NotesShala Class 11 · Physics

Chapter04

Laws of Motion

Force, Inertia and Newton's First Law · Momentum, Second and Third Laws · Equilibrium, Friction and Circular Motion

Reading key Definition Note Remember Derivation Strategy Formulas sit on graph paper · coloured words are the key terms

Part 1

Force, Inertia and Newton's First Law

Motion is everywhere in nature. For large (macroscopic) bodies, we can see it easily. At the atomic and nuclear level, we cannot see motion, but it is still there. A motion can be as simple as a ball thrown up into the air. To study any motion, we must study the forces that control it.

1.1Force

Definition

Force is an external push or pull on a body. It changes, or tries to change, the body's state of rest or of uniform motion. It can also change the body's direction of motion.

SI Unit

newton (N)

Dimensional Formula

$[\text{MLT}^{-2}]$

A force on a body can produce four types of effects:

iStarts Motion

It can change the state of rest of a body into motion.

iiChanges Speed

It can change the speed of a body. The body then moves faster or slower.

iiiChanges Direction

It can change the direction of motion of a body.

ivChanges Shape

It can change the shape of a body.

Note

Key Points About Force

A bar magnet attracts an iron nail without touching it, from some distance. So forces such as gravitational and magnetic forces can act without contact.
A force always has a point of application. So force is a polar vector.
The vector sum of all the forces acting on a body is called the net force.

1.2Aristotle's Fallacy

Think of a box sliding with constant velocity on a slab of ice. It is in uniform motion along a straight line. Is an external force needed to keep it moving? Aristotle's fallacy and Galileo's law of inertia answer this question.

Concept

Aristotle's View (384 BC – 322 BC)

Aristotle was a Greek philosopher. He believed that a moving body needs an external force to keep it moving.

e.g. An arrow keeps flying after it leaves the bow. Aristotle's reason was that the air behind it pushes it forward all the time.

What Aristotle's law claimsUniform motion needs an external force

Aristotle's idea comes from a simple fact. Resistive forces are always present, and they stop every motion. So a force seems necessary to balance the resistive force.

e.g. A rolling ball slows down and stops because of friction.

A ball on a horizontal floor: fast → slow → rest, due to friction

Concept

Galileo Corrects Aristotle

In nature, opposing forces are always present. Solids have friction and fluids have viscous forces. Nearly two thousand years after Aristotle, Galileo found the correct answer. An external force is required only to cancel friction. If there were no friction, a body in uniform motion would go on moving with no external force at all. In this way, Galileo proved the law of inertia.

1.3Galileo's Single Inclined Plane

Galileo first studied objects moving on an inclined plane. He made three observations:

iMoving Down the Plane

The speed of the object increases.

iiMoving Up the Plane

The speed decreases. The object retards.

iiiMoving on a Horizontal Plane

The speed stays constant.

Note

A level (horizontal) plane gives the object no acceleration and no retardation. So the object keeps a constant speed on it.

1.4Galileo's Double Inclined Plane

Galileo saw that the bob of a swinging pendulum always rises to the same height on both sides of its mean position. He then did another experiment with a double inclined plane. It has two inclined planes facing each other.

iEqual Slopes

The ball rolls down one plane and climbs up the other. Because of friction, it reaches almost the same height. With no friction (ideal case), the final height equals the initial height.

iiSecond Slope Reduced

Now make the second plane less steep. The ball again rises to the same height, but it has to roll a longer distance to get there.

iiiSecond Slope Zero

The second plane is now horizontal. In the ideal case, the ball travels an infinite distance. It moves forever with uniform velocity. This is called perpetual motion.

Concept

Conclusion: the Law of Inertia

In the ideal case, nothing slows the ball on a horizontal second plane. It goes on with uniform velocity without any push from outside. From this, Galileo arrived at the law of inertia. A body stays at rest, or keeps moving uniformly, until an external force acts on it.

1.5Inertia

Galileo's experiments led to the law of inertia. When no external force acts, a body at rest and a body moving with constant velocity behave alike. The two states are equivalent.

Definition

Inertia is the natural (inherent) property of a body due to which it resists any change in its state, whether that state is rest or uniform motion along a straight line. The word inertia means resistance to change. The idea of inertia was first given by Galileo.

Types of Inertia

iInertia of Rest

Inertia of rest is the tendency of a body to stay at rest.

e.g. A passenger stands in a train that is at rest. When the train starts suddenly, the passenger falls backward. The feet move forward with the train. But inertia keeps the upper body at rest.

Train starts with a jerkPassenger falls backward

iiInertia of Motion

Inertia of motion is the tendency of a moving body to keep moving uniformly along a straight line.

e.g. A passenger stands in a moving bus. The driver brakes hard and the bus stops suddenly. The passenger falls forward. The feet stop along with the bus. But inertia keeps the upper body moving.

Bus stops with a jerkPassenger falls forward

iiiInertia of Direction

Inertia of direction is the inability of a body to change its direction of motion by itself.

e.g. An umbrella keeps us dry in the rain. The raindrops fall vertically downward. They cannot change their direction by themselves to wet us.

Raindrops falling verticallyDo not turn by themselves

1.6Newton's First Law of Motion

Sir Isaac Newton (1642–1727) made a careful, step-by-step study of motion. He took Galileo's ideas further and stated three laws of motion. We call them Newton's laws of motion.

Law

Newton's first law of motion states that a body at rest remains at rest, and a body in uniform motion keeps moving uniformly along a straight line, until an external force acts on it and changes this state.

Note

The First Law Defines Force Qualitatively

In the state of rest and in uniform motion, the acceleration is zero. So the first law can be written as:

\vec{F}_{\text{net}} = 0 \;\;\Rightarrow\;\; \vec{a} = 0

So a body has zero acceleration whenever the net external force on it is zero. Its acceleration can be non-zero only when a net external force acts. In this way, the first law defines force qualitatively.

Three Parts of the First Law

iBody at Rest

A body at rest stays at rest. It begins to move only when an external force acts on it.

e.g. A book lies at rest on a table. Two forces act on it. Its weight $w$ acts downward due to gravity. The normal force $R$ from the table acts upward. The book stays at rest, so the net force on it is zero. So $w = R$ .

Book at rest on a table:

w = R

iiBody in Motion

A moving body keeps its uniform speed and its straight path until an external force acts on it.

e.g. A rolling ball soon stops on rough ground because of friction. It rolls farther on a smooth floor. With zero friction, the ball would keep rolling forever.

Less friction means a longer roll

iiiChange of Direction

A body in uniform straight-line motion cannot turn by itself. Its direction changes only when a force acts normal (perpendicular) to its motion.

e.g. The direction of motion of the Moon changes all the time. The gravitational pull of the Earth on the Moon gives the force needed for this change.

Velocity

v

is along the orbit. The Earth's pull

F

points to the Earth

Concept

The First Law Defines Inertia

By the first law, a body by itself can neither start moving, nor stop, nor turn. This property is inertia. So the first law defines inertia, and it is also called the law of inertia.

Part 2

Momentum, Second and Third Laws

2.1Linear Momentum

Definition

The linear momentum of a body is the product of its mass $m$ and its velocity $\vec{v}$ . It measures the quantity of motion in the body. It is written as $\vec{p}$ .

Momentum

\vec{p} = m\vec{v}

Nature

Vector quantity.

Unit

$\text{kg m s}^{-1}$ (SI); $\text{g cm s}^{-1}$ (CGS).

Dimensional Formula

$[\text{MLT}^{-1}]$

2.2Variation of Momentum

Momentum changes with both the mass and the velocity of a body. The three cases below show how:

IEqual Masses (

v_1 > v_2

)

Two bodies have the same mass $m$ but different velocities $v_1$ and $v_2$ . Then:

p_1 = mv_1,\;\; p_2 = mv_2 \;\Rightarrow\; \dfrac{p_1}{p_2} = \dfrac{v_1}{v_2}

Equal massesMomentum ∝ velocity

p \propto v

(

m

constant)

IIEqual Velocities (

m_1 > m_2

)

A heavier body ( $m_1$ ) and a lighter body ( $m_2$ ) move with the same velocity, which is $v$ for both. Then:

p_1 = m_1 v,\;\; p_2 = m_2 v \;\Rightarrow\; \dfrac{p_1}{p_2} = \dfrac{m_1}{m_2}

Equal velocitiesMomentum ∝ mass

p \propto m

(

v

constant)

IIIEqual Momenta (

p

constant)

Two bodies have equal linear momenta:

p_1 = p_2 \;\Rightarrow\; m_1 v_1 = m_2 v_2 \;\Rightarrow\; \dfrac{v_2}{v_1} = \dfrac{m_1}{m_2}

Equal momentaVelocity ∝ 1 / mass

p = \text{constant}

v

falls as

m

rises

2.3Momentum in Everyday Life

Many common observations depend on momentum:

iTruck and Car

To bring a truck and a car to equal speeds in equal times, the truck needs a bigger force. Stopping the truck from that speed also takes a bigger opposing force. The reason is that the truck is heavier.

iiFast Bullet

A fast bullet from a gun easily goes through human tissue. At a moderate speed, the same bullet causes very little harm. So, for the same mass, a higher speed needs a bigger force to stop the body.

iiiLight and Heavy Stones

Drop a light stone and a heavy stone from the top of a building. Catching the light stone is easier. So mass plays a key part in deciding how a force changes a body's motion.

ivRotating Stone

A stone tied to a string is whirled in a horizontal circle at uniform speed. The magnitude of its momentum stays the same. But its direction keeps changing.

A force is needed to change this momentum vector. The hand gives this force through the string. A greater force is needed for a greater speed or a smaller radius. In both cases, the rate of change of momentum is greater.

The size of

\vec{p}

stays the same, but its direction keeps changing

Result

All these examples point to one quantity, momentum, which is the product of mass and velocity. Momentum decides how much force is required to bring a body to rest.

2.4Newton's Second Law of Motion

Law

Newton's second law of motion states that the external force on a body is directly proportional to how fast its momentum changes, that is, to its rate of change of momentum. The momentum changes in the direction of the applied force.

Setup — Given Information

A body of mass $m$ is acted on by an external force $\vec{F}$ for a small time $\Delta t$ .
Its velocity changes from $\vec{v}$ to $\vec{v} + \Delta\vec{v}$ .
So the change in momentum is $\Delta\vec{p} = m\,\Delta\vec{v}$ .

Derivation

By the second law, the force is proportional to the rate of change of momentum:
$\vec{F} \propto \dfrac{\Delta\vec{p}}{\Delta t} \;\Rightarrow\; \vec{F} = k\,\dfrac{\Delta\vec{p}}{\Delta t}$
Here $k$ is the constant of proportionality.
Let $\Delta t \to 0$ . The ratio becomes a derivative:
$\vec{F} = k\,\dfrac{d\vec{p}}{dt}$
For a body of constant mass $m$ , and since $\dfrac{d\vec{v}}{dt} = \vec{a}$ (acceleration):
$\vec{F} = k\,\dfrac{d(m\vec{v})}{dt} = k\,m\,\dfrac{d\vec{v}}{dt} = k\,m\vec{a}$
Choose the units so that $k = 1$ :
$\vec{F} = m\vec{a}$

Newton's Second Law

\vec{F} = m\vec{a}

$\therefore\; 1\ \text{unit force} = 1\ \text{unit mass} \times 1\ \text{unit acceleration}$

Note

Unit Force

One unit of force gives a body of unit mass an acceleration of one unit. Force has both size and direction, so it is a vector. Its SI unit is the newton (N).

Units of Force

Kind · System	Unit	Meaning
Absolute · SI	newton (N)	$1\ \text{N} = 1\ \text{kg} \times 1\ \text{m s}^{-2}$ . It gives an acceleration of $1\ \text{m s}^{-2}$ to a mass of $1\ \text{kg}$ .
Absolute · CGS	dyne	$1\ \text{dyne} = 1\ \text{g} \times 1\ \text{cm s}^{-2}$ . It gives an acceleration of $1\ \text{cm s}^{-2}$ to a mass of $1\ \text{g}$ .
Gravitational · SI	kg-wt (kgf)	$1\ \text{kg-wt} = 1\ \text{kgf} = 9.8\ \text{N}$ . It gives an acceleration of $9.8\ \text{m s}^{-2}$ to a mass of $1\ \text{kg}$ .
Gravitational · CGS	g-wt (gf)	$1\ \text{g-wt} = 1\ \text{gf} = 980\ \text{dyne}$ . It gives an acceleration of $980\ \text{cm s}^{-2}$ to a mass of $1\ \text{g}$ .

Relation between Newton and Dyne

1\ \text{N} = 1000\ \text{g} \times 100\ \text{cm s}^{-2} = 10^{5}\ \text{g cm s}^{-2} = 10^{5}\ \text{dyne}

Remember

One gravitational unit equals $g$ times the absolute unit of the same system. Gravitational units are also known as practical units.

e.g. A body of mass $10\ \text{kg}$ has a weight of $10\ \text{kg-wt}$ .

Second Law in Component Form

The force, momentum and acceleration vectors can be written in terms of their rectangular components:

\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k} \qquad \vec{p} = p_x\hat{i} + p_y\hat{j} + p_z\hat{k} \qquad \vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}

From $\vec{F} = \dfrac{d\vec{p}}{dt} = m\vec{a}$ , equate the components along the three axes:

Along x

F_x = \dfrac{dp_x}{dt} = m a_x

Along y

F_y = \dfrac{dp_y}{dt} = m a_y

Along z

F_z = \dfrac{dp_z}{dt} = m a_z

Note

These three equations are the second law in component form. Along each axis, force = rate of change of momentum = mass × acceleration, all taken along that axis.

2.5Impulse

Definition

Impulse measures the effect of a large force that acts for a very short time and produces a finite change in momentum. It is equal to the average force multiplied by the time for which the force acts. Its symbol is $\vec{I}$ .

Impulse

\vec{I} = \vec{F}_{\text{av}} \times t = \vec{F}_{\text{av}}\,\Delta t

Nature

Vector quantity. Its direction is the same as the force.

SI Unit

$\text{newton-second }(\text{N}\cdot\text{s})$

Dimensional Formula

$[\text{MLT}^{-1}]$

Impulse–Momentum Theorem

Setup — Given Information

Let a force $\vec{F}$ act on a body between the times $t_1$ and $t_2$ .
$\vec{F}_{\text{av}}$ is the average force during this interval.

Derivation

Impulse in a very small time $dt$ :
$d\vec{I} = \vec{F}\,dt$
Integrate over the whole interval:
$\vec{I} = \displaystyle\int_{t_1}^{t_2} \vec{F}\,dt$
For the average force:
$\vec{I} = \vec{F}_{\text{av}}\displaystyle\int_{t_1}^{t_2} dt = \vec{F}_{\text{av}}(t_2 - t_1) = \vec{F}_{\text{av}}\,\Delta t$
By Newton's second law:
$\vec{F} = \dfrac{d\vec{p}}{dt} \;\Rightarrow\; \vec{F}\,dt = d\vec{p}$
Integrate both sides:
$\displaystyle\int_{t_1}^{t_2}\vec{F}\,dt = \int_{p_1}^{p_2} d\vec{p} \;\Rightarrow\; \vec{I} = \vec{p}_2 - \vec{p}_1$

Impulse–Momentum Theorem

\vec{I} = \vec{p}_2 - \vec{p}_1 = \Delta\vec{p}

The impulse given by a force equals the total change in momentum it causes.

2.6Applications of Impulse

In daily life, the time of contact is often changed on purpose:

iBall Bouncing off a Wall

The wall pushes the ball for a very short time, yet it reverses the ball's momentum. Force and time are hard to measure alone. Their product $F\times t$ (the change in momentum) can be measured.

Contact lasts a very short timeLarge force

iiAthlete Stopping Slowly

After a fast race, an athlete should slow down gradually, not stop at once. Stopping then takes more time, so the force on the body is smaller.

Gradual stop (time ↑)Smaller force

iiiBuffers in Train Bogies

Railway bogies have buffers. During shunting, they make each impact last longer. A longer impact time means a smaller force, so there are no severe jerks.

Longer impact (buffers)Smaller force

ivCricketer Lowering Hands

A cricketer moves the hands down while catching a fast ball. The ball takes more time to stop, so a smaller retarding force gives the same change in momentum. The hands are not hurt.

Hands loweredLess force, no injury

Lowering the hands makes the stopping time longer, so the force on the hands is smaller

2.7Newton's Third Law of Motion

Press a coiled spring with your hand. The force of your hand compresses the spring. The compressed spring also pushes back on your hand. In the same way, the Earth pulls a stone downward. The stone also pulls the Earth with an equal and opposite force. We do not notice it because the Earth is very massive. So forces always occur in pairs. They come from the mutual interaction of two bodies.

Law

Newton's third law of motion states that when one body exerts a force on another (the action), the second body exerts a force on the first that is equal in size and opposite in direction (the reaction).

Forces in nature always come in pairs. Body $B$ acts on body $A$ with some force. Body $A$ then acts on body $B$ with a force of the same size in the opposite direction. In symbols:

\vec{F}_{AB} = -\,\vec{F}_{BA}

Here $\vec{F}_{AB}$ is the force on $A$ by $B$ , and $\vec{F}_{BA}$ is the force on $B$ by $A$ .

Action–reaction pair on bodies

A

and

B

Important Points About the Third Law

iAny Kind of Force

The third law is true for every type of force. The action–reaction pair can be mechanical, gravitational, electric or any other kind.

iiTwo Different Bodies

The action acts on one body and the reaction acts on the other body. They never act on the same body. If they did, they would add up to zero, and no body could ever accelerate.

iiiThey Never Cancel

Even though action and reaction are equal and opposite, they cannot cancel. Each one acts on a different body.

ivNo Action Without Reaction

A force cannot act alone, so there is no action without a reaction. In a game of tug of war, one side can pull hard only because the other team holds and pulls the opposite end. If that end were left free, the team could not apply any force.

2.8Examples of the Third Law

The third law explains many everyday events:

iWalking

When we walk, each foot pushes the ground backward. This push is the action. The ground pushes back on the foot with an equal and opposite reaction $R$ . Its vertical component $V = R\sin\theta$ supports our weight. Its horizontal component $H = R\cos\theta$ moves us forward.

Reaction

R

on the foot and its two components

iiRotatory Lawn Sprinkler

A rotatory lawn sprinkler is a good example of the third law. Water rushes out through the bent nozzles. In return, the water pushes each nozzle backward with an equal and opposite force. These backward pushes turn the sprinkler the other way, and it sprays water all around.

Top view of a rotating sprinkler

2.9Apparent Weight in a Lift

Think of a man of mass $m$ standing on a weighing machine inside a lift. He presses on the machine with his actual weight $mg$ . The machine pushes him back with a reaction $R$ , and its reading shows this value of $R$ .

Definition

Apparent weight is the reaction $R$ that the surface of contact exerts on a person. Let us see how $R$ compares with $mg$ in different cases.

iLift Accelerating Upward

The net force on the man acts upward:

R - mg = ma \;\Rightarrow\; R = m(g+a)

R = m(g+a)

Apparent weight increases ↑

R > mg

iiLift Accelerating Downward

The net force on the man acts downward:

mg - R = ma \;\Rightarrow\; R = m(g-a)

R = m(g-a)

Apparent weight decreases ↓

R < mg

iiiLift at Rest or with Uniform Velocity

The acceleration is zero, $a = 0$ . So:

R - mg = m\times 0 \;\Rightarrow\; R = mg

R = mg

Equal to the actual weight

R = mg

ivFree Fall (

a = g

)

Suppose the cable holding the lift snaps. The lift then falls freely, so $a = g$ . Then:

R = m(g-a) = m(g-g) = 0

R = 0

Weightlessness

2.10Conservation of Momentum

This law follows from the second and third laws of Newton.

Principle

The law of conservation of momentum states that the total momentum of a system never changes with time as long as no external force acts on it, that is, when $\vec{F}_{\text{ext}} = 0$ .

Note

In an isolated system, the particles push and pull each other with internal forces. These forces change the momentum of each particle. But they come in equal and opposite pairs (Newton's third law). So they cancel in pairs, and the total momentum does not change.

Proof for Two Particles

Consider two particles that exert forces on each other. We will prove that their total linear momentum does not change when no external force acts. The result holds whatever the strength of the forces between them.

Setup — Given Information

Particle 1 has mass $m_1$ and momentum $\vec{p}_1$ . Particle 2 has mass $m_2$ and momentum $\vec{p}_2$ .
Forces $\vec{F}_1$ and $\vec{F}_2$ act on them. In a small time $\Delta t$ , their momenta change by $\Delta\vec{p}_1$ and $\Delta\vec{p}_2$ .
Every force has two parts. The external part comes from bodies outside the system. The internal part is the force that the two particles exert on each other.

Derivation

The total momentum is the vector sum of the two momenta:
$\vec{p} = \vec{p}_1 + \vec{p}_2 \qquad \text{…(i)}$
In the small time $\Delta t$ , the total change in momentum is:
$\Delta\vec{p} = \Delta\vec{p}_1 + \Delta\vec{p}_2$
Divide by $\Delta t$ and take the limit $\Delta t \to 0$ :
$\dfrac{d\vec{p}}{dt} = \dfrac{d\vec{p}_1}{dt} + \dfrac{d\vec{p}_2}{dt}$
By the second law, each rate of change of momentum equals the force on that particle:
$\dfrac{d\vec{p}}{dt} = \vec{F}_1 + \vec{F}_2 \qquad \text{…(ii)}$
Split each force into its external and internal parts:
$\vec{F}_1 = \vec{F}_{1,\text{ext}} + \vec{F}_{1,\text{int}} \quad \text{and} \quad \vec{F}_2 = \vec{F}_{2,\text{ext}} + \vec{F}_{2,\text{int}}$
Put these in (ii) and regroup:
$\dfrac{d\vec{p}}{dt} = (\vec{F}_{1,\text{ext}} + \vec{F}_{2,\text{ext}}) + (\vec{F}_{1,\text{int}} + \vec{F}_{2,\text{int}}) \qquad \text{…(iii)}$
The two internal forces form an action–reaction pair, so by the third law they cancel:
$\vec{F}_{1,\text{int}} = -\vec{F}_{2,\text{int}} \;\Rightarrow\; \vec{F}_{1,\text{int}} + \vec{F}_{2,\text{int}} = 0$
So only the external forces remain:
$\dfrac{d\vec{p}}{dt} = \vec{F}_{1,\text{ext}} + \vec{F}_{2,\text{ext}} = \vec{F}_{\text{ext}}$
An isolated system has no external force on it, so $\vec{F}_{\text{ext}} = 0$ . Therefore:
$\dfrac{d\vec{p}}{dt} = 0 \;\Rightarrow\; \vec{p} = \text{constant}$
Hence the total momentum never changes with time. It stays conserved.

Law of Conservation of Momentum

\vec{F}_{\text{ext}} = 0 \;\;\Rightarrow\;\; \vec{p} = \text{constant}

Whenever $\vec{F}_{\text{ext}} = 0$ , the linear momentum of the system is conserved, whatever the internal forces may be.

Applications of Conservation of Momentum

iRecoil of a Gun

As the gun fires, it pushes the bullet forward with a force $\vec{F}$ . By the third law, the bullet pushes the gun back with $-\vec{F}$ for the same time $\Delta t$ . Gun and bullet were at rest at first, so their total momentum stays zero:

\vec{p}_b + \vec{p}_g = 0 \;\Rightarrow\; \vec{p}_g = -\,\vec{p}_b

So the gun recoils (moves backward). Its momentum is equal and opposite to that of the bullet.

iiExplosion of a Bomb

Before it explodes, a bomb at rest has zero momentum. Suppose it breaks into two equal pieces. The pieces move apart in exactly opposite directions at equal speeds. Their momenta cancel, so the total momentum is still zero.

The explosion forces act inside the bomb, so they are internal forces. No force from outside acts on the bomb.

Part 3

Equilibrium, Friction and Circular Motion

3.1Equilibrium of a Particle

Definition

Concurrent forces are forces that act at a single point or on a single particle. Such forces are in equilibrium when their resultant (vector sum) is zero.

Conditions for Equilibrium

iTwo Concurrent Forces

A pair of forces $\vec{F}_1$ and $\vec{F}_2$ balance each other only when they have the same size and opposite directions:

\vec{F}_1 = -\,\vec{F}_2

Equal (tick marks) and opposite forces

iiThree Concurrent Forces

For three forces, find the resultant of any two by the parallelogram law. The forces balance when this resultant is equal and opposite to the third force:

\vec{F}_1 + \vec{F}_2 = -\,\vec{F}_3 \;\Rightarrow\; \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 0

The resultant of

\vec{F}_1

and

\vec{F}_2

balances

\vec{F}_3

iii

n

Concurrent Forces

Many forces ( $n$ of them) keep a particle in equilibrium when, drawn head to tail in order, they form a closed polygon:

\vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots + \vec{F}_n = 0

So the sum of the components along each axis is zero:

\sum F_x = 0, \qquad \sum F_y = 0, \qquad \sum F_z = 0

Here, for example, $\sum F_x = F_{1x} + F_{2x} + \cdots + F_{nx}$ . $F_{1x}, F_{1y}, F_{1z}$ are the rectangular components of $\vec{F}_1$ along the $x$ , $y$ and $z$ directions.

3.2Lami's Theorem

Law

Lami's theorem states that when three concurrent forces $\vec{F}_1$ , $\vec{F}_2$ and $\vec{F}_3$ keep a body in equilibrium, each force divided by the sine of the angle between the other two forces gives the same value.

Lami's Theorem

\frac{F_1}{\sin\alpha} = \frac{F_2}{\sin\beta} = \frac{F_3}{\sin\gamma}

\alpha

Angle between

\vec{F}_2

and

\vec{F}_3

\beta

Angle between

\vec{F}_3

and

\vec{F}_1

\gamma

Angle between

\vec{F}_1

and

\vec{F}_2

Each force sits opposite to its own angle. For example, $\alpha$ lies between the other two forces, $\vec{F}_2$ and $\vec{F}_3$ .

Three concurrent forces at angles

\alpha

\beta

\gamma

Proof

Setup — Given Information

The forces $\vec{F}_1$ , $\vec{F}_2$ and $\vec{F}_3$ balance one another. So, drawn head to tail in the same order, they form the sides of a triangle $ABC$ .
The interior angles of the triangle are $\pi-\alpha$ , $\pi-\beta$ and $\pi-\gamma$ .

Forces as the sides of

\triangle ABC

Derivation

Apply the law of sines to $\triangle ABC$ :
$\dfrac{F_1}{\sin(\pi-\alpha)} = \dfrac{F_2}{\sin(\pi-\beta)} = \dfrac{F_3}{\sin(\pi-\gamma)}$
Since $\sin(\pi-\theta) = \sin\theta$ :
$\dfrac{F_1}{\sin\alpha} = \dfrac{F_2}{\sin\beta} = \dfrac{F_3}{\sin\gamma}$
This is Lami's theorem.

3.3Common Forces in Mechanics

The many forces in mechanics are of two broad types: non-contact forces and contact forces.

Non-Contact Force

Gravity pulls every object on the Earth. In mechanics, the gravitational force is the only force that acts without contact.

Contact Forces

All other common forces come from touching a solid or a fluid. Such contact forces between two bodies obey Newton's third law.

Parts of a Contact Force

Normal Reaction

N

The component of the contact force at right angles (normal) to the touching surfaces.

Friction

f

The component of the contact force along (parallel to) the touching surfaces.

A block with normal reaction

N

and friction

f

Forces Due to Fluids

Buoyant Force

A fluid pushes upward on a solid placed in it. This push equals the weight of the fluid displaced.

Viscous Force and Air Resistance

The viscous force opposes motion through a fluid. The viscous force and air resistance are also contact forces.

Upward buoyant force on an immersed body

3.4Tension and Spring Force

Strings and springs also exert contact forces. These are the tension in a string and the spring force:

Tension in a String

Hang a body of mass $m$ from a string. The string pulls it upward with a force called tension $T$ , which balances its weight:

T = mg

Here $g$ is the acceleration due to gravity and $T$ is the tension in the string.

Tension

T = mg

in the string

Spring Force

When an external force squeezes or stretches a spring, the spring pushes or pulls back with a restoring force. For small displacements, this force is proportional to the displacement:

F = kx

Here $k$ is the spring constant and $x$ is the displacement. The restoring force acts opposite to the displacement.

Restoring force

F = kx

, opposite to the displacement

Note

Among the forces of mechanics, only gravity acts without contact.
For a heavy (not light) rope or string, include its weight as well.

3.5Friction

Definition

Friction is the force that comes into play along the surface of contact when one body slides, or tends to slide, over another. It opposes their relative motion.

Push a wooden block gently along a horizontal table. It moves a little and then comes to rest. Its velocity has changed. Newton's second law tells us that some retarding force acts on it. This force is friction.

Friction acts along (tangential to) the touching surfaces. It always points against the relative motion.

Friction points against the motion

Types of Friction

1Internal Friction

It acts between two layers of a liquid that move relative to each other. Its other name is the viscosity of the liquid.

2External (Contact) Friction

It acts between two touching bodies that move, or are about to move, over each other. Its three kinds are static, limiting and kinetic friction.

(i) Static Friction

Push a block lying on a table with a small force $F$ . The block stays where it is, because a friction force $f_s$ appears and balances the push.

Static friction ( $f_s$ ) is the friction on a body that has not yet started to slide over another body. It opposes the impending motion, that is, the motion that would start if there were no friction.

Static friction adjusts itself to match the push, so it is a self-adjusting force. If nothing pushes the block, static friction is zero.

Static friction

f_s

balances the applied force

F

(ii) Limiting Friction

Slowly increase the push. Static friction $f_s$ grows with it, and the block stays at rest.

At one particular push, the block is just about to slide. Static friction has now reached its largest value. Limiting friction $f_{s(\max)}$ is this maximum value of static friction.

f_s \le f_{s(\max)}

Limiting friction on a body about to move

3.6Laws of Limiting Friction

aNature of Surfaces

Limiting friction is decided by the nature (kind) of the surfaces in contact and by how rough they are.

bDirection

It acts along the contact surfaces (tangential), against the direction in which the body is about to move.

cIndependent of Area

Changing the area of contact does not change it, provided the normal reaction is unchanged.

dProportional to Normal Reaction

It increases in direct proportion to the normal reaction $R$ . Hence:

f_{s(\max)} \propto R \;\Rightarrow\; f_{s(\max)} = \mu_s R

Coefficient of Static Friction

Definition

The coefficient of static friction ( $\mu_s$ ) of two surfaces is their limiting friction divided by the normal reaction. It is the constant in law (d).

Coefficient of Static Friction

\mu_s = \frac{f_{s(\max)}}{R} = \frac{\text{Limiting friction}}{\text{Normal reaction}}

For a block on a horizontal surface, $R = w = mg$ ( $m$ = mass of the block, $g$ = acceleration due to gravity).

3.7Angle of Friction

Definition

The angle between the normal reaction and the resultant of the normal reaction and the limiting friction is called the angle of friction ( $\theta$ ).

Setup — Given Information

$OA$ = normal reaction $R$ , $OB$ = limiting friction $f_{s(\max)}$ and $OC$ = their resultant.
$\theta$ is the angle between the normal reaction $R$ and the resultant $OC$ .

OC

is the resultant of

R

and

f_{s(\max)}

. It makes angle

\theta

with

R

Derivation

In $\triangle AOC$ :
$\tan\theta = \dfrac{AC}{OA} = \dfrac{OB}{OA} = \dfrac{f_{s(\max)}}{R}$
But $\dfrac{f_{s(\max)}}{R} = \mu_s$ . So:
$\tan\theta = \mu_s \;\Rightarrow\; \theta = \tan^{-1}(\mu_s)$

Angle of Friction

\tan\theta = \mu_s \;\;\Rightarrow\;\; \theta = \tan^{-1}(\mu_s)

So the tangent of the angle of friction gives the coefficient of static friction.

3.8Angle of Repose

Definition

The angle of repose ( $\alpha$ ) is the smallest angle with the horizontal to which an inclined plane must be tilted so that a body resting on it just starts sliding down. It depends on the materials and nature of the two surfaces.

Setup — Forces on the Body

Put a body of mass $m$ on an inclined plane. Tilt the plane until the body is just about to slide. That angle is $\alpha$ .
Its weight $mg$ acts vertically downward.

Body on an incline at the angle of repose

\alpha

Derivation

Limiting friction acts up the incline. It balances the part of the weight along the incline:
$f_{s(\max)} = mg\sin\alpha \qquad \text{…(i)}$
The normal reaction balances the part of the weight perpendicular to the incline:
$R = mg\cos\alpha \qquad \text{…(ii)}$
Divide (i) by (ii):
$\dfrac{f_{s(\max)}}{R} = \dfrac{mg\sin\alpha}{mg\cos\alpha} \;\Rightarrow\; \mu_s = \tan\alpha$

Angle of Repose

\mu_s = \tan\alpha \qquad\text{and}\qquad \theta = \alpha

The tangent of the angle of repose gives the coefficient of static friction. It also shows that the angle of friction and the angle of repose are equal.

3.9Kinetic Friction

Definition

Kinetic friction (also called dynamic friction) is the friction between two surfaces that are already sliding over each other. It acts against their relative motion.

It takes over once the applied force becomes slightly more than the limiting friction. Its value is slightly smaller than the limiting friction.

Laws of Kinetic Friction

aConstant Value

It acts against the relative motion. For a given pair of surfaces, its value is fixed (constant). This value depends on the nature of the surfaces.

bIndependent of Area

A bigger or smaller area of contact makes no difference, if the normal reaction is the same.

cIndependent of Velocity

Its value is the same at any speed of sliding. It does not depend on velocity.

dProportional to Normal Reaction

It grows in direct proportion to the normal reaction $R$ . Hence:

f_k \propto R \;\Rightarrow\; f_k = \mu_k R

Coefficient of Kinetic Friction

\mu_k = \frac{f_k}{R} = \frac{\text{Kinetic friction}}{\text{Normal reaction}}

Divide the kinetic friction by the normal reaction to get $\mu_k$ .

Remember

Kinetic friction is smaller than the limiting friction, so $f_k < f_{s(\max)}$ . Therefore $\mu_k < \mu_s$ . For the same two surfaces, $\mu_k$ is always smaller than $\mu_s$ .

3.10Types of Kinetic Friction

There are two kinds of kinetic friction: sliding and rolling.

iSliding Friction

Sliding friction is the friction on a body that slides along another surface.

e.g. When a flat block is dragged across a flat table top, sliding friction opposes it.

iiRolling Friction

Rolling friction is the friction on a body that rolls along another surface.

e.g. Rolling wheels, circular discs, rings and spheres all feel rolling friction.

Laws of Sliding Friction

It acts against the applied force. Its value is constant and is fixed by the kind of surfaces that slide on each other.
It grows in direct proportion to the normal reaction $R$ .
The area of contact does not affect it, provided the normal reaction stays unchanged.
Its value is not affected by velocity.

Why Rolling Friction Is Small

Concept

When a wheel rolls without slipping, its rim and the road never rub against each other. At each instant, the point of the wheel that touches the road is at rest relative to the road. So, ideally, no sliding (static) friction acts on the wheel.

Only rolling friction remains to be overcome, and it is far smaller than sliding friction. The wheel replaces sliding with rolling, so it cuts down friction greatly. For this reason, the wheel is counted among the greatest inventions.

Cause of Rolling Friction

A rolling wheel rests on a tiny area of the road, so the pressure there is large. The road gets pressed into a depression under the wheel, and a small bump rises just ahead of it.

The depression, the bump and the adhesive force between the surfaces together hold the wheel back. This retarding force is rolling friction.

Depression and bump slow the wheel

3.11Methods of Reducing Friction

Friction can be reduced in these ways:

iLubrication

Lubricants like oil and grease fill the tiny hollows (irregularities) of the surfaces and make them smoother. This lowers the kinetic friction in a machine.

iiPolishing

A polished surface is smoother, so there is less friction.

iiiBall Bearing

In a ball bearing, hard steel balls sit between two co-axial cylinders, $P$ and $R$ .

As the wheel turns, the balls roll. The wheel no longer slides on the axle. It rides on the rolling balls instead. Rolling friction takes the place of sliding friction, so less power is wasted. The power dissipation falls.

Parts of a ball bearing

ivAir Cushion

A thin layer (cushion) of air is kept between two solid surfaces that move over each other. The air stops them from touching and rubbing.

vStreamlining

Streamlining the shape of a body that moves through air greatly reduces the friction due to air.

3.12Dynamics of Circular Motion

A body moves in a circle of radius $r$ with uniform speed $v$ . Its acceleration is $\dfrac{v^2}{r}$ , directed towards the centre. By Newton's second law, a force must produce this acceleration. This force is the centripetal force.

Definition

Centripetal force is the force that keeps a body of mass $m$ moving on a circular path. It always points towards the centre of the circle. The quantity $\dfrac{v^2}{r}$ is called the centripetal (radial) acceleration.

Centripetal Force

F_c = \frac{mv^2}{r} = mr\omega^2

since $v = \omega r$ , where $m$ = mass, $v$ = speed, $r$ = radius, $\omega$ = angular velocity

Sources of Centripetal Force

Tension

When a stone on a string is whirled in a circle, the tension in the string supplies the centripetal force.

Gravitation

A planet goes round the Sun because the Sun's gravitational pull supplies the centripetal force.

Friction

When a car turns on a flat (level) road, friction between its tyres and the road supplies the centripetal force.

3.13Car on a Level Circular Road

A car of weight $mg$ moves with speed $v$ round a level curve of radius $r$ . It tends to skid outward. Friction from the road on the tyres supplies the centripetal force that it needs.

(i) Weight

mg

Vertically downward

(ii) Normal reaction

R

Vertically upward

(iii) Friction

F

Towards the centre

Car on a level circular road (rear view)

Setup — Given Information

A car of mass $m$ moves with speed $v$ on a level circular road of radius $r$ .
The car does not accelerate up or down (in the vertical direction).

Derivation

The vertical forces balance:
$R = mg$
Static friction supplies the centripetal force. It can be at most the limiting friction $F$ of the road. So:
$\dfrac{mv^2}{r} \le F$
The limiting (maximum) friction is:
$F = \mu_s R = \mu_s mg$
Hence:
$\dfrac{mv^2}{r} \le \mu_s mg \;\Rightarrow\; v \le \sqrt{\mu_s r g}$

Maximum Safe Speed on a Level Road

v_{\max} = \sqrt{\mu_s\, r\, g}

This is the highest speed at which a vehicle can go round a level curve without skidding ( $\mu_s$ = coefficient of static friction).

3.14Car on a Banked Road

Definition

Banking of a road is the raising of the outer edge of a curved road above its inner edge. The angle through which the road is raised is called the angle of banking ( $\theta$ ).

Banking reduces the wear and tear of the tyres. The road slopes up towards its outer edge. Then the horizontal component of the normal reaction gives the needed centripetal force.

Forces on the car: weight $mg$ (downward), normal reaction $R$ (perpendicular to the road) and friction $F$ . Friction points down the slope, since the car tends to slide outward. Their components are:

$R\cos\theta$ (upward) and $R\sin\theta$ (horizontal, towards the centre)
$F\cos\theta$ (horizontal, towards the centre) and $F\sin\theta$ (downward)

Car on a road banked at angle

\theta

Setup — Given Information

A car of mass $m$ moves with speed $v$ on a curved road of radius $r$ , banked at angle $\theta$ .
The car tends to slide outward, so friction on it points down the slope. At the top speed, friction is limiting: $F = \mu_s R$ .

Derivation

The car does not move vertically. So the upward and downward components are equal:
$R\cos\theta = mg + F\sin\theta \qquad \text{…(i)}$
The horizontal components together give the centripetal force:
$R\sin\theta + F\cos\theta = \dfrac{mv^2}{r} \qquad \text{…(ii)}$
Put $F = \mu_s R$ in (i):
$R\cos\theta = mg + \mu_s R\sin\theta \;\Rightarrow\; R(\cos\theta - \mu_s\sin\theta) = mg$
$\therefore\; R = \dfrac{mg}{\cos\theta - \mu_s\sin\theta} \qquad \text{…(iii)}$
Put $F = \mu_s R$ in (ii):
$R\sin\theta + \mu_s R\cos\theta = \dfrac{mv^2}{r} \;\Rightarrow\; R(\sin\theta + \mu_s\cos\theta) = \dfrac{mv^2}{r} \qquad \text{…(iv)}$
Put $R$ from (iii) into (iv):
$\dfrac{mg(\sin\theta + \mu_s\cos\theta)}{\cos\theta - \mu_s\sin\theta} = \dfrac{mv^2}{r}$
Solve for $v^2$ . The mass $m$ cancels:
$v^2 = \dfrac{rg(\sin\theta + \mu_s\cos\theta)}{\cos\theta - \mu_s\sin\theta}$
Divide the numerator and the denominator by $\cos\theta$ :
$v^2 = \dfrac{rg(\tan\theta + \mu_s)}{1 - \mu_s\tan\theta}$
This is the maximum safe speed. If the car goes faster than this, it skids outward (up the slope).

Maximum Speed on a Banked Road

v_{\max} = \sqrt{\frac{rg(\mu_s + \tan\theta)}{1 - \mu_s\tan\theta}}

Special Case — Frictionless Banked Road

Optimum Speed (

\mu_s = 0

)

With no friction ( $\mu_s = 0$ ), the car does not slide at this speed, even on a smooth road:

v = \sqrt{rg\tan\theta}

Angle of Banking

For the least wear and tear of the tyres, the angle of banking should be:

\theta = \tan^{-1}\!\left(\dfrac{v^2}{rg}\right)

Note

Suppose the road is smooth. If the speed is less than $\sqrt{rg\tan\theta}$ , the car tends to slide down the slope, and the radius decreases. If the speed is more, the car tends to slide up the slope.

3.15Solving Problems in Mechanics

Most mechanics problems involve an assembly of bodies, not a single body. The bodies push and pull on one another, and gravity acts on each of them.

Definition

A free body diagram (FBD) is a drawing of one chosen system on its own, with every force on it marked. These forces come from the rest of the assembly and from any other outside agencies.

Problem Solving Strategy

Steps to Solve a Mechanics Problem

Draw the assembly. Make a rough sketch of the whole assembly. Show its parts, links, supports, etc.
Choose a system. Pick one convenient part of the assembly and treat it as the system.
Draw the free body diagram. Draw the chosen system by itself. Mark every force on it, from the rest of the assembly and from outside agencies. This drawing is the free body diagram.
Mark the known forces. On the FBD, show the magnitude and direction of each known (given) force. The remaining forces are unknowns. Use the laws of motion to find them.
Repeat if needed. Do the same for other parts of the system if required. Solving the equations of all the parts together gives the required results.

End of Chapter 4 · notesshala.in