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Laws of Motion

Ch. 4Std 11

Easy Overview

Why does a book sitting on a table stay there until you push it? Why does your body lurch forward when a bus suddenly stops? And why do you feel pushed back into your seat when the bus accelerates? The answers are Newton's three laws of motion - the most fundamental rules about how things move or do not move. These three laws, published in 1687, are the foundation of classical mechanics. They explain everything from why a cricket ball follows a curved path after being hit, to why a rocket can travel in the vacuum of space, to why a passenger in a turning car feels flung sideways. This chapter goes deep into Newton's laws - what they really mean, how to apply them in different situations, and the practical nuances like friction that make real-world motion different from ideal physics problems. Once you master these laws, you will see the world differently - every motion around you will make physical sense.

Newton's First Law - Inertia

Newton's first law says: an object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by an unbalanced external force. This is the law of inertia. Galileo first came up with this idea - he imagined a ball rolling down one incline and up another, and realized that if there were no friction, the ball would roll forever. Inertia is the tendency of objects to resist changes in their state of motion. More mass = more inertia. That is why pushing a heavy box is harder than pushing a light one. The first law also defines what we mean by an inertial frame of reference - a frame where the law holds. A bus accelerating is not an inertial frame - things inside seem to move without any obvious force acting on them.

Newton's Second Law - F = ma

Newton's second law quantifies force: the net force on an object equals its mass times its acceleration: F_net = ma. In formal terms: F = dp/dt, where p = mv is momentum. Force is the rate of change of momentum. If mass is constant, F = ma. This is a vector equation - force and acceleration are always in the same direction. A larger mass needs more force for the same acceleration. The same force produces less acceleration on a larger mass. This law is the workhorse of mechanics - any time you need to find how something moves under forces, start with F = ma. The key is identifying all forces acting on the object and vector-adding them to get the net force.

Newton's Third Law - Action and Reaction

Every action has an equal and opposite reaction. If you push on a wall, the wall pushes back on you with the same force. If Earth pulls on you (gravity), you pull on Earth with the same force. The forces always come in pairs - they act on DIFFERENT objects. This last point is crucial and commonly misunderstood. When you punch a wall, your hand exerts a force on the wall. The wall exerts an equal and opposite force on your hand - that is why your hand hurts. The two forces act on different objects (hand and wall), so they do not cancel each other. Action-reaction pairs are always equal in magnitude, opposite in direction, and act on different bodies.

Free Body Diagrams

A free body diagram (FBD) is your best friend for solving force problems. Isolate the object of interest. Represent it as a dot. Draw arrows for ALL forces acting on it - gravity (mg, downward), normal reaction (perpendicular to surface), tension (along the string, away from the object), friction (opposing relative motion or tendency), applied forces. Label each force clearly. Choose coordinate axes (typically along the direction of acceleration). Break forces into components along those axes. Apply F_net = ma along each direction separately. FBDs force you to be systematic - most mistakes in force problems come from missing a force or getting its direction wrong. Always draw the FBD before writing any equations.

Connected Bodies - Tension and Pulleys

When objects are connected by strings over pulleys, they move together. The tension in an ideal string (massless, inextensible) is the same throughout. For an Atwood machine (two masses connected by a string over a pulley): m_1g - T = m_1a (if m_1 > m_2), T - m_2g = m_2a. Solving: a = (m_1 - m_2)g/(m_1 + m_2), T = 2m_1m_2g/(m_1 + m_2). For more complex systems, the same approach works: isolate each body, write F = ma for each, and solve the system. The constraint equation (relationship between accelerations) comes from the geometry of the strings.

Friction - Static and Kinetic

Friction opposes relative motion (or tendency) between surfaces in contact. Static friction f_s prevents motion from starting. It is self-adjusting - it matches the applied force up to a maximum: f_s(max) = mu_s N, where mu_s is the coefficient of static friction. Kinetic friction f_k acts when surfaces are sliding: f_k = mu_k N. Usually mu_k < mu_s - it is harder to start sliding than to keep sliding. Friction direction always opposes relative motion. A common mistake: friction always opposes motion - false! When you walk, friction with the ground pushes you forward. When a car accelerates, road friction pushes it forward. Without friction, you could not walk, cars could not move, and nails would not stay in walls.

Angle of Friction and Angle of Repose

Angle of friction lambda = arctan(mu) - the angle between normal reaction and the resultant of normal reaction and friction. Angle of repose alpha is the maximum incline angle at which a body stays put without sliding: tan alpha = mu_s. Conveniently, they are equal. If you slowly increase incline angle, the block starts sliding when tan theta = mu_s. This gives an experimental method for mu_s - tilt the surface until sliding starts and measure the angle. The acceleration down an incline: a = g(sin theta - mu_k cos theta). For mu_k = 0, a = g sin theta - it is just the component of gravity along the slope.

Applications of Friction

Rolling friction is much smaller than sliding friction - that is why wheels are efficient. When a wheel rolls without slipping, static friction acts at the point of contact (which is momentarily at rest). For braking, stopping distance s = v^2/(2 mu g). Higher mu = shorter stopping distance. ABS prevents wheel lockup, maintaining static friction (stronger than kinetic) between tire and road. In banking, friction helps: max safe speed on a banked curve with friction is higher than without. Friction is also why you can not drive on icy roads - low mu means minimal force for acceleration, braking, and turning.

Dynamics of Circular Motion

In circular motion, the net force toward center provides centripetal acceleration: F_c = mv^2/r. This can come from tension, gravity, friction, normal reaction, or combinations. In vertical circular motion, net force changes at each point because gravity's direction relative to center changes. At top: mg + T = mv^2/r. At bottom: T - mg = mv^2/r. If centripetal force is removed, the object moves in a straight line tangent to the circle - not outward. The centrifugal force you feel is not real - it is your body's inertia wanting to go straight while the car forces you into a circular path.

Inertial and Non-Inertial Frames

An inertial frame is where Newton's first law holds - objects at rest stay at rest, moving objects keep constant velocity unless acted upon. A frame at rest or moving with constant velocity is inertial. A non-inertial frame is accelerating - a car speeding up, a rotating merry-go-round. In non-inertial frames, objects appear to accelerate without apparent force. To make Newton's laws work, we invent pseudo-forces. In a linearly accelerating frame, pseudo-force = -ma (opposite to acceleration). In a rotating frame, centrifugal force appears: F_cf = m omega^2 r outward. These are not real forces - they are mathematical corrections.

Impulse and Conservation of Momentum

Impulse J = F dot delta t = delta p (change in momentum). A large force for short time can produce same impulse as small force for long time. Airbags save lives by increasing stopping time, reducing force on passengers. Cricketers pull hands back while catching to increase stopping time and reduce impact force. Conservation of momentum: in absence of external forces, total momentum of system stays constant. Derived from Newton's third law - internal forces come in equal-opposite pairs, canceling out when summing momentum changes of all bodies.

Collisions - Elastic and Inelastic

In elastic collisions, both kinetic energy and momentum are conserved. Objects bounce off with no energy loss. In perfectly inelastic collisions, objects stick together - momentum conserved, kinetic energy not (converted to heat, sound, deformation). For 1D elastic collision: v_1f = ((m_1 - m_2)/(m_1 + m_2))v_1i + (2m_2/(m_1 + m_2))v_2i. Equal masses exchange velocities. For inelastic: m_1v_1i + m_2v_2i = (m_1 + m_2)v_f. Coefficient of restitution e = (relative speed after)/(relative speed before). e = 1 for elastic, e = 0 for perfectly inelastic.

Key Points

  • •First law: objects resist changes in motion (inertia). Defines inertial frames.
  • •Second law: F_net = ma = dp/dt. Force causes acceleration in same direction.
  • •Third law: action-reaction are equal, opposite, act on different bodies. Never cancel.
  • •Free body diagrams: isolate object, draw all forces, resolve components, apply F = ma in each direction.
  • •Connected bodies: apply F = ma to each, use constraint equations for connections.
  • •Static friction: f_s = mu_s N. Self-adjusting. Kinetic friction: f_k = mu_k N. mu_k < mu_s.
  • •Angle of repose = angle of friction = arctan(mu). Block slides when theta > arctan(mu_s).
  • •Centripetal force F_c = mv^2/r. Not a separate force - provided by tension, friction, gravity, etc.
  • •Non-inertial frames: pseudo-forces appear - -ma (linear), m omega^2 r (centrifugal).
  • •Impulse J = F delta t = delta p. Same impulse from large force/short time or small force/long time.
  • •Conservation of momentum: total momentum of isolated system constant. Derived from third law.
  • •Elastic collision: momentum and KE conserved. Inelastic: only momentum. Perfectly inelastic: objects stick.
  • •Coefficient of restitution e = relative speed after / relative speed before. e = 1 elastic, e = 0 perfectly inelastic.
  • •Friction can be necessary for motion - walking, car acceleration depend on friction.

Practice Questions

  • State Newton's laws. Explain how third law differs from action-reaction misconception.
  • Two masses 5 kg and 3 kg connected by string over pulley. Find acceleration and tension.
  • A 10 kg block on rough surface with mu_s = 0.4, mu_k = 0.3. A 30 N force is applied. Will it move? If yes, find acceleration.
  • Derive expressions for velocities after 1D elastic collision between two bodies.
  • A 1500 kg car at 20 m/s collides with stationary 1000 kg car, they stick. Find common velocity and KE loss.
  • Explain why a cricketer pulls hands back while catching. Use impulse-momentum.
  • A 2 kg mass moves in circle of radius 0.5 m at 3 m/s. Find centripetal force. What provides it on a string?
  • A block slides down a 30 degree incline with mu_k = 0.2. Find acceleration and time to slide 5 m from rest.