Basic Engineering Mechanics

Difference between speed and velocity:

Speed: The rate at which a moving body describes its path is called as speed or the rate of change of distance with respect to time is termed as speed.

E.g.; If the length of the path followed by a moving body between two points ‘P’ and ‘Q’ is 20 km and the time taken to travel this path is 2 hrs, then the speed of the body is 20 km/2 hr = 10 km/hr. The path followed by the body may be curved or a straight line and its shape have no concern with the speed of the body.

Speed is purely a scalar quantity and hence it has magnitude only.

Velocity: The rate of change of displacement with respect to time is called as velocity.

E.g.; If ∆x represents the displacement of the body in time interval ∆t, then the velocity of the body is defined as

                           v = ∆x/∆t

In the limiting case when ∆t →0, v = Lim_∆t→0 ∆x/∆t = dx/dt

Velocity is also expressed by both magnitude and direction just like displacement. Hence velocity is a vector quantity.

Relative velocity: How to find relative velocities in different cases:-

Relative velocity: The velocity of a body or with respect to another body is termed as relative velocity.

How to find relative velocities in different cases:

   Case 01. When two bodies are moving along a straight line in the same direction (say body ‘A’ and ‘B’), the magnitude of the relative velocity of body ‘A’ with respect to ‘B’ is just equal to the difference of the magnitudes of their velocities, i.e. magnitude of the velocity of body ‘A’ minus magnitude of velocity of body ‘B’.

                           Therefore,   v_AB = v_A – v_B  

   Case 02. When two bodies are moving along a straight line in the opposite direction (say body ‘A’and ‘B’), the magnitude of the relative velocity of body ‘A’ with respect to ‘B’ is just equal to the sum of the magnitudes of their velocities.

                  Therefore,   v_AB = v_A + v_B

   Case 03. Motion of a body on a body in the same direction: If a train is moving with velocity ‘v₁’ and a man is running inside it in the same direction with a velocity ‘v₂’, then,

           Relative velocity of man with respect to earth is = v₁ + v₂

       

    Case 04. Motion of a body on a body in the opposite direction: If a train is moving with velocity ‘v₁’ and a man is running inside it in a direction opposite to that of train with a velocity ‘v₂’, then, 

                Relative velocity of man with respect to earth is = v₁ - v₂

       

Newton's Laws of Motion:-

Motion:- When a body changes its position with respect to other bodies, then body is called as to be in motion.

The entire subject of rigid body mechanics is based on three fundamental law of motion given by an American scientist Newton.

01. Newton’s first law of motion:-  According to Newton’s first law, every body continues in its state of rest or of uniform motion in a straight line, unless it is compelled by some external force to change the state.

Newton’s first law can be divided into two parts

Inertia: - This inherent properties of the bodies that they do not change their state unless acted upon by an external force, is called inertia. It is because of inertia that a body cannot change its state of rest or of uniform motion by itself.

Force: - force may be called as an agent that causes or tends to cause acceleration.                        Therefore, Force is a pull or push which generates or tends to generate motion in a body at rest and stops or tends to stop a body which is in motion, increases or decreases the magnitude of the velocity of the moving body or change the direction of motion of the moving body.

The unit of force is Newton.

1-Newton: It is magnitude of force, which develops an acceleration of 1 m/s² in 1 kg mass of the body.

02.Newton’s second law of motion: - According to Newton’s second law, the time of change of momentum of a body is directly proportional to the external force applied on it and the change in momentum takes place in the direction of force.

Therefore, if a body of mass ‘m’ moving with a velocity ‘v’ has got a momentum‘mv’,

        then , F ∝ d(mv)/dt

    Or.        F = k × d(mv)/dt = m × dv/dt = ma

(Here, k = 1, constant of proportionality and dv/dt = a, acceleration produced in the direction of the body)

Mass: The properties of matter by which the action of one body can be compared with that of another is defined as mass.

                            So,     m = ₰ × v

Where, ₰ = Density of body and v = Volume of the body

(Weight of a body is the force with which the body is attracted towards the center of the earth.)

Momentum: It is defined as the total quantity of motion contained in a body and is measured as the product of the mass and its velocity.

                           Momentum = Mass × Velocity = m × v

03.Newton’s third law of motion: - It states that to every action, there is an equal and opposite reaction. I.e. The force of action and reaction between interacting bodies are equal in magnitude, opposite in direction and have the same line of action. Therefore,

In general, if a body ‘A’ exerts force ‘F_AB‘(which may be gravitational, electrical or magnetic etc.) on a body B, then the body ‘B’ will exert a force ‘F_BA‘ on body ‘A’,

                         such that, F_AB = - F_BA

Line of action: The straight line along which a force is directed is called the line of action of the force.                                  

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