Revision Notes Chapter 5 Laws Of Motion Class 11 Notes

Newton's Second Law and the Mechanics of Momentum

Newton's Second Law is the core quantitative framework of dynamics. It states that the rate of change of linear momentum of a body is directly proportional to the applied external force, and this change takes place in the direction of the force. Mathematically, linear momentum is defined as the product of mass and velocity, written as $\vec{p} = m\vec{v}$.

The applied force is expressed as:
$\vec{F} \propto \frac{d\vec{p}}{dt} \implies \vec{F} = k \frac{d(m\vec{v})}{dt}$
In SI units, the constant of proportionality $k = 1$. Under classical mechanics, where the mass $m$ of the system remains constant, the equation simplifies directly to:
$\vec{F} = m\frac{d\vec{v}}{dt} = m\vec{a}$
When force is applied for a very short duration, it gives rise to the concept of Impulse. Impulse is the measure of the total effect of a force and is defined as the product of the average force and the time interval, which equals the total change in momentum:
$\vec{I} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$
This is known as the Impulse-Momentum Theorem, a critical conceptual tool for collision analysis.

Core Definitions & Terminology

Inertia

The inherent property of a body by virtue of which it resists any change in its state of rest or uniform motion in a straight line.

Linear Momentum (p)

The quantity of motion contained in a body, mathematically defined as the vector product of mass and velocity (p = mv).

Impulse (I)

The product of a large force acting on a body for a short interval of time, equivalent to the change in momentum produced in that interval.

Limiting Friction

The maximum value of static friction force that comes into play when a body is just on the verge of sliding over another surface.

Centripetal Force

The net force directed toward the center of curvature that causes an object to follow a circular path.

Pseudo Force

An imaginary force (like centrifugal force) applied to a body to make Newton's laws applicable in a non-inertial (accelerating) frame of reference.

Friction Comparison: Static Friction vs. Kinetic Friction

How to Draw and Solve Free Body Diagrams (FBD)

1. Isolate the Body — Choose the specific object of interest and mentally disconnect it from all surrounding contacting surfaces or bodies.
2. Identify Non-Contact Forces — Draw the force of gravity (mg) acting vertically downwards from the center of mass of the isolated body.
3. Identify Contact Forces — Examine the contact points. Add Normal Reaction (N) perpendicular to contact planes, Tension (T) pulling away along strings, and Friction (f) opposing intended/actual motion.
4. Establish Coordinate Axes — Select Cartesian axes (X and Y). If the body is on an incline, align one axis parallel to the incline to simplify vector resolution.
5. Resolve Forces and Solve Equations — Split non-aligned forces into components (e.g., mg cos θ and mg sin θ). Write separate equilibrium or acceleration equations: ΣFx = max and ΣFy = may.

Must-Remember Formulas & Crucial Concepts

• Newton's First Law defines force and inertia qualitatively; Newton's Second Law measures force quantitatively; Newton's Third Law states action and reaction forces are equal, opposite, and act on different bodies.
• Law of Conservation of Linear Momentum: In the absence of an external force, the total momentum of an isolated system remains conserved: Σp_initial = Σp_final.
• Apparent weight of a man in a lift accelerating upwards is R = m(g + a); if accelerating downwards, R = m(g - a); during free fall, R = 0.
• The maximum velocity for safe turning on a level circular road without slipping is v_max = sqrt(\mu_s g R).
• For a banked circular road of radius R and angle θ, the optimum speed (no friction required) is v_0 = sqrt(g R tan θ).
• The maximum safe speed on a banked road with friction is: v_max = sqrt( g R [ (\mu_s + tan \theta) / (1 - \mu_s * tan \theta) ] ).
• Recoil velocity of a gun is given by: V = - (m * v) / M, where m and v are mass and velocity of bullet, and M is the mass of the gun.

Solved Revision Problems

• {"title":"Problem 1: Acceleration of Connected Masses","description":"Two blocks of masses m1 = 3 kg and m2 = 2 kg are connected by a light string passing over a frictionless, massless pulley (Atwood machine). Find the acceleration of the system.","solution":"1. Identify the forces: m1 is pulled down by m1g and up by Tension T. m2 is pulled down by m2g and up by Tension T.\n2. Write the equations of motion assuming m1 (larger mass) moves down:\nFor m1: m1g - T = m1a\nFor m2: T - m2g = m2a\n3. Add both equations:\n(m1 - m2)g = (m1 + m2)a\n4. Calculate 'a' using g = 10 m/s^2:\na = [(3 - 2) / (3 + 2)] 10 = (1/5) 10 = 2 m/s^2."}
• {"title":"Problem 2: Friction Threshold Check","description":"A wooden block of mass 5 kg lies on a horizontal floor. The coefficient of static friction is 0.5. A horizontal force of 20 N is applied. Will the block move, and what is the frictional force?","solution":"1. Find Normal Reaction: N = mg = 5 10 = 50 N.\n2. Calculate Maximum Limiting Static Friction: f_max = \\mu_s N = 0.5 50 = 25 N.\n3. Compare Applied Force with f_max:\nThe applied force (20 N) is less than the limiting friction (25 N).\n4. Conclusion: The block does not move. Because static friction is self-adjusting, the actual friction force acting on the block is equal to the applied force, which is 20 N (not 25 N)."}

Common Exam Traps and Marking Cues

The Static Friction Trap: Never automatically substitute $f = \mu_s N$ as the friction force unless the problem states the body is on the verge of sliding. If the applied force is less than the limiting value, the friction force is simply equal to the applied force.

Action-Reaction Pairs: When writing justifications for Newton's Third Law, remember that action and reaction forces never cancel each other because they act on different bodies. You will lose marks if you assert they equilibrate a single body.

Inclined Planes: Always resolve gravity into $mg \cos \theta$ perpendicular to the incline and $mg \sin \theta$ parallel to the incline. Do not write $N = mg$ as a default; on an incline, $N = mg \cos \theta$.

Quick Revision Checks

• Why is Newton's second law called the real law of motion? Because both the first and third laws of motion can be derived from the second law. The first law is contained in it (when net force is zero, acceleration is zero), and the third law can be proved using the conservation of momentum which is established by the second law.
• Can a body have a constant speed and still have a non-zero net force acting on it? Yes, in uniform circular motion. The speed of the body remains constant, but its direction changes continuously, meaning it undergoes centripetal acceleration, which requires a non-zero net centripetal force.
• Why are shock absorbers installed in vehicles? Shock absorbers increase the time interval of the jerk or impact ($Δt$). Since impulse $I = F * Δt$ is constant for a given change in momentum, increasing the duration ($Δt$) significantly reduces the average force ($F$) experienced by the passengers.
• Why is pulling a lawnmower easier than pushing it? When pulling, the vertical component of the pulling force acts upwards, reducing the effective normal reaction ($N = mg - F\sinθ$) and thus lowering the friction. When pushing, the vertical component acts downwards, increasing the normal reaction ($N = mg + F\sinθ$) and increasing the friction.

Frequently Asked Questions