Dynamics  Physics:

Dynamics  physics is the branch of classical mechanics that deals with the study of forces and their effect on the motion of objects. It focuses on understanding how and why objects move and what causes the changes in their motion.

While kinematics describes the motion of objects (without considering the forces causing the motion), dynamics explains the causes of that motion.

Force:

Force is an interaction that causes an object to change its velocity (acceleration). The primary goal of studying force in dynamics is to understand how forces influence the motion of objects.

Linear Momentum

Linear momentum (often referred to simply as momentum) of an object is the product of its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.
Formula:

The SI unit of linear momentum is kg·m/s.

Impulse:

Impulse can be defined Large force acts on a body for a short period of time is called impulse. Impulse of a force is defined as the product of the average force and the time for which the force acts. i.e.

$$\text{Impulse (I)}=\text{Average force}\times \text{time}(t)$$ 
I=F_av×t

Impulse is given by the area enclosed by the time-force graph: i.e. Impulse = Area of ABCD

• Example of Impulse are kicking a ball, collision of two bodies.
• Impulse gives the measurement of net effect of force.
• Impulse is mathematically equal to the change in momentum.
• Impulse is a vector quantity.
• Its SI unit is Newton-second (Ns).
• Its dimensional formula is [M¹L¹T⁻¹]

Relation between Linear Momentum and Impulse

The relationship between impulse and momentum is given by:

Impulse=Change in Momentum

This is sometimes called the Impulse-Momentum Theorem, which states that the impulse applied to an object is equal to the change in its momentum.

Newton’s Laws of Motion First Law:

Newton’s  first Law of Motion states that, “Everybody in the universe continues its state of rest or motion with uniform velocity unless an external force is applied on it.”For example,

• A book placed on the table remains in the state of rest unless an external force is applied to it.
• If a football is kicked on the ground, it moves with uniform velocity. But after some time, it comes to rest.It is because of the opposing frictional force
• It is also called the Law of Inertia.

Newton’s Second Law of Motion:

It states that, “The rate of change of momentum of a body is directly proportional to the force applied on it.”
Consider a body of mass ‘m’ moving with velocity ‘u’ on a linear path along straight line. After time ‘t’, due to the application of force ‘f’, its final velocity becomes v.
Now
Initial momentum of the body = mu
Final momentum of the body = mv
Change of momentum of the body = mv – mu
Rate of change of momentum of the body = (mv – mu)/t

from Newton’s Second Law

F ∝ (mv – mu)/t
                                                                    F = k * m(v – u)/t [where k is proportionality constant]
                         f = kma ……….1) α [∴ a = v – u/t]

Where a is the acceleration of the body.

If m = 1kg, a = 1 m/s², then F = 1N
From equation (i), k = 1
Again from equation F = m.a
Newton’s second law of motion measures force.

Newton’s Third Law of Motion:

It states that, “To every action, there is always an equal and opposite reaction.”
For examples:
• A block resting on a table.
• Pushing a block.
• Swimming
• Flight of rocket
• Walking

Second Law is the Real Law of Motion
For the First Law
We know from Newton’s second law of motion, F = ma
If F = 0 then ma = 0 ⇒ m ≠ 0, so, a = 0
i.e. v – u = 0
t
⇒ v = u
∴ v = constant

For the Third Law
Consider two bodies A and B moving along the same straight line. As a result of collision, their momentum will change.
Let, ‘t’ be the time of impact, then change in momentum of A is,
ΔPₐ = Fₐ × t
Similarly change in linear momentum of B is
ΔPᵦ = Fᵦ × t
So, total change in linear momentum of A and B is
ΔPₐ + ΔPᵦ

ΔP = Fₐ × t + Fᵦ × t
   = t (Fₐ + Fᵦ) — (i)
If no external force acts on the system, then ΔP = 0.
⇒ 0 = (Fₐ + Fᵦ) t
 Fₐ + Fᵦ = 0
∴ Fₐ = -Fᵦ which is Newton’s third law of motion.

Applications of Newton’s Laws of Motion
1.Horse cart system

Let,
m = mass of a horse
m’ = mass of a cart
g = acceleration due to gravity
f = force exerted by horse on the ground
R = normal reaction given by the ground to the horse.
T = Tension on the rope.
V = Vertical component of R.
H = Horizontal component of R.
F = Frictional force between the tyres of cart and road.
a = Acceleration of horse cart system
R’ = Normal reaction given by road to the cart.
m’g = Weight of the cart
mg = Weight of the horse.

For the balanced motion of horse cart system, the equation for horse can be written as
H – T = ma — (i)

For the cart
T – F = m’a — (ii)

Adding equations (i) and (ii),
H – F = (m + m’) a

From (i)
a = (H – F) / (m + m’)
T = (m’H + mF) / (m + m’)

2.Mass – Pulley System:
Let us consider a mass pulley system where two unequal masses m₁ and m₂ are connected to the two ends of an inextensible string passing over a smooth frictionless pulley as shown in the figure. The lighter mass m₁ moves up and the heavier mass m₂ moves down with the same acceleration a. Let T be the tension in the rope.

For mass m₁
T – m₁g = m₁a —- (i)

For mass m₂
m₂g – T = m₂a —- (ii)

Adding equations (i) and (ii)

m₂g – m₁g = m₁a + m₂a
(m₂ – m₁)g = (m₁ + m₂)a
a = (m₂ – m₁)g / (m₁ + m₂)

When two blocks of masses m₁ and m₂ are tied to end of a string passing over a frictionless pulley.

For mass m₁,m₁a = T ….(i)

For mass m₂,
m₂a = m₂g – T ….(ii)

Adding (i) and (ii),
a(m₁ + m₂) = m₂g

When two blocks of masses m₁ and m₂ are tied to the end of a string passing over a frictionless pulley. Here m₁ be the mass on inclined plane and m₂ be mass of another block, θ is the angle of inclination of slope.

If m₁gsinθ> m₂g

For mass m₁,
m₁a = T – m₁g sinθ ….(i)

For mass m₂,
m₂a = m₂g – T ….(ii)

Adding (i) and (ii),
m₁a + m₂a = m₂g – m₁g sinθ
a(m₁ + m₂) = m₂g – m₁g sinθ

When two blocks of masses m1​ and m2​ are tied to the end of a string passing over a frictionless pulley.

Friction
Friction is the opposing force that is set up between the surfaces of contact when one body slides or rolls or tends to do so on the surface of another body.

Causes of Friction
➺ Classical View
The interlocking of projections of the objects which are kept in contact is the cause of the friction.

➺ Modern View
Intermolecular force of attraction between the surfaces in contact is the cause of the friction.

Types of Friction
➺ Static Friction
The force of friction that comes into play between the surfaces of two bodies before the actual motion starts is called static friction.

➺ Kinetic Friction / Sliding Friction
The frictional force acting between two surfaces when one surface is in steady motion over the other surface is called kinetic friction.
It is always less than the limiting static friction.

➺ Rolling Friction
Objects like wheel, cylinder, etc. will start rolling instead of sliding, when given a push. In such case, friction is called rolling friction.
Rolling friction is much smaller than sliding friction.

Laws of Solid Friction

1. The value of the frictional force depends upon the nature of the two surfaces in contact and their state of roughness.
2. The force of friction is tangential (parallel) to the two surfaces in contact and acts opposite to the direction in which the body would start moving.
3. The value of the frictional force between two surfaces is directly proportional to the normal reaction between the two surfaces.
   i.e

The constant of proportionality is called the coefficient of friction.
∴ Coefficient of friction is defined as the ratio of the frictional force to the normal reaction.

Angle of Repose:

The angle of repose is defined as the angle θ of the inclined plane at which a body placed on it just begins to slide down.
Fc = mgsinθ — (i)
R = mgcosθ — (ii)

Dividing (i) by (ii)

∴ The coefficient of limiting friction is equal to the tangent of the angle of repose.

Numerical

An iron block of mass 10 kg rests on a wooden plane inclined at 30° to the horizontal. It is found that the least force parallel to the plane which causes the block to slide up the plane is 100 N. Calculate the coefficient of sliding friction between the wood and the iron.

Solution

Applied least force (Fa) = 100 N
Mass (m) = 10 kg
Angle of inclination (θ) = 30°
Acceleration (a) = 0 m/s²

From figure:

R=mg cos30∘
F′=μR= μmgcos30∘
For least force,
Fa​=F′+mgsin30∘ where F′ is frictional force

Acceleration of a Body Sliding on an Inclined Plane :

In ΔABO

OB=mgcos⁡ϕ
OA=mg
BA=mgsinϕ
At the equilibrium condition,

Now,

Principle of Conservation of Linear Momentum:

The principle of Conservation of Linear Momentum states that “If no external forces act on a system, the linear momentum of the system remains constant.”

Consider a body of mass m moving with initial velocity u along a straight line. Its linear momentum,

p=mu
We know, force on the system:

Compiled by: Er.Basant Kumar Yadav