Mechanics

Mechanics in physics is the branch that deals with the motion of objects and the forces acting on them. It includes concepts like displacement, velocity, acceleration, and force. Mechanics is divided into kinematics (study of motion) and dynamics (study of forces), forming the foundation for understanding physical systems.

Kinematics

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion. It involves analyzing quantities such as displacement, velocity, and acceleration. Kinematics provides mathematical descriptions of motion in one, two, or three dimensions, helping to predict an object’s future position.

Motion

Motion refers to the change in the position of an object with respect to time. We can define any motion with the help of the following terms: 

• Displacement: The shortest distance between the initial and final position of the object, along with the direction.
• Distance: The total path length travelled by the object, regardless of direction.
• Speed: The rate at which distance is covered. It is a scalar quantity. 

Speed= Δ DistanceΔ Time

Here, the symbol Δ (delta) must be read as “the change in”. The speed is a change in distance divided by a change in time. Therefore,  Δ Distance represents the change in distance and Δ Time represents the change in time.

• Velocity: The rate of change of displacement. It is a vector quantity and has both magnitude and direction. 

Velocity= Δ DisplacementΔ Time

• Acceleration: The rate at which velocity changes with time. It is a vector quantity. 

Acceleration= ΔVelocityΔTime

Graphical Representation of Motion

Graphs are a useful way to visually represent motion. The most common types of motion graphs are:

1. Position-Time Graph (Displacement-Time Graph): This shows how the position of an object changes over time.
   • The slope of the graph represents the velocity.
   • A straight line with a constant slope indicates uniform motion (constant velocity).
   • A curve indicates non-uniform motion (changing velocity).
2. Velocity-Time Graph: This shows how the velocity of an object changes over time.
   • The slope of the graph represents acceleration.
   • A horizontal line indicates constant velocity (zero acceleration).
   • The area under the curve gives the displacement.
3. Acceleration-Time Graph: This shows how acceleration changes over time.
   • The area under the curve gives the change in velocity.
   • A constant positive or negative value indicates uniform acceleration.

Equations of Motion 

In physics, there are three key equations of motion that describe the motion of an object under constant acceleration. These are:

• First Equation of Motion:

v=u+at

Where: u is the initial velocity, v is the final velocity and a is the acceleration.

This is directly derived from the definition of acceleration.

We know that Acceleration = ΔVelocityΔTime

• Acceleration = Final velocity v– initial velocityuΔ Time
• a = v-ut
• v=u+at

The first equation comes from the fact that the slope of a velocity-time graph gives acceleration. If the velocity increases linearly with time, the change in velocity is v−u=atv – u = atv−u=at, leading to the first equation.

Graphical Intuition of the Accelerated Motion

In order to develop an intuition on how to measure the distance covered by an object that is accelerating, we must understand how to represent this motion on a velocity-time graph. Let’s draw a graph in which the x-axis represents time (t) and the y-axis represents velocity (v). The graph for uniformly accelerated motion is a straight line, starting from the initial velocity u (on the y-axis at t=0) and increasing with a constant slope, representing constant acceleration a. The slope of the graph is equal to the acceleration a. Since acceleration is defined as the change in velocity per unit time: slope=ΔvΔt=a\text{slope} = \frac{\Delta v}{\Delta t} = aslope=ΔtΔv​=a   Area Under the Graph: The area under the velocity-time graph represents the displacement (s) travelled by the object. The area under the graph is a trapezoid with bases u (initial velocity) and v (final velocity) and height t (time).

• Second Equation of Motion:

s=ut+12at²

Where s is the displacement, u is initial velocity, a is acceleration and t is the time for which acceleration was applied.

• Third Equation of Motion:

From the first equation

• v=u+at
• t=v-ua

Substituting this ‘t’ into the 2nd equation, we get: 

• • s=u.( v-ua) + 12a(v-ua)² 

• 2.a.s = v2 – u2 
  

v²=u²+2as

These equations can also be derived graphically from a velocity-time graph.

• The second equation is derived by finding the area under the velocity-time graph, which gives the displacement.
• The third equation is a result of eliminating time from the first two equations.

Uniform Circular Motion

Uniform Circular Motion refers to the motion of an object moving in a circle at a constant speed. Even though the speed remains constant, the direction of motion changes continuously, which means the object is constantly accelerating.

• Centripetal Acceleration: An object moving in a circle experiences an acceleration directed toward the centre of the circle, called centripetal acceleration. The magnitude of centripetal acceleration is given by:

a=v²r

Where r is the radius of the circle

• Centripetal Force: The force that causes centripetal acceleration is called centripetal force. It is essentially a pseudo force that is not put by any external entity but is experienced by the object just by virtue of its tendency to move in a straight line in a uniform motion.
• Angular Velocity: The rate at which an object moves through an angle in a given time.  

Dynamics

Dynamics is the branch of physics that deals with the study of forces and their effects on the motion of objects. We shall study the forces that cause motion in this section.

In Kinematics, we understood the motion but did not pay any attention to the forces that can cause motion. ‘Dynamics’ explores how forces cause changes in velocity (acceleration) and include concepts like Newton’s laws of motion, work, energy, and momentum, which are fundamental to understanding physical systems.

Force

Force is a vector quantity that causes an object to change its state of motion or shape. It is measured in newtons (N) and can cause an object to accelerate, decelerate, or deform. Forces can be classified into contact forces (like friction, tension, and normal force) and non-contact forces (like gravitational, electromagnetic, and nuclear forces). 

According to Newton’s Second Law, the force acting on an object is the product of its mass and acceleration (F = ma). Force is a fundamental concept in dynamics and is essential for understanding the motion and behaviour of physical objects.

Laws of Motion

Newton studied Galileo’s ideas on force and motion and presented 3 fundamental laws.

• First law of motion: It is also known as the Law of inertia 

“A body continues to be in a state of rest or in  Uniform motion unless compelled to change that state by an applied unbalanced force.”

Inertia

Inertia is the tendency of an object to remain in its state of motion (or rest). Inertia is measured by Mass. The greater the mass the more difficult it becomes to move an object.

• For example, a planet does not change its motion and keeps on moving in the same orbit year after year for millions of years. 

• Galilio’s Law of Inertia conveys the same principle in other words: “If there is no net force acting on an object, the object remains in the same state of motion”. 

• The second law of motion: 

“Rate of change of momentum of an object is proportional to applied unbalanced force in the direction of force.”

Momentum

Momentum is defined as the product of an object’s mass and its velocity.  Momentum=m . v Where ‘m’ is the mass and ‘v’ is the velocity of the object.   It is the measure of how difficult would it be to change the motion of an object. The greater the velocity or the mass, the greater would be the difficulty in changing its motion.

In fact, the rate of change in momentum is defined as the force.

• F= Change in MomentumChange in Time
• F= mv-mut
• F= m (v-u)t =m. (v-u)t
• F=ma

• Third law of Motion: It is also known as the law of action and reaction. 

“For every action, there is equal and opposite reaction.”

Conservation of Momentum

The Conservation of Momentum is a fundamental principle in physics stating that the total momentum of an isolated system remains constant if no external forces act upon it. 

It means that:

Mass X velocity = Constant

This law is particularly useful in analyzing collisions, whether elastic (where kinetic energy is also conserved) or inelastic (where kinetic energy is not conserved but momentum is). 

Applications of conservation of momentum

The conservation of momentum is crucial in understanding the behaviour of objects in interactions, such as car crashes, rocket propulsion, or particle physics experiments. For example: 

• In rocket propulsion, as gas is ejected in the opposite direction by the engine, the rocket moves forward, demonstrating Newton’s third law of motion.

• When billiard balls collide, the total momentum before and after the collision remains constant. This helps in predicting the motion and final velocities of the balls.
• When a person jumps off a stationary boat, the boat moves in the opposite direction to conserve momentum. The combined momentum of the person and boat before and after the jump is zero.
• When fireworks explode, they eject gases and particles outward. The momentum of the particles is balanced by an equal and opposite momentum of the fireworks shell, following the principle of conservation of momentum.
• Airplane Propulsion: In an aeroplane, the engines expel air backwards. The conservation of momentum leads to a forward thrust on the aeroplane, allowing it to take off and stay in flight.
• In a gunshot, when the bullet is fired forward, the gun experiences an equal and opposite momentum, pushing it backwards (recoil).  

The conservation of momentum is a fundamental law of nature. It has never been found to be violated.

Pressure

Thrust/Area.

• The bag strap is broad.
• Injection middle 
• Barometer.
• Pascal’s law: same pressure at sea at the same depth.
• At high pressure, the boiling point increases.
• Buoyancy: Upward force exerted by water on an immersed object is known as upthrust or buoyant force. This phenomenon is known as buoyancy.
• Relative density: The density of a substance relative to the density of water.

Work and Energy

It is the measure of energy transfer that occurs when an object is moved over a distance by an external force.

• The scientific conception of work: product of force & displacement. 
• Energy: Capacity to do work.

Forms of energy: 

• Kinetic Energy: Energy possessed by an object by virtue of its motion.

Ek = 1/2 mv2

• Potential Energy: Energy stored in an object due to work done on it.
  • Potential energy due to height: 

Ep = force X displacement = mg X h = mgh

• Law of Conservation of Energy = Energy can neither be created nor be destroyed, it can change from one form to another  OR Total energy is always constant.
• Power: Rate of doing work