Laws of Motion: Class 11 Physics · Chapter 4

4.1 Introduction

In Chapters 2 and 3 we described motion: where a body is, how fast it goes and how its velocity changes. We never asked why it moves as it does. That question, about the causes of motion, is the subject of this chapter, and its answer is the foundation of all of mechanics (गति के नियम, laws of motion).

Everyday experience suggests a first answer. A carrom coin moves only when the striker hits it, and a rolling ball stops when a fielder blocks it. Sometimes the push or pull needs contact, as with the striker; sometimes it acts across empty space, as when a mango drops from a branch or a magnet tugs a pin towards itself. So starting and stopping both need a force from outside. But what about a coin that is already gliding steadily across a well-powdered board? Does something have to keep pushing it?

4.2 Aristotle's fallacy

The Greek philosopher Aristotle (384–322 BCE) answered yes. In his scheme, a body moves only for as long as something keeps acting on it; take the mover away and the motion dies. He even claimed that air rushing in behind a flying arrow keeps shoving it forward. Put simply, the Aristotelian rule says: to keep a body moving, an outside force must keep acting on it.

It is easy to see why this felt obvious. A vegetable seller's handcart keeps rolling only while the seller keeps pushing; the moment he lets go, it slows and stops. On Earth, things left alone always seem to come to rest.

The flaw is that the cart is never left alone. The road and the axles exert friction on it, and the air drags on it. While the cart rolls at a steady speed, the seller's push only cancels these opposing forces, so the net force on the cart is zero (Fig. 4.1). When he lets go, friction is the only horizontal force left, and it slows the cart. Remove friction and drag entirely, and the cart would roll on with no push at all. Aristotle's observations were accurate; what his bookkeeping lacked was a force. The breakthrough came from asking what would happen if that missing force could be switched off, and the person who asked it was Galileo.

4.3 The law of inertia

Early in the seventeenth century, Galileo Galilei timed balls rolling on smooth, gently sloping tracks and spotted a pattern. Downhill, a ball gains speed; uphill, it loses speed; and the steeper the track, the faster the change. A track that slopes neither way should then cause no change at all, provided friction could be removed: the ball would roll on at a constant velocity.

A thought experiment with two facing ramps makes the same point (Fig. 4.2). Let a ball go from some height on the left ramp. It runs down, crosses the bottom and climbs the right ramp until it is almost back at its starting level; the smoother the ramps, the closer it gets, so a frictionless ball would reach that level exactly. Now tilt the right ramp lower. To regain the same level the ball has to travel further along it. Lay the right ramp flat, and the level can never be regained, so the ball should roll on without end. Real friction stops it eventually, but the trend leaves little doubt about the ideal case.

This overturned two thousand years of thinking. Sitting still and gliding steadily are the same state as far as forces go: either way, the forces on the body add up to zero. What needs a force is a change of velocity. The handcart seller in Section 4.2 pushes only to cancel friction. Every body tends to keep doing what it is already doing, staying put or cruising straight on at a steady speed; this tendency is called inertia (जड़त्व).

4.4 Newton's first law of motion

Galileo had shown what was wrong with the old mechanics; the new one was built largely by Isaac Newton (1642–1727), whose Principia (1687) set out three laws of motion. Newton took Galileo's law of inertia as his first law.

We use the law in two directions. Sometimes we know the forces cancel and conclude that the acceleration is zero: a dust grain drifting in deep space, far from every star, feels essentially no force, so it keeps whatever velocity it has. More often we cannot list every force in advance, but we can see that a body is not accelerating, and then the law tells us the forces on it must add up to zero. On Earth gravity always acts, and moving things meet friction and drag, so a body at rest or in uniform motion is not free of forces; its forces simply cancel.

Take a phone lying on a desk. Two forces act on it: its weight $W$ downwards and the push of the desk, the normal force $N$, upwards. Nobody measured $N$ beforehand; the desk supplies whatever push is needed, and it adjusts itself to the load. The correct chain of reasoning runs one way only: the phone is observed to be at rest, so by the first law the net force is zero, so $N=W$. Saying the phone stays at rest because $N$ equals $W$ gets the logic backwards.

Now consider a car pulling away from a traffic light, then cruising at a steady speed on a straight road. While it gains speed it accelerates, so a net external force must act on it. Its engine and gears are internal: on their own they could no more accelerate the car than you can lift yourself by pulling on your shoelaces. The engine turns the wheels, the tyres push backwards on the road, and the road pushes the car forwards through friction (Section 4.9). Once the car cruises steadily, that forward friction exactly balances air drag and rolling resistance, and the net force is zero again.

Inertia explains a familiar jolt in a bus (Fig. 4.3). When a standing passenger's bus starts suddenly, friction from the floor drags the feet forward with the bus, but the upper body tends to stay where it was, so the passenger tips backwards relative to the bus until the muscles bring the body along. When the bus brakes hard, the feet are stopped by friction while the upper body carries on forwards, and the passenger lurches ahead. Holding a handrail helps because the arm can then supply the force the upper body needs.

4.5 Newton's second law of motion

The first law covers the special case of zero net force. The second law handles every other case: it tells us how much a given net external force changes a body's motion. To state it we need one new quantity.

Momentum

Multiply a body's mass by its velocity and you get its momentum (संवेग):

Because velocity is a vector, so is momentum, and it points the same way as the velocity. Why is this particular product worth naming? Several everyday observations point to it:

• Mass matters. Pushing a loaded tempo up to walking pace takes far more effort, or far more time, than pushing an empty scooter to the same pace. Stopping the tempo is just as much harder.
• Speed matters. A tennis ball lobbed gently is easy to stop with your palm; the same ball arriving at $40\mathrm{m}\mathrm{/}\mathrm{s}$ stings. For one mass, more speed means more force to stop it in a given time.
• Time matters. Jump off a wall onto a sand pit and you hardly feel it; land stiff-legged on concrete and your knees jar. The change in momentum is the same, but sand and bent knees spread it over a longer time, so the force is smaller.
• The same force for the same time gives the same momentum change. Apply one steady force for one second to a light trolley and to a heavy one, both starting at rest. The light trolley ends up faster, but both end up with exactly the same momentum. This is the key clue to the second law.

Momentum can also change in direction alone. An athlete spinning a hammer (a steel ball on a wire) before the throw keeps its speed steady for a moment, yet its direction turns all the time, and the athlete must keep pulling inwards on the wire to make it turn. Spin it faster, or on a shorter wire, and the pull needed grows. A faster change in momentum, whether in size or in direction, needs a larger force. We return to this pull in Section 4.10.

The SI unit of force, the newton, is the force that gives a $1\mathrm{k}\mathrm{g}$ mass an acceleration of $1\mathrm{m}\mathrm{/}{\mathrm{s}}^2$: $1\mathrm{N}=1\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}$. An average apple weighs roughly $1\mathrm{N}$.

Four features of the second law deserve attention before we use it.

1. It contains the first law. Put $𝐅=0$ in (4.5) and you get $𝐚=0$.
2. It is a vector law. It stands for three separate equations, one for each axis ((4.6)). A force changes only the velocity component along its own direction; the component at right angles is untouched. For a projectile, gravity acts vertically, so the horizontal velocity stays constant. The force is always along the acceleration, but it may point along the velocity, against it, at right angles to it or at any angle.
3. It applies to systems too. For a rigid body, or any group of particles, $𝐅$ means the total external force and $𝐚$ the acceleration of the centre of mass (Chapter 6). Forces between parts of the system are internal and are left out.
4. It is local in time. The acceleration at an instant is fixed by the force at that same instant and place. A body keeps no memory of forces that acted on it earlier (Fig. 4.4).

Impulse

A bat strikes a ball for only a few milliseconds, and during that time the force rises and falls in a way that is very hard to record. The ball's velocity before and after, however, is easy to measure, and the second law links the two: adding up force × time over the whole contact gives the change in momentum. This product is called the impulse of the force:

When the force varies, the impulse is the area under the force–time graph. The same area can be a tall, narrow spike or a low, wide hump (Fig. 4.5): this is why bending your knees on landing, a cricketer letting the hands give as the ball arrives, a car's crumple zone or a helmet's foam lining all reduce the force by stretching out the time.

A force that is very large but lasts only an instant is called an impulsive force. It is not a different kind of force: apart from its size and brevity, the second law treats it exactly like a gentle push. Because the contact time is so short, the body hardly moves while it acts, so we can treat its position as fixed during the impact.

4.6 Newton's third law of motion

The second law tells us what a force does; the third tells us where forces come from. Every force on a body is exerted by some other body, and the influence always runs both ways. Step off a small boat onto a jetty and the boat drifts backwards as you move forwards. With bodies that do not touch, the two-way nature is harder to see. The Earth pulls a falling coconut down; Newton's claim is that the coconut pulls the Earth up just as hard. For a $1.2\mathrm{k}\mathrm{g}$ coconut that force is about $12\mathrm{N}$, which gives the Earth, of mass $6\times {10}^{24}\mathrm{k}\mathrm{g}$, an acceleration of only about $2\times {10}^{-24}\mathrm{m}\mathrm{/}{\mathrm{s}}^2$. That is why we never notice it.

So a force is never a one-sided affair; it is an interaction between two bodies, and the two sides of it are always equal in size and opposite in direction. Newton put it this way:

Because Newton's wording is so short, it is often misread. Three points prevent most mistakes:

1. The two forces act on different bodies. When you study the motion of A alone, only ${𝐅}_{\text{AB}}$ matters, so the pair can never cancel each other in A's equation of motion. Only when A and B are treated together as one system do the two become internal forces that add to zero. In the same way, all the internal forces inside a body cancel in pairs, which is what lets the second law work for a whole body.
2. Neither comes first. The two forces arise together and vanish together; one does not cause the other. Either may be called the action.
3. Action and reaction are just forces. The words add nothing else. It is clearest to name every force as "the force on A by B".

Fig. 4.6 shows the law at work. Two skaters on smooth ice push palms. Each pushes the other with the same force, so the lighter skater, with less inertia, gets the larger acceleration. They glide apart in opposite directions, and since the forces act for the same time, they end up with equal and opposite momenta: the $40\mathrm{k}\mathrm{g}$ skater moves off $1.5$ times as fast as the $60\mathrm{k}\mathrm{g}$ one.

4.7 Conservation of momentum

Put the second and third laws together and a powerful result drops out. Fire a cannon. While the powder burns, the cannon pushes the ball forward with a force $𝐅$, and the ball pushes the cannon back with $-𝐅$, for the same short time $\mathrm{\Delta }t$. The ball's momentum changes by $𝐅\mathrm{\Delta }t$ and the cannon's by $-𝐅\mathrm{\Delta }t$. Both started at rest, so afterwards the ball's momentum and the cannon's recoil momentum add up to zero, just as before the shot.

4.8 Equilibrium of a particle

A particle is in equilibrium (साम्यावस्था) when the net external force on it is zero. By the first law it is then either at rest or moving with constant velocity. With two forces this simply means they are equal and opposite:

With three forces acting at one point, equilibrium needs their vector sum to vanish:

Read geometrically, this says that the resultant of any two of the forces (by the parallelogram law) is equal and opposite to the third. Equivalently, if we draw the three forces head to tail, they close into a triangle (Fig. 4.8). The idea extends to any number: a particle is in equilibrium under $n$ forces exactly when their arrows, joined head to tail, form a closed polygon.

For calculations we usually resolve along axes instead. (4.11) then splits into one scalar equation per axis:

For an extended body this is only half the story. A body can feel zero net force and still start to spin, so its equilibrium also needs zero net torque, which we study in Chapter 6. A particle has no size, so for it zero net force is enough.

4.9 Common forces in mechanics

Gravity is special: it acts between bodies that are not touching, across empty space. Every object near the Earth feels it, and it also steers the Moon round the Earth and the planets round the Sun. (Charged and magnetised bodies also feel electric and magnetic forces at a distance; we leave them aside here.)

Every other force we meet in mechanics is a contact force: it exists only while one object touches another, solid or fluid, and it comes in third-law pairs. Where two solid surfaces press together, it is useful to split the contact force into two parts (Fig. 4.10). The part perpendicular to the surfaces is the normal reaction; the part along them is friction. Fluids exert contact forces too: an object in water or air feels an upward buoyant force that matches the weight of the fluid it pushes aside, and a body moving through a fluid feels a viscous drag, which in air we call air resistance.

Two other contact forces appear in almost every problem. A spring that is stretched or squashed pushes or pulls back towards its natural length. For small deformations the restoring force is proportional to the displacement $x$ from the natural length:

The minus sign says the force always opposes the displacement. A string behaves like a spring with an enormous $k$: it stretches almost not at all, and the pull it exerts is called tension. In problems we usually take a string to be massless and inextensible. Then the tension is the same all along it, and a smooth, light pulley changes only the direction of the pull, not its size.

Physicists recognise four fundamental forces: gravitational, electromagnetic, strong and weak. The strong and weak forces act only inside atomic nuclei, so in mechanics only gravity and electromagnetism matter. Remarkably, every contact force, from the normal reaction and friction to tension, spring forces and drag, is electrical at heart: it comes from the forces between the charged electrons and nuclei of the atoms in the two surfaces. Working them out atom by atom would be hopelessly complicated, so mechanics describes each one by a simple law found from experiment.

4.9.1 Friction

Lean on a steel almirah standing on the floor: nothing happens. Lean harder: still nothing. Since the almirah does not accelerate, the first law says the horizontal forces on it must balance, so the floor is pushing back on it with a force exactly equal to yours. That force is static friction $f_s$. With no push there is no static friction at all; as you push harder it grows to match you. It opposes the impending motion, the sliding that would happen if the floor were frictionless.

Static friction cannot grow without limit: push hard enough and the almirah starts to slide. Experiments show that the largest possible static friction does not depend on the area of contact and is, to a good approximation, proportional to the normal force:

The constant ${\mu }_s$, the coefficient of static friction, depends only on the pair of surfaces in contact. The rule for static friction is therefore an inequality:

Once sliding starts, the friction force drops below the static maximum. This kinetic friction (or sliding friction) also does not depend on the contact area, is nearly independent of the speed, and follows a similar law:

Because ${\mu }_k<{\mu }_s$, it takes more force to start a body sliding than to keep it sliding (Fig. 4.11). Once it is moving, the net force on it is $F-{\mu }_{k}N$, so a block can glide at a steady speed under a push smaller than the one that started it, and it slows to rest at ${\mu }_{k}g$ once the push stops. Try both regimes in the friction lab below (Fig. 4.12).

Unlike Newton's law of gravitation, these friction rules are not deep laws of nature. They are rules of thumb distilled from measurements, and real surfaces can depart from them noticeably. For most calculations, though, they are all we need.

Notice what friction actually opposes: relative motion between the surfaces, not motion as such. A crate on the open bed of a delivery truck speeds up with the truck. The only horizontal force on the crate is friction from the truck bed, and it points forwards, in the direction of motion. If the bed were frictionless, the truck would slide out from under the crate. Static friction opposes that impending slip, and in doing so drags the crate forward with the truck.

Rolling friction

Drag a brick across the floor and then roll a ball of the same weight: the ball keeps going far longer. In the ideal case a hard wheel rolling on a hard road touches it at a single point that is momentarily at rest on the road, so nothing slides and there is no sliding friction at all. Real wheels still slow down, because the tyre flattens a little and the road dents a little. They now touch over a small patch instead of a point, and the contact force on that patch tilts slightly backwards (Fig. 4.14). Its backward component is rolling friction. For the same load it is typically hundreds to a thousand times smaller than sliding friction, which is why loads move so much more easily on wheels or rollers than when dragged.

Friction is often a nuisance. In a machine, sliding parts waste power as heat and wear each other away. Engineers fight it with lubricating oil, with ball bearings that turn sliding into rolling, and with a thin cushion of air between surfaces, as in a hovercraft or an air-hockey table.

Just as often, friction is exactly what we need. You can walk only because the ground's friction pushes your foot forward; people and animals need an external force to accelerate, just like carts. The same friction from the road speeds up a car, steers it round a bend and, through its brakes, brings it to a stop. On a road covered with oil or ice, a car can do none of these.

4.10 Circular motion

In Chapter 3 we found that a body moving at constant speed $v$ round a circle of radius $R$ accelerates towards the centre, with magnitude $v^2\mathrm{/}R$. By the second law, a net force must point towards the centre too. Its size is

This inward net force is called the centripetal force (अभिकेंद्र बल). The name describes a job, not a new kind of force. In every circular motion some real force, exerted by a real object, does the job: the tension in a string for a whirled stone, the Sun's gravity for a planet, the Earth's gravity for the Moon or a satellite, friction for a car on a flat bend, and part of the normal force for a car on a banked track or a rider on a maut ka kuan (wall of death). Never draw "centripetal force" as an extra arrow beside these; it is their resultant towards the centre.

Motion of a car on a level road

Motion of a car on a banked road

Relying on friction alone is risky, so highway curves and racetracks are banked: the outer edge of the road is built higher than the inner edge. The normal force, being perpendicular to the road, then tilts towards the centre of the curve, and its horizontal component helps to supply the centripetal force (Fig. 4.15).

At the optimum speed $v_0$ the tilted normal force alone provides $m{v}^2\mathrm{/}R$, so the tyres need no sideways friction at all and wear least. Drive faster than $v_0$ and friction must act down the slope; drive slower and the car tends to slide down the bank, so friction acts up the slope. Repeating the derivation with $f$ reversed gives the lowest safe speed:

If $\tan \theta \le {\mu }_s$ the right-hand side is zero or negative, so there is no lower limit: a car can even be parked on the bank without sliding down.