In this article, we are sharing engineering mechanics short notes, which can be used for revision for GATE Mechanical Engineering or any other competitive exam in Mechanical Engineering.

**MOTION UNDER GRAVITY**

Newton’s law of Universal gravitation states that any two particles or point masses attract each other along the line connecting them with a mutual force whose magnitude is directly proportional to the product of the masses and inversely proportional to the square of the distance between the particles.

For the configuration shown in fig. the gravitational force F between two masses m1 and m2 at distance r is given by

Where G is the universal constant of gravitation, equal to **.**

**EARTH’S GRAVITY**

Assuming earth to be stationary and spherical body of radius R, the gravitational force of the earth (due to its mass M) acting on a body of mass m, placed at a height h above the surface of the earth, is given by **.**

This force is the weight of the body equal of the body equal to mg. therefore, acceleration due to gravity g is derived as **.**

The value of g is approximately 9.81 m/s^{2}**.**

In engineering applications, g is usually considered as a constant and the weight force is assumed to be directly perpendicular to the earth’s surface.

When a particle is projected vertically upward, there is a retardation upon it due to earth’s attraction. This retardation is denoted by – g. When a particle falls down under gravity, it possesses an acceleration equal to g.

**Projectile**

The particle projected under gravity other than vertical is called a projectile. The angle of projection is the angle of initial velocity with horizontal plane. The path described by the particle is called trajectory. The range of projectile is the distance between the point of projection and the point where trajectory meets any horizontal plane through the projection.

Let a particle is projected upward at an angle from horizontal at initial velocity of u.

The following are the features of a projectile:

**Flight time **

**Maximum height **

**Range **

**The range is maximum if **

**Vertical Projection**

Consider a particle of is projected vertically upward at initial velocity u, and let it reaches upto height h where velocity v become zero. Using , we get .

The time t taken in the reaching to the height h is determined as , .

**DEPENDENT MOTION OF PARTICLES**

In some types of engineering application, motion of particles is dependent upon others. Two blocks inter-connected with an inextensible spring over a pulley represent the most simple situation of dependent motion.

The relationship between dependent velocities can be found using constant length of the inextensible string.

**NEWTON’S LAW OF MOTION**

The problems of mechanics can be solved by applying Newton’s laws of motion, described as follows:

**1) FIRST LAW OF MOTION**: - Newton’s first law of motion states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.

The first law of motion is normally taken as the definition of inertia. If there is no net force acting on an object, then the object will maintain a constant velocity. If that velocity is zero, then the object remains at rest. If an external force is applied, the velocity will change because of the force.

**2) SECOND LAW OF MOTION:-** Newton’s second law of motion states that if the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force. The law explains how the velocity of an object changes when it is subjected to an external force. The law defines a force to be equal to changes in momentum per unit time.

For an object with a constant mass m, Newton’s second law of motion states that the force F is the product of an object’s mass and its acceleration a.

For an externally applied force, the changes in velocity depends on the mass of the object. A force will cause a change in velocity, and likewise, a change in velocity will generate a force. The above equation works in both ways.

**3) THIRD LAW OF MOTION**:- Newton’s third law of motion states that for every action in nature there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B also exerts an equal forces on object A.

The third law of motion can be used to explain the generation of lift by a wing and the production of thrust by a jet engine.

**WORK AND ENERGY**

**ENERGY**

The energy of a body is its capacity of doing work. Energy is possessed by a body, while the work is done by force on a body when it has a displacement in the direction of the forces. If position vector is denoted by r, then work dw, a scalar quantity, is defined as the dot product of force F and displacement vector dr.

**Modes of Mechanical Energy**

Energy can be in several forms like mechanical energy, electrical energy, heat, sound, pressure. The present context is of mechanical energy possessed by a body due to its position or motion. Hence, mechanical energy can be of two types : potential and kinetic energy.

**1) Potential energy:-** The energy which a body possess by virtue of its position or configuration is called potential energy. Few examples to clarity the concept of potential energy are following:

(a) If a body of mass m is raised through a height h above of datum level, then the work done on it by the gravitational force is written as

This energy is stored in the body as potential energy. In coming down to the original position, the body is capable of doing work equal to mgh.

(b) If a spring is twisted through an angle ϴ by application of a torque varying from zero in the beginning to T in the end, the work done by the average torque T2 is written as

This is the potential energy of the spring due to its configuration.

(c) If a spring of stiffness k is stretched, the force F acting on it does not remain constant, but increases with displacement x undergoes by the spring. At any time,

Therefore, average force acting on the spring is

Hence, the work done by the average force F for displacement x of the spring is written as

This is the potential energy of the spring due to its configuration.

Gravity force, elastic spring, and torsional spring are examples of conservative forces. Thus, potential energy is the measure of the amount of work done by a conservative force in moving a body from one position to another .

**2) Kinetic Energy:-** The energy which a body possess by virtue of its motion is called kinetic energy. It is measured by the amount of work required to bring the body to rest.

Let a body of mass m moving with velocity v be brought to rets by the application of a constant force F which causes a retardation – a. if s is the distance through which the body moves in this period, the kinetic energy is given by the work done by the force F on the body. Using third equation of linear motion,

Therefore, kinetic energy is determine as

A system of particles or body can have both forms of mechanical energy. During motion or change in the amount or direction forces, one form of energy gets converted into another form.

**Principle of work and Energy**

The Principle of work and energy states that the work done by all of the external forces and couples as a rigid body moves from position 1 to position 2 is equal to the change in the potential energy of the body:

**Principle of conservation of Energy**

The principle of conservation of energy states that the total amount of energy in the universe is constant; energy can neither be created nor be destroyed although it can be converted into various forms.

The principle of conservation of energy can be appropriate stated as when a particle moves under the action of conservative forces, the sum of kinetic energy and potential energy of the particle remains constant. If the potential energy and kinetic energy are denoted by U and T, respectively, the principle can be stated for a system between two instances 1 and 2 as

The principle of conservation of energy is generally applied to solve the problems involving forces, displacement and velocities. The principle can be applied to each element of a structure or body separately. The problems involving energy dissipation through friction and damping can be solved by considering suitable sign of the energy component of the system.

**D’ALEMBERT’S PRINCIPLE**

According to the D’Alembert’s Principle, the external forces acting on a body and the resultant inertia forces on it are in equilibrium. It is indeed, a restatement of Newton’s Second Law of motion but it suggest that the term (- ma) can be considering as a fictitious forces, often called d’Alemberts force F or the inertia force. Accordingly, the net external force F actually acting on the body and the inertia force Fi together keep the body in a states of fictitious equilibrium.

The d’Alemberts principle tends to give the solution procedure of a dynamic problem, an appearance like that of a static problem, and the above equation becomes equation of dynamic equilibrium.

**IMPULSE AND MOMENTUM**

**Linear Momentum**

Momentum (p) is a measure of the tendency of an object to keep moving once it is set in motion. Let a particle of mass m move with a velocity v and acceleration a. using Newton’s law of motion, the force acting on the body is given by

The rate of change of momentum is

This equation states that the rate of change of momentum is equal to the applied force. This statement is known as the principle of linear momentum. The law is also known as Euler’s first law. If there are no forces applied to a system, the total momentum of the system remains constant; the law in this case is known as the law of conservative of momentum.

**Impulse – Momentum Principle**

If a constant force F acts for time t on a body, the product F × t is called the impulse of the given force. Similarly, if a torque T acts on a body for time t, then the angular impulse is T × t.

Let a constant force F acts on a body of mass m for time and changes its velocity from u to v under acceleration a. then, impulse is given by

Therefore, impulse – momentum principle states that the compound of resultant linear impulse along any direction is equal to change in the component momentum in that direction.

**LAW OF RESTITUTION**

Impact is the collision of two particles for a very short period of time that results into relatively large impulsive forces exerted between the particles. An impact is called central or line impact when direction of motion of the mass centre of the two colliding particles is in a single line, otherwise, it is called oblique impact.

The law of restitution states that the velocity of separation of two moving bodies which collide with each other bears a constant ratio with their velocity approach. The constant of proportionality is called coefficient of restitution, denoted by e. This property first discovered by **Newton’s law of restitution.**

Consider two particles moving with initial velocities u1 and u2 towards each other. These particles on centre line and after impact, their respective velocities become v_{1} and v_{2}.

The coefficient of restitution is expressed as the ratio of relative velocities of the particles separation just after impact to the relative velocity of the particle approach just before impact:

Experiments show that e varies appreciably with impact velocity as well as with the size and shape of colliding particles, ranging from 0 to 1. The value of the coefficient of restitution has got physical meaning. The collision can be classified into two types:

**1) Elastic collision:- **A perfect elastic collision occurs without loss of kinetic energy of the particles. Thus, for elastic collision e = 1.

**2) Inelastic collision:** - A inelastic collision or plastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision.

Momentum is conserved in inelastic collisions, however, the kinetic energy in the collision is converted into other forms of energy. For inelastic collision e = 0.

The principle of work and energy cannot be used for the analysis of impact problems because it is impossible to know the variation in the internal forces of deformation and restitution during the collision. The energy loss can be calculated as the change in kinetic energy of the particle.

**PRINCIPLE OF VIRTUAL WORK**

When the point of application of a force is imagined to be displaced through a differential distance in the direction of the force, the imaginary work done by the force is called virtual work.

The principle of virtual work states that the work done on a rigid body or a system of rigid bodies in equilibrium is zero for any virtual displacement compatible with the constraints on the system.

Virtual displacement is an imaginary infinitesimal displacement. A differential virtual displacement is denoted by δ to distinguish it from differential displacement generally denoted by d.

The method of virtual work is explained below:

1) Consider a rod AB which can rotate about a fulcrum O. A vertical load F1 is applied at end A. it is required to calculated the vertical force F2 to be applied at end B to keep the rod in current position. In virtual work method, the body is assumed to be virtually displaced. For

present case, let the rod undergo a virtual rotation through angle δϴ about the fulcrum O to assume the new position A’B’. The total virtual work during this rotation is given by

According to the principle of virtual work, total work must be zero, therefore,

2) Consider a lazy tong mechanics. The joint A has a pin which is free to slide inside the vertical groove provided in the frame. The joint E, has a torsional spring to keep the mechanics in equilibrium under the external force F applied at the hinge joint E.

The magnitude of the moment M required to keep the mechanism in equilibrium can be determined using the method of virtual work. The horizontal distance x of joint E from the AB is

The virtual rotation of link BC for virtual displacement of δx of the joint E is given by

The reaction at the joints A and B will not cause any work, the total virtual work done by moment M and external force F must be zero:

In apllying the method of virtual work, it is necessary to only calculate the displacements of the points of application of the forces, and hence a problem of equilibrium is converted into one of geometry, which is comparitively easier to solve. There is no solid advantage of applying principle of virtual work in equilibrium problems.

We hope you will find these Engineering Mechanics short notes helpful. For problems and questions on engineering mechanics, feel free to visit here.

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Shubham SharmaThanks for these notes.