{- 1}]$
Momentum is a vector quantity.

3. Newton's Second Law of Motion

Statement: The rate of change of momentum of a body is directly proportional to the net external force applied on it, and takes place in the direction of that force.

If a body of mass $m$ moves with velocity $v$ , its momentum is $p = m v$ . According to the Second Law:

F \propto \frac{d p}{d t} \Rightarrow F = k \frac{d p}{d t}

By choosing units such that unit force produces unit rate of change of momentum, the constant $k = 1$ , giving:

F = \frac{d p}{d t}

For a body with constant mass, this reduces to the familiar form:

F = \frac{d}{d t} (m v) = m \frac{d v}{d t} = m a

Note:

F = m a

is valid only when mass is constant. For variable-mass systems like rockets, the general form

F = d p / d t

must be used.

Impulse

When a force acts on a body for a short time $Δ t$ , the change in momentum is called impulse:

J = F_{a v g} \cdot Δ t = Δ p

SI unit: N·s (same as kg·m/s). Impulse is a vector along the direction of the average force.

4. Newton's Third Law of Motion

Statement: To every action, there is an equal and opposite reaction. The two forces always act on different bodies.

If body A exerts a force $F_{A B}$ on body B, then body B exerts a force $F_{B A}$ on body A such that:

F_{A B} = - F_{B A}

Action and reaction act on different bodies, so they never cancel out. This is a common source of confusion.

Solved Example 1

Figure 1: Force F applied at angle θ decomposes into horizontal (F cos θ) and vertical (F sin θ) components.

A block of mass $m = 10$ kg is pulled by a force $F = 100$ N at an angle $θ = 3 0^{\circ}$ with the horizontal along a smooth horizontal surface. What is the acceleration of the block? (Take $g = 10 m / s^{2}$ .)

Solution:

Decompose $F$ along the $x$ and $y$ axes.

Along the vertical (no acceleration): $\sum F_{y} = N + F sin θ - m g = 0$

Along the horizontal: $\sum F_{x} = F cos θ = ma$

a = \frac{F cos θ}{m} = \frac{100 cos 3 0 ^{\circ}}{10} = \frac{100 \cdot 3 /2}{10} = 53 m / s^{2}

The block accelerates at $53 m / s^{2} \approx 8.66 m / s^{2}$ towards the direction of pull. Since $F sin θ < m g$ , the block does not lift off the surface.

Solved Example 2

Figure 2: Junction P in equilibrium under three tensions. The 45° angle is measured at the wall between the vertical (dashed) and the cord.

Tension in a horizontal cord attached to a junction P is 30 N. From P, one cord runs to a wall at $4 5^{\circ}$ above horizontal (tension $T_{2}$ ), and another hangs vertically supporting body B of weight W. Find W.

Solution:

Isolate P. Forces: horizontal 30 N, tension $T_{2}$ at $4 5^{\circ}$ , tension $T_{1}$ downward.

Equilibrium at P:

$\sum F_{x} = 0 : 30 - T_{2} sin 4 5^{\circ} = 0 \Rightarrow T_{2} = 302 N$

$\sum F_{y} = 0 : T_{2} cos 4 5^{\circ} - T_{1} = 0 \Rightarrow T_{1} = 30 N$

Since body B is in equilibrium, $T_{1} = W$ , giving $W = 30 N$ .

Solved Example 3

Figure 3: Horizontal cord from m1 wraps over the top of a wall-mounted pulley and drops vertically to hanging m2.

A block of mass $m_{1}$ on a frictionless horizontal table is connected by a light cord over a small frictionless pulley to a hanging block of mass $m_{2}$ . Find the acceleration of the system and the tension in the cord.

Solution:

Both blocks share the same magnitude of acceleration $a$ (cord is inextensible), and tension $T$ is uniform.

For $m_{1}$ (horizontal): $T = m_{1} a$ ... (1)

For $m_{2}$ (vertical): $m_{2} g - T = m_{2} a$ ... (2)

Adding: $m_{2} g = (m_{1} + m_{2}) a$ , so

a = \frac{m _{2} g}{m _{1} + m _{2}}, T = \frac{m _{1} m _{2} g}{m _{1} + m _{2}}

Solved Example 4

Figure 4: Two blocks in contact — external force F pushes m₁, and contact force N transmits to m₂.

Two blocks of masses $m_{1} = 2$ kg and $m_{2} = 1$ kg are in contact on a smooth horizontal surface. A horizontal force $F = 3$ N is applied on $m_{1}$ . Find the contact force between the blocks.

Solution:

Let contact force between the blocks be $N$ . Both blocks accelerate together with acceleration $a$ .

Treating the system as a whole: $F = (m_{1} + m_{2}) a \Rightarrow a = \frac{F}{m _{1} + m _{2}} = \frac{3}{3} = 1 m / s^{2}$

Isolating $m_{2}$ (only force on it horizontally is $N$ ): $N = m_{2} a = 1 \cdot 1 = 1 N$

Contact force $N = 1 N$ .

5. Spring Force (Hooke's Law)

When a spring is stretched or compressed, the restoring force is directly proportional to the deformation and directed opposite to it:

F = - k x

Here $k$ is the spring constant (or force constant), measured in N/m. The negative sign indicates that the force always tries to restore the spring to its natural length.

Large $k$ = stiff spring; small $k$ = soft spring.
For a spring cut into two equal halves, each piece has spring constant $2 k$ .
Springs in series: $\frac{1}{k _{e q}} = \frac{1}{k _{1}} + \frac{1}{k _{2}}$ . In parallel: $k_{e q} = k_{1} + k_{2}$ .

6. Frame of Reference

A frame of reference is a coordinate system used to describe the position and motion of a body.

Inertial Frame

A frame in which Newton's First Law holds is called an inertial frame. A frame at rest or moving with constant velocity relative to distant stars is inertial. In an inertial frame, no fictitious forces are needed.

Non-Inertial Frame and Pseudo Force

A frame that is accelerating is a non-inertial frame. To apply Newton's Second Law inside such a frame, we introduce a pseudo (fictitious) force:

F_{p se u d o} = - m a_{0}

where $a_{0}$ is the acceleration of the frame with respect to an inertial frame. The pseudo force acts on every body of mass $m$ inside the accelerating frame, opposite to the frame's acceleration.

Pseudo force is not a real force. It has no reaction pair. It only appears when we choose to work inside a non-inertial frame.

Solved Example 5

Figure 5: In the accelerating wedge frame, pseudo force ma acts leftward on the block; combined with mg, it balances against normal reaction from the incline.

A block of mass $m$ is placed on a smooth inclined plane of angle $θ$ . With what horizontal acceleration $a$ should the wedge move so that the block does not slide relative to the incline? All surfaces are smooth.

Solution:

In the wedge's (non-inertial) frame, the block is at rest. Forces on the block: weight $m g$ downward, normal reaction $R$ perpendicular to the incline, pseudo force $ma$ opposite to the wedge's acceleration.

Perpendicular to the incline (block does not slide): $R = m g cos θ + ma sin θ$

Along the incline (block does not slide): $m g sin θ = ma cos θ$

a = g tan θ

Solved Example 6

Figure 6: Pendulum in an accelerating car tilts backward at angle θ from the vertical, where tan θ = a0/g.

A pendulum of mass $m$ hangs from the ceiling of a car accelerating with $a_{0}$ on a horizontal road. Find the angle the string makes with the vertical.

Solution:

In the car's frame, the bob is stationary. Forces: tension $T$ along the string, weight $m g$ down, pseudo force $m a_{0}$ opposite to $a_{0}$ .

$T sin θ = m a_{0}$ ... (i)

$T cos θ = m g$ ... (ii)

Dividing (i) by (ii):

tan θ = \frac{a _{0}}{g} \Rightarrow θ = tan^{- 1} (\frac{a _{0}}{g})

7. Centripetal and Centrifugal Force

A body moving with constant speed in a circle is continuously accelerated towards the centre of the circle. This acceleration is called centripetal acceleration:

a_{c} = \frac{v ^{2}}{r} = ω^{2} r

The net radial force producing this acceleration is the centripetal force:

F_{c} = m a_{c} = \frac{m v ^{2}}{r} = m ω^{2} r

Centripetal force is not a new type of force — it is the name given to whichever real force (tension, friction, gravity, normal reaction) provides the required radial acceleration.

Centrifugal Force

In a rotating (non-inertial) frame, an outward pseudo force $m ω^{2} r$ appears on the body. This is the centrifugal force. It is a fictitious force introduced only to make Newton's laws applicable in the rotating frame.

Total Acceleration in Non-Uniform Circular Motion

When speed also varies, total acceleration has two components:

a = a_{T} + a_{R}

where $a_{T}$ is tangential (along velocity, changes speed) and $a_{R}$ is radial/centripetal (perpendicular to velocity, changes direction).

Solved Example 7

Find the ratio of the radius of curvature at the highest point of a projectile's trajectory to that just after projection, if the angle of projection is $θ = 3 0^{\circ}$ .

Solution:

Let $v_{0}$ be initial speed. At projection point O, the component of gravity normal to velocity is $g cos θ$ . At the highest point P, gravity is entirely normal to velocity, and speed is $v_{p} = v_{0} cos θ$ .

Using $r = v^{2} / a_{⊥}$ :

$r_{0} = \frac{v _{0}^{2}}{g cos θ}$ (at O)

$r_{p} = \frac{v _{0}^{2} cos ^{2} θ}{g}$ (at P)

Ratio:

\frac{r _{p}}{r _{0}} = cos^{3} θ = cos^{3} 3 0^{\circ} = \frac{3 3}{8}

Solved Example 8

A 1200 kg car rounds a level unbanked curve of radius 200 m at 72 km/h. Find the minimum coefficient of friction between tyres and road so the car does not skid. ( $g = 10 m / s^{2}$ .)

Solution:

On an unbanked road, friction alone provides the centripetal force. Convert $v = 72$ km/h $= 20$ m/s.

Friction condition: $μ m g \geq \frac{m v ^{2}}{r}$

μ_{min} = \frac{v ^{2}}{g r} = \frac{20 \cdot 20}{10 \cdot 200} = 0.2

8. Rocket Propulsion (Variable Mass)

A rocket ejects gas at exhaust speed $u$ (relative to itself), losing mass at rate $d M / d t$ . By conservation of momentum, this gives the rocket a forward push called thrust:

F_{t h r u s t} = u \frac{d M}{d t}

If $M_{0}$ is initial mass and $t$ is time since launch, the instantaneous acceleration of the rocket is:

a = \frac{u ( d M / d t )}{M _{0} - ( d M / d t ) t} - g

(subtract $g$ if gravity is acting).

Recoil of a Gun

When a gun of mass $m_{g}$ fires a bullet of mass $m_{b}$ with velocity $v_{b}$ , conservation of momentum gives the gun's recoil velocity:

v_{g} = \frac{m _{b} v _{b}}{m _{g}}

Common Mistakes to Avoid

Watch out

Treating action-reaction as balancing forces on the same body. They act on different bodies and never cancel out.
Using $F = ma$ for variable-mass systems. For rockets, always use $F = d p / d t$ .
Forgetting the pseudo force sign. It always points opposite to the frame's acceleration.
Assuming tension is always equal to weight. Tension equals weight only when the string is vertical and the body is in equilibrium.
Confusing centripetal force with a new force. It's a label for the net radial force, not an independent force.
Ignoring vector nature of momentum. Momentum can be conserved along one axis and not another; always resolve.
Applying Newton's laws in a non-inertial frame without adding pseudo force. Either switch to an inertial frame or add $- m a_{0}$ .

Frequently Asked Questions

Q1. What is the difference between Newton's First and Second Laws?

The First Law is qualitative: it defines inertia and identifies when no net force acts (body stays at rest or moves uniformly). The Second Law is quantitative: it tells us how much a body accelerates for a given net force, via $F = m a$ . The First Law can be viewed as a special case of the Second when $F = 0$ .

Q2. Why do action and reaction not cancel each other?

Action and reaction act on different bodies. For example, when you push a wall, you exert force on the wall; the wall exerts an equal, opposite force on you. Since the two forces act on different objects, they never appear in the same free-body diagram and cannot cancel.

Q3. What is a pseudo force and when do I need it?

A pseudo (or fictitious) force is $F_{p se u d o} = - m a_{0}$ , introduced only when solving a problem inside a non-inertial (accelerating) frame. It has no physical origin and no reaction pair. Once added, Newton's Second Law $F = m a$ works inside that frame as if it were inertial.

Q4. Is centripetal force a real force?

Centripetal force is a name given to the net inward radial force required for circular motion. The actual providers are real forces like tension (whirling a stone on string), friction (car on a curve), gravity (satellite orbit), or normal reaction (banked road). It is not a separate fundamental force.

Q5. How does $F = ma$ break down for a rocket?

$F = ma$ assumes mass is constant. A rocket continuously ejects fuel, so its mass changes with time. The general form $F = d p / d t = m (d v / d t) + v (d m / d t)$ must be used. The extra term $v (d m / d t)$ is what produces rocket thrust.

Q6. What is impulse and how is it related to momentum?

Impulse $J$ is the product of average force and the time it acts: $J = F_{a v g} Δ t$ . By Newton's Second Law, this equals the change in momentum: $J = Δ p$ . This is why sports players "follow through" — they extend $Δ t$ to increase the momentum change on the ball.

Q7. Why does momentum conservation not require an inertial frame condition explicitly?

Momentum conservation follows from Newton's Third Law: internal action-reaction pairs cancel in the total momentum. If external net force on the system is zero, total momentum is constant. This holds in any inertial frame; in non-inertial frames, momentum is conserved only if pseudo forces on the system sum to zero (rare).

Q8. When solving pulley problems, why is the tension the same throughout an ideal string?

In JEE/NEET problems, strings are usually assumed light (massless) and inextensible, and pulleys are frictionless. A massless string cannot support a tension difference (else it would have infinite acceleration), and a frictionless pulley cannot change tension across it. Under these idealizations, tension is uniform along the string.

Q9. What is the difference between mass and weight?

Mass is the amount of matter in a body (measured in kg), a scalar and a measure of inertia. Weight is the gravitational force on the body, $W = m g$ , a vector measured in newtons. Mass is invariant; weight changes with location (e.g. lower on the Moon).

Previous year questions on Newton's Laws Of Motion

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1. What is a Force?

Contact vs Field (Non-contact) Forces

2. Newton's First Law of Motion (Law of Inertia)

Linear Momentum

3. Newton's Second Law of Motion

Impulse

4. Newton's Third Law of Motion

5. Spring Force (Hooke's Law)

6. Frame of Reference

Inertial Frame

Non-Inertial Frame and Pseudo Force

7. Centripetal and Centrifugal Force

Centrifugal Force

Total Acceleration in Non-Uniform Circular Motion

8. Rocket Propulsion (Variable Mass)

Recoil of a Gun

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is the difference between Newton's First and Second Laws?

Q2. Why do action and reaction not cancel each other?

Q3. What is a pseudo force and when do I need it?

Q4. Is centripetal force a real force?

Q5. How does F=ma break down for a rocket?

Q6. What is impulse and how is it related to momentum?

Q7. Why does momentum conservation not require an inertial frame condition explicitly?

Q8. When solving pulley problems, why is the tension the same throughout an ideal string?

Q9. What is the difference between mass and weight?

Previous year questions on Newton's Laws Of Motion

Ready to master Laws of Motion?

Q5. How does $F = ma$ break down for a rocket?