The motion of an object, whether in a straight line, plane, or space, is influenced by the forces acting upon it. Newton's laws of motion and gravitation are relevant to everyday life situations and can be understood through the concepts of force and motion.

Stationary Position and Motion

Definition of Stationary Position and Motion

Everything around us is either at rest or in motion relative to us. An object is considered to be in motion if it changes its position with respect to its surroundings over time. Conversely, if an object does not change its position relative to its surroundings, it is considered to be at rest or in a stationary position.

Examples of motion include a speeding car, a moving train, and the moon orbiting the Earth.

Relative Nature of Stationary Position and Motion

Stationary position and motion are relative terms. An object can be at rest with respect to one object while simultaneously being in motion relative to another object. For example, a passenger in a moving train is at rest with respect to fellow passengers but is in motion relative to observed objects or trees outside the train.

Scalar and Vector Quantities

A physical quantity that has only magnitude is known as a scalar quantity. Examples include mass, speed, time, work, and distance.

A physical quantity that has both magnitude and direction is known as a vector quantity. Examples include weight, velocity, force, acceleration, and displacement.

Types of Motion

There are broadly six types of motion:

1. Rectilinear Motion: In rectilinear motion, a particle or body moves along a straight line.
   Example: A car moving on a straight road.
2. Circular Motion: Circular motion occurs when a particle moves in a circle.
   Example: Earth revolving around the sun.
3. Oscillatory Motion: In oscillatory motion, a body moves to and fro about a fixed point in a definite interval of time.
   Example: Motion of a pendulum.
4. Rolling Motion: Rolling motion is a combination of rotational and translational motion.
   Example: A wheel rolling on a road.
5. Periodic Motion: If a body repeats its motion at regular intervals of time, it is in periodic motion. Circular and oscillatory motions are examples.
   Example: Rotation and revolution of planets around the sun.
6. Random Motion: In random motion, a body changes its direction arbitrarily without a fixed pattern.
   Example: Motion of an ant or mosquito.

Distance, Displacement, Speed, Velocity, and Acceleration

Distance and Displacement

Distance is the length of the actual path traveled by an object between its initial and final position. Displacement is the difference between the final and initial position of an object, measured in a straight line from the initial position to the final position.

• SI unit for both distance and displacement is meter (m).
• Displacement can be positive, negative, or zero. Displacement is zero when the initial and final positions are the same.
• Distance covered by an object can never be zero.
• Odometer in automobiles measures the distance covered by them.

Example 1

A car moves on a circular track with a radius of 10 m. Find the distance covered and displacement of the car after one complete revolution.

Speed and Velocity

Speed is the distance covered by an object in a given interval of time. Velocity is the displacement of an object in a given interval of time.

• Velocity is a vector quantity, while speed is a scalar quantity.
• SI unit for both speed and velocity is meter/second (m/s or m·s^-1).
• Speed is always equal to or greater than the magnitude of velocity.

Uniform and Non - uniform Motion

A body is in uniform motion if it covers equal distances in equal intervals of time. In nonuniform motion, the velocity changes with time, i.e., either the speed, direction, or both change.

• For uniform motion along a straight line, displacement and distance covered are the same and in the same direction.
• No force is required to maintain the motion of an object in uniform motion.

Average Speed and Average Velocity

Average speed is the total distance traveled divided by the total time taken. Average velocity is the total displacement divided by the total time taken.

• Average velocity is zero when the displacement of the object is zero, irrespective of the time interval.
• Average speed of an object can never be zero.

Relative Velocity

Relative velocity is the rate of change of position of one object with reference to another object.

• When both objects are moving in the same direction: Relative velocity =    V₁ - V₂
• When both objects are moving in opposite directions: Relative velocity =   V₁ + V₂

Acceleration

Acceleration is the rate of change of velocity of an object.

Acceleration:

• Acceleration is a vector quantity.
• SI unit of acceleration is meter/second² (m/s²).
• An object has positive acceleration if its velocity increases without a change in direction.
• An object has negative acceleration if its velocity decreases without a change in direction.
• An object has zero acceleration if it is at rest or moving with uniform velocity.
• An object has uniform acceleration if its velocity changes by the same amount in equal intervals of time.
• An object has non-uniform acceleration (variable acceleration) if its velocity changes by different amounts in equal intervals of time.

Graphical Representation of Motion

Distance-Time Graph

• It is a graph plotted between the distance covered and time taken.
• For uniform speed, the graph is a straight line.
• For non-uniform speed, the graph is a non-linear curve with varying slopes.

Displacement-Time Graph

• The slope of the displacement-time graph gives the velocity of an object.
• For uniform velocity, the graph is a straight line.
• For non-uniform velocity, the graph is a non-linear curve.

Speed-Time Graph

• In a speed-time graph, the area under the curve gives the distance traveled by the object.
• A straight line parallel to the time axis indicates constant speed.

Velocity-Time Graph

• The slope of the velocity-time graph gives the acceleration of the object.
• For constant acceleration, the graph is a straight line.
• The area under the velocity-time graph gives the displacement of the object.

Velocity-Time Graphs for Accelerated Motion

Zero Acceleration

If the object has zero acceleration, its velocity does not change with time.

Positive Acceleration

• If the object starts from rest, the velocity-time graph is a straight line starting from the origin.
• If the object has some initial velocity, the velocity-time graph is a straight line starting from a point.

Negative Acceleration

If the object has negative acceleration, its velocity decreases linearly with time.

• If the initial velocity is positive, the velocity-time graph is a straight line with a negative slope.

Equations of Motion with Constant Acceleration

• When an object moves with constant acceleration, the following equations can be used:

First Equation of Motion

• v = u + at
• Where:
  • u is the initial velocity of the object,
  • v is the final velocity of the object,
  • a is the constant acceleration,
  • t is the time elapsed.

Second Equation of Motion

• 
• Where:
  • s is the distance covered by the object,
  • u is the initial velocity of the object,
  • t is the time elapsed,
  • a is the constant acceleration.

Third Equation of Motion

• 
• Where:
  • u is the initial velocity of the object,
  • v is the final velocity of the object,
  • a is the constant acceleration,
  • s is the distance traveled by the object.

Freely Falling Object

• A freely falling object near the Earth's surface experiences constant acceleration due to gravity.
• Acceleration due to gravity (g) is approximately \( 9.8 \, \text{m/s}^2 \) directed vertically downwards.
• If upward is taken as positive, then acceleration due to gravity is taken as negative.

Equations of Motion due to Gravity

• When a body is projected vertically upwards with initial velocity u:
• v = u - gt
• (where y is the vertical distance)
• 
• 

Force

• Force is an external agency that can change or tends to change the state of rest or motion or direction of a body.
• It can also deform the shape of an object.
• The SI unit of force is Newton (N) and the CGS unit is dyne.
• Force is a vector quantity, having both magnitude and direction.

Types of Forces

• Net force: When forces act in the same direction, they are added. When they act in opposite directions, they are subtracted.
• Balanced Force: If two equal forces act on an object in opposite directions, they cancel each other out, resulting in no net force (F = 0).
• Unbalanced Force: If the net force on an object is not zero, it is termed as unbalanced force, which can change both the shape and motion of an object.

Types of Forces

1. Non-contact Forces

Forces acting between bodies without any physical contact:

1. Gravitational Force: Any two bodies with masses attract each other with a force given by:
   Where G  is the universal gravitation constant  m₁ & m₂ are the masses, and  r  is the distance between them.
2. Electromagnetic Forces: Combination of electric and magnetic forces:
   • Magnetic Force: Associated with magnets.
   • Electrostatic Force: Between particles with charges q₁ and q₂ , given by , where k is the Coulomb constant 8.85 x 10¹² , C²Nm² .
3. Nuclear Force: Strongest force, short-range, acts between nucleons (neutrons and protons) in the nucleus.
4. Weak Force: Short-range force involved in reactions with protons, neutrons, and electrons.

Gravitational force is the weakest among these forces.

2. Contact Forces

Forces acting only when objects are in physical contact:

1. Muscular Force: Force exerted by muscles.
2. Force of Friction: Arises between surfaces of two objects in contact during relative motion:
   • Static Friction: Prevents motion before it starts.
   • Limiting Friction: Maximum static friction.
   • Kinetic Friction: Opposes motion of one body sliding or rolling over another.
   • 
     
     • Sliding Friction: Between surfaces sliding over each other.
     • Rolling Friction: Between surfaces rolling over each other.
   • Friction acts tangentially to the surface and helps in various activities but can also cause wear, tear, and energy loss.
   • Static friction is greater than kinetic friction.
3. Tension: Force transmitted axially by means of a string or cable.

3. Centripetal Force

Force required to move a body in a circular path with uniform speed:

• Magnitude:  , where m  is mass,  v  is speed, and   r is radius of the circle.
• Applications:
  • Electrons orbiting a nucleus experience electrostatic attraction.
  • Planetary motion involves gravitational attraction between sun and planets.
  • During vehicle turns, centripetal force equals friction force between wheels and road.
  • In circular motion, tension in a string provides centripetal force.
  • Satellites orbiting planets experience gravitational force as centripetal force.

4. Centrifugal Force

Apparent force in non-inertial frames, opposite to centripetal force:

• Magnitude: , where omega  is angular velocity and r  is radius.
• Direction: Radially outward from the centre of circular motion.
• Applications:
  • People feel pushed outward on a merry-go-round.
  • Centrifuge devices separate substances based on mass by rotating at high speeds.

Circular Motion

Motion along a circular path with angular variables:

1. Angular Position: Angle  θ in radians describes particle position relative to circle centre.
2. Angular Velocity:  , SI unit radian/second.
3. Angular Acceleration:, SI unit radian/second².

Non-uniform Circular Motion

If the speed of a particle in circular motion is not constant, it is termed as non-uniform circular motion.

For example, a satellite orbiting a planet has uniform circular motion.

Oscillatory Motion

In oscillatory motion, a particle moves back and forth or up and down about a fixed point called equilibrium position. It is a periodic motion.

A special case of oscillatory motion is Simple Harmonic Motion (SHM), where:

• Displacement in SHM:   where  ω = 2 π v.
• Time Period (T): Time taken for one complete oscillation.
• Frequency (v) : Number of oscillations per second.
• Amplitude (A): Maximum displacement from the equilibrium position.
• In SHM, the total mechanical energy (kinetic + potential) remains constant.

Motion of Simple Pendulum

A simple pendulum consists of a small heavy bob suspended from a rigid support through a light inextensible string. It executes SHM.

Newton’s Laws of Motion

Newton’s First Law of Motion

If the sum of all forces acting on a particle is zero, the particle remains at rest or moves with uniform velocity (law of inertia).

Linear Momentum

Linear momentum (p) is the product of mass (m) and velocity (v) of a body.

Formula:  p = mv.

It is a vector quantity with SI unit kg·m/s.

Newton’s Second Law of Motion

Force (F) acting on an object is equal to the rate of change of its momentum  

This is applicable when mass  (m) is constant.

Law of Conservation of Linear Momentum

If no external force acts on a system, the total momentum of the system remains constant:

Kinetic energy K E of a body: 

or 

Newton’s Third Law of Motion

For every action, there is an equal and opposite reaction.

Examples: Walking, firing a bullet, swimming.

Impulse of a Force

Impulse is the product of the force and the time interval. If a constant force (F) is applied to a body for a short interval Δ t, then the impulse of the force F  will be F x Δ t.

The impulse given to the body is equal to the change in the momentum of the body.

Example: Vehicles like cars and buses are equipped with shock absorbers to increase the time duration of jerks caused by roads. Train bogies have buffers for the same reason.

Pressure

Pressure is the force acting per unit area of a surface.

Formula: 

SI unit of pressure is Pascal (Pa). Other units include N/m² or Kg/ms². Pressure is a scalar quantity.

Characteristics of Pressure:

• Both liquids and gases contained in a container exert pressure on the walls of the container.
• Pressure is maximum at the base of the container and the pressure is the same at two points in the same horizontal level:   

P_A <  P_B  = P_C  <  P_D ,  where ( P_A , P_B P_C ,P_D ) are pressures at points A, B, C, D respectively.

• If the pressure over a liquid increases, its boiling point also increases.

Pascal’s Law

Pascal’s law states that if the pressure in a liquid changes at a particular point, the change is transmitted throughout the entire liquid without being diminished.

Hydraulic lifts, used to raise heavy loads, are based on this principle.

Atmospheric Pressure

The pressure exerted by the air around us is known as atmospheric pressure. Despite its magnitude, we do not feel this pressure because it is balanced by the pressure inside our bodies.