NCERT Class 11 Physics Chapter 5 Notes Laws of Motion - Download PDF

NCERT Notes for Class 11 Chapter 4: Download PDF

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NCERT Notes for Class 11 Chapter 4

The NCERT Notes for Class 11 Chapter Laws of Motion explains all the topics in simple language, suitable for learning. They will help enhance your knowledge and prepare you better for exams.

Aristotle’s Fallacy

Aristotle, a Greek philosopher, posed a view that for a body to be in its motion and keep moving, something external is required. He said that motion had to be caused by a force. To explain why an arrow kept flying after the bow string was no longer pushing on it, he said that the air rushed around behind the arrow and pushed it forward.

Aristotle gave his law of motion which may be phrased as "An external force is required to keep a body in motion." But this is wrong, because an arrow shot in a vacuum chamber does not instantly drop to the floor as soon as it leaves the bow. Most of the Aristotlion ideas on motion are now known to be wrong and need not concern as the flaw in Aristotle's argument can be understood by taking an example that a ball rolling on a floor comes to rest after a while due to the external force of friction on the ball by the floor, which opposes its motion. Now, to keep that ball moving on the floor, we require to give some external force on it to move. But when the ball is moving in uniform motion, there is no net external force acting in its direction of motion. This is due to the fact that the force given by us to move a ball cancels with the force of friction by the floor. In simple language, we may say that to keep a ball in uniform motion, we would not be required to apply any external force in the absence of friction. This is the reason that why we need external sources to overcome opposing forces like friction (solids) and viscous drag (for fluids) which are always present in natural world.

The Law of Inertia

As we have seen that, before Galileo, it was thought that a force is required to keep a body moving with uniform velocity. Galileo observed that speed of a ball increases, as it rolls down an inclined plane.

The speed of that ball decreases as it is rolled up that inclined plane.

So, what should happen if it is rolled on a horizontal plane? As this case is just in between the situations discussed, the result must also be in between, i.e., speed of ball should remain constant.

Explanation :
When you move down speed increases.
When you move up speed decreases.
Moving horizontally (i.e., neither up nor down) speed should remain constant (neither increase nor decrease.

Newton’s First Law of Motion (Law of Inertia)

Statement:

If no external force acts on a body, it stays at rest or keeps moving in a straight line with the same speed.

This is called the Law of Inertia.

It explains:

• No force → no change in motion.
• If something changes (starts/stops/turns), a force must be acting.

Types of Inertia:

1. Inertia of Rest: An Object stays still unless something moves it.
2. Inertia of Motion: A Moving object keeps going unless stopped.
3. Inertia of Direction: The Object keeps moving in the same direction unless forced to turn.

Balanced and Unbalanced Forces

Balanced Forces:

• Two equal forces pulling/pushing in opposite directions
• Motion doesn’t change
  Example: In a tug of war, both teams pull equally → The rope doesn’t move.

Unbalanced Forces:

• Forces are not equal
• They cause objects to move, stop, or change shape
  Example: In tug of war, if one team pulls harder → the rope moves towards them.

Linear Momentum

• Linear momentum is a vector quantity (has both magnitude and direction).

• It’s the product of an object’s mass (m) and velocity (v).
  $\vec{p}=m \vec{v}$

• SI Unit: kg·m/s

• CGS Unit: g·cm/s

• Dimension: [MLT^-1]

Note: If two objects have the same momentum but different masses, the lighter object will have a higher velocity.

Newton’s Second Law of Motion

The rate of change of momentum of a body is directly proportional to the applied external force, and this change happens in the direction of the force.

In Simple Words:

Force is what causes momentum to change.
More force → faster change in motion.

Mathematically:

Let a body of mass m move with velocity v.
Its momentum: p = m × v

If a force (F) is applied, then:

$\vec{F}=\frac{d p}{d t}=\frac{d}{d t}(m \vec{v})=m \frac{d \vec{v}}{d t}=m \vec{a} \quad\left(\right.$ from $\left.a=\frac{d \vec{v}}{d t}\right)$

This formula tells us:
Force = mass × acceleration. It’s one of the most important equations in Physics!

Impulse

Impulse is the effect of a force applied to an object for a short time, causing a sudden change in its motion. It helps explain how quickly an object speeds up, slows down, or changes direction when hit or pushed. For example, when you kick a football or hit a cricket ball with a bat, you give it an impulse that changes its speed and direction instantly. Even airbags in cars use the concept of impulse—they increase the time of impact during a crash to reduce the force on passengers. In simple words, impulse shows how a quick force can create a big change in how something moves.

$\vec{I}=\int_{t_1}^{t_2} \vec{F} d t$

• Impulse is a vector quantity, so it has both magnitude and direction.
• Dimension of impulse: [MLT^-1]

• SI unit- Newton-second or Kg-m-s^-1 and CGS unit- Dyne-second or gm-cm-s^-1

Newton’s Third Law of Motion

Newton's Third Law of Motion says that for every action, there is an equal and opposite reaction. This means that if one object applies a force on another, the second object pushes back with the same amount of force but in the opposite direction. If we say $\vec{F}_{AB}$ is the force that object B applies on object A (action), and $\vec{F}_{BA}$ is the force that object A applies on object B (reaction), then these forces are equal in size but opposite in direction, written as:

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

This law helps explain things like how a swimmer pushes water backward and moves forward, or how a rocket launches by pushing gas downward and lifting upward.

Conservation of Momentum

According to the Law of Conservation of Linear Momentum, when no external forces act on an isolated system, its total linear momentum remains constant. In other words, the total momentum of a closed system of interacting objects remains constant over time, assuming no external forces influence it.

According to this law, $\vec{F}=\frac{d \vec{p}}{d t}$

In the absence of external forces, If F equals 0, then p is constant.

i.e, $P_{\text {system }}=P_1+P_2+P_3+P_4+\ldots \ldots \ldots . .=$ constant

Applications of Law of Conservation of Linear Momentum

(i) Recoil velocity of a Gun: When a bullet is fired from a gun, the gun recoils or gives a kick in backward direction.

Let $\quad \vec{v}_1 \rightarrow$ velocity of bullet after firing
$\vec{v}_2 \rightarrow$ velocity of gun after firing
$m_1 \rightarrow$ mass of the bullet
$m_2 \rightarrow$ mass of the gun
According to the law of conservation of linear momentum,
Linear momentum before firing = Linear momentum after firing

$
0=m_1 \vec{v}_1+m_2 \vec{v}_2
$

or

$
\begin{aligned}
& m_2 \vec{v}_2=-m_1 \vec{v}_1 \\
& \vec{v}_2=-\frac{m_1}{m_2} \vec{v}_1
\end{aligned}
$

where $\vec{v}_2 \rightarrow$ recoil velocity of a gun, negative sign shows that the gun moves backward.
(ii)Rocket propulsion: Before firing the rocket, total momentum of the system is zero because the rocket is in the state of rest. When it is fired, chemical fuels inside the rocket are burnt and the hot gases are passed through nozzle with greater speed. According to law of conservation of momentum, the total momentum after firing must be equal to zero. As the hot gases gain momentum to the rear on leaving the rocket, the rocket acquires equal momentum in the upward (i.e.) opposite direction.

Equilibrium of a Particle

In mechanics, a body is in equilibrium when it is at rest or moving with constant velocity in an inertial frame of reference. A hanging lamp, a suspension bridge, an aeroplane flying straight and level at a constant speed - all are examples of equilibrium situations.

The essential physical principle is Newton's first law: When a particle is at rest or is moving with constant velocity in an inertial frame of reference, the net force acting on it, i.e., the vector sum of all the forces acting on it, must be zero.

$
\Sigma \vec{F}=0 \quad \text { (Particle in equilibrium vector form) }
$

We most often use this equation in component form :

$
\Sigma \vec{F}_x=0 ; \quad \Sigma \vec{F}_y=0
$

and $\Sigma \vec{F}_z=0$
Equilibrium under concurrent forces (i.e., those forces which act on same particle at same time) may be seen as

In Fig. (a),

$
\vec{F}_1+\vec{F}_2=0
$

In Fig. (b)-(i) and (b)-(ii),

$
\vec{F}_1+\vec{F}_2+\vec{F}_3=0
$

Common Forces in Mechanics

In mechanics, we come across many types of forces: Some of common forces are given below:
1. Weight : The weight of a body is the gravitational force with which the earth pulls the body.
2. Spring Force : When a spring is extended, it pulls the body attached to its ends towards its centre and if compressed it pushes the body away from its centre.
If the extension or compression is not too large, the force exerted by the spring is proportional to the change in its length from its natural length.
i.e., $F \propto x$, where $x$ is the elongation or compression
or $\quad \vec{F}=-k \vec{x}$
where $k$ is a positive constant called spring constant or force constant. Negative sign indicates that the force is opposite to the displacement from the unstretched or uncompressed state.

3. Tension Force: The pulling force transmitted through a string, rope, or wire

4. Normal Reaction: Normal reaction is a contact force between two surfaces in contact, which is always perpendicular to the surfaces in contact. The following diagrams show normal reaction between two surfaces

Block pushes ground downward with force $R$ and ground pushes the block back with force $R$.

Here $m_1$ pushes $m_2$ towards left by force $R$ and $m_2$ pushes $m_1$ towards right by force $R$

5. Friction

Friction is a force that occurs at the point of contact between two surfaces and works to oppose relative or approaching motion between them.

Friction can be categorized into three main types:

• Static Friction: Acts on stationary objects, preventing them from initiating motion. The force of static friction increases with the applied force until motion is initiated.
  $f_{\mathrm{s}} \leq \mu_{\mathrm{s}} N$
• The maximum static frictional force $\left(\mathrm{f}_{\mathrm{s}}\right)_{\max }=\mu_{\mathrm{s}} N$
• Kinetic friction: Kinetic friction is the force that arises between two surfaces when they are moving relative to one another.

  The kinetic frictional force ($f_{\mathrm{k}}$) is proportional to the normal force ($N$) and can be expressed as $f_{\mathrm{k}}=\mu_{\mathrm{k}} N$

• Rolling friction: Rolling friction is the force that opposes the motion of an object as it rolls across a surface. It occurs when there is relative motion between the surface and the part of the object that makes contact with it. it is similar to kinetic friction.

Factors Affecting Friction :

1. Nature of the medium of contact between two bodies :
   Roughness: It increases friction between the two surfaces.
   Smoothness: It reduces the friction.

2. Normal reaction: The force of friction also depends on the normal reaction also. As the normal reaction increases, the interlocking between two surfaces in contact increases as they press harder against each other and hence, friction increases.
3. Area of contact : Force of friction is independent of area of contact.

Angle of Friction: The angle $(\lambda)$ between the resultant maximum possible contact force $(F)$ and the normal force $(N)$ is called the angle of friction.

From the figure, clearly $\tan \lambda=\frac{f}{N}=\frac{\mu_s N}{N}$
i.e., $\tan \lambda=\mu_{\mathrm{s}}$
or $\lambda=\tan ^{-1}\left(\mu_s\right)$

Angle of repose $(\alpha)$ : It is the maximum angle of inclination $(\alpha)$ of a rough inclined plane with the horizontal such that the block kept on it remains at rest.

At angle of repose,

$
\begin{aligned}
& \text { Driving force = Limiting friction } \\
& m g \sin \alpha=\mu_s N \\
& m g \sin \alpha=\mu_s m g \cos \alpha \\
& \tan \alpha=\mu_s
\end{aligned}
$
Again, we have

$
\begin{aligned}
& \tan \theta=\mu_s(\theta=\text { angle of friction }) \\
\therefore \quad & \text { Angle of friction }=\text { Angle of repose }
\end{aligned}
$

Circular Motion

• Angular Displacement(θ): The angular displacement of an object moving around a circular path is defined as the angle traced out by the radius vector at the center of the circular path over time. This is a vector quantity.θ= s/r, where s is the arc length and r is the radius.
• Angular Velocity (ω): The angular velocity of an object in circular motion is defined as the rate at which its angular displacement changes over time.
  ω= Δθ/Δt, where Δθ is the change in angular displacement, and Δt is the change in time.
• Angular Acceleration (α): The angular acceleration of an object in circular motion is defined as the rate at which its angular velocity changes over time. α= Δt/Δω, where Δω is the change in angular velocity, and Δt is the change in time
• Uniform Circular Motion: Uniform circular motion occurs when a point object moves along a circular path at a constant speed.

  In this type of motion, the speed remains constant while the direction changes continuously, resulting in circular motion. a_c=v²/r, where r is the radius

Solving Problems in Mechanics

(i) Draw a diagram showing schematically the various parts of the assembly of bodies, the links, supports, etc.
(ii) Choose a convenient part of the assembly as one system.
(iii) Draw a separate diagram which shows this system and all the forces on the system by the remaining part of the assembly. Include also the forces on the system by other agencies. Do not include the forces on the environment by the system. A diagram of this type is known as 'a free-body diagram'. (Note this does not imply that the system under consideration is without a net force).
(iv) In a free-body diagram, include information about forces (their magnitudes and directions) that are either given or you are sure of (e.g., the direction of tension in a string along its length). The rest should be treated as unknowns to be determined using laws of motion.
(v) If necessary, follow the same procedure for another choice of the system. In doing so, employ Newton's third law. That is, if in the free-body diagram of $A$, the force on $A$ due to $B$ is shown as $\mathbf{F}$, then in the free-body diagram of $B$, the force on $B$ due to $A$ should be shown as $-\mathbf{F}$.

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