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Unit1 - Subjective Questions

MEC107 • Practice Questions with Detailed Answers

Define Engineering Mechanics and classify it into its main branches.

Engineering Mechanics is the branch of science that deals with the behavior of a body when it is at rest or in motion under the action of forces.

It is mainly classified into:

Statics: Deals with bodies at rest under the action of forces.
Dynamics: Deals with bodies in motion. It is further divided into:
- Kinematics: Study of motion without considering the forces causing the motion.
- Kinetics: Study of motion considering the forces that cause the motion.

Explain the concept of a Rigid Body and a Particle in Mechanics.

Rigid Body: A rigid body is defined as a definite quantity of matter, the parts of which are fixed in position relative to one another. It does not deform under the action of external forces. In reality, no body is perfectly rigid, but the deformations are often negligible compared to the overall motion.
Particle: A particle is a body of infinitely small volume but possessing mass. It is considered as a point mass, meaning its dimensions are negligible in the analysis of its motion.

State Newton's Three Laws of Motion.

Newton's Laws of Motion are the fundamental principles of mechanics:

First Law: Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external force to change that state.
Second Law: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. Mathematically: $F = m \cdot a$ .
Third Law: To every action, there is always an equal and opposite reaction.

Define Force and state its key characteristics.

Force is an agent which produces or tends to produce, destroys or tends to destroy motion. It is a vector quantity.

The key characteristics (or elements) of a force are:

Magnitude: The quantity of the force (e.g., 50 N).
Direction: The line of action of the force and the angle it makes with a reference axis.
Nature (or Sense): Whether the force is a push or pull.
Point of Application: The exact point at which the force acts on the body.

What is a System of Forces? List the various types of force systems.

When two or more forces act on a body simultaneously, they constitute a System of Forces.

Types of Force Systems:

Coplanar Forces: All forces lie in the same plane.
Non-Coplanar (Space) Forces: Forces lie in different planes.
Concurrent Forces: Lines of action of all forces meet at a single point.
Non-Concurrent Forces: Lines of action do not meet at a single point.
Collinear Forces: Lines of action of all forces lie along the same straight line.
Parallel Forces: Lines of action are parallel to each other (can be like or unlike).

State and explain the Principle of Transmissibility of forces.

The Principle of Transmissibility states that the point of application of a force can be transmitted to any other point along its line of action without changing the effect of the force on the rigid body, provided the new point is strictly connected to the body.

Explanation: If a push force of 10 N is applied at point A on a rigid block, it will have the exact same translational and rotational effect as a pull force of 10 N applied at point B, as long as A and B lie on the exact same line of action.

State the Parallelogram Law of Forces and write the expressions for resultant magnitude and direction.

The Parallelogram Law of Forces states that if two forces acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through their point of intersection.

Let $P$ and $Q$ be two forces acting at an angle $\theta$ .

Magnitude of Resultant ( $R$ ):
$R = \sqrt{P^2 + Q^2 + 2PQ \cos \theta}$
Direction (Angle $\alpha$ with force $P$ ):
$\tan \alpha = \frac{Q \sin \theta}{P + Q \cos \theta}$

Explain the Triangle Law and Polygon Law of Forces.

Triangle Law of Forces: If two forces acting on a body are represented in magnitude and direction by the two sides of a triangle taken in order, their resultant is represented by the third side of the triangle taken in the opposite order.
Polygon Law of Forces: If a number of coplanar concurrent forces are represented in magnitude and direction by the sides of a polygon taken in order, their resultant is represented in magnitude and direction by the closing side of the polygon, taken in the opposite order.

How do you resolve a force into rectangular components in a 2-D plane?

Resolving a force means splitting it into components along mutually perpendicular axes (usually $x$ and $y$ ).

Let a force $F$ make an angle $\theta$ with the positive x-axis.

The horizontal component along the x-axis is:
$F_x = F \cos \theta$
The vertical component along the y-axis is:
$F_y = F \sin \theta$

The original force can be written in vector form as $\vec{F} = F_x \hat{i} + F_y \hat{j}$ . The magnitude is $F = \sqrt{F_x^2 + F_y^2}$ .

Define the Moment of a Force and state its physical significance.

The Moment of a force is the turning effect produced by a force on a body about a point or an axis.

Mathematically, the moment $M$ about a point is the product of the magnitude of the force ( $F$ ) and the perpendicular distance ( $d$ ) from the point to the line of action of the force:
$M = F \times d$

SI Unit: Newton-meter (N·m).
Physical Significance: It measures the tendency of a force to cause a body to rotate about a specific point or axis. It is classified as clockwise (usually negative) or counter-clockwise (usually positive).

State Varignon's Theorem (Principle of Moments).

Varignon's Theorem states that the moment of a resultant of a system of coplanar forces about any point in their plane is equal to the algebraic sum of the moments of the individual forces about the same point.

Mathematical Expression:
$R \times d = \sum (F_i \times d_i)$
where $R$ is the resultant force, $d$ is the perpendicular distance from the point to the resultant, $F_i$ are individual forces, and $d_i$ are their respective perpendicular distances from the point.

Define a Couple and list its characteristics.

A Couple is formed by two equal, opposite, and non-collinear parallel forces. It produces purely rotational motion without any translation.

Characteristics:

The algebraic sum of the forces forming a couple is zero.
The translational effect of a couple is zero.
The moment of a couple is constant about any point in its plane and is equal to the product of the magnitude of one of the forces and the perpendicular distance between their lines of action ( $M = F \times d$ ).
A couple can only be balanced by another equal and opposite couple.

Distinguish between a Moment and a Couple.

Feature	Moment	Couple
Definition	Turning effect of a single force about a point.	Two equal and opposite parallel forces producing pure rotation.
Translational Effect	Can produce both translation and rotation.	Produces ONLY rotation (zero translation).
Dependence on Point	Value depends on the point about which it is taken.	Moment of a couple is independent of the point of rotation; it is the same everywhere in the plane.
Resultant Force	Resultant force is not zero.	Resultant force is exactly zero.

Describe the analytical method to find the resultant of a system of coplanar concurrent forces.

To find the resultant of a system of coplanar concurrent forces analytically:

Resolve Forces: Resolve all forces into their horizontal ( $x$ ) and vertical ( $y$ ) components.
Sum Components: Find the algebraic sum of all horizontal components, $\sum F_x$ , and the algebraic sum of all vertical components, $\sum F_y$ .
Magnitude of Resultant: Calculate the magnitude of the resultant force ( $R$ ) using the Pythagorean theorem:
$R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$
Direction of Resultant: Calculate the angle $\theta$ the resultant makes with the x-axis:
$\tan \theta = \left| \frac{\sum F_y}{\sum F_x} \right|$
The quadrant of the resultant is determined by the signs of $\sum F_x$ and $\sum F_y$ .

What is meant by Equilibrium of a system of forces?

A body is said to be in Equilibrium if the resultant of all the forces and moments acting on it is zero. Consequently, the body remains in its state of rest or uniform motion.

For a system of forces, equilibrium implies that:

The forces do not tend to move the body in any direction (Translational equilibrium: $\sum F = 0$ ).
The forces do not tend to rotate the body about any point (Rotational equilibrium: $\sum M = 0$ ).

State Lami's Theorem and its mathematical formulation.

Lami's Theorem states that if three coplanar concurrent forces acting on a point keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

Mathematical Formulation:
Let $P, Q,$ and $R$ be three forces in equilibrium. Let $\alpha$ be the angle between $Q$ and $R$ , $\beta$ be the angle between $P$ and $R$ , and $\gamma$ be the angle between $P$ and $Q$ . Then:
$\frac{P}{\sin \alpha} = \frac{Q}{\sin \beta} = \frac{R}{\sin \gamma}$

What is a Free Body Diagram (FBD)? Explain its significance.

A Free Body Diagram (FBD) is a sketch of an isolated body (or a portion of a body) free from all its surroundings, showing all the external forces and reactive forces acting on it.

Significance:

It is the most important step in solving mechanics problems.
It helps visualize all the forces (like weight, normal reactions, friction, applied forces, and tensions) acting on the body.
It translates physical situations into a mathematical model, allowing the application of equilibrium equations ( $\sum F_x = 0, \sum F_y = 0, \sum M = 0$ ) directly from the diagram.

State the Equations of Equilibrium for Coplanar Concurrent and Coplanar Non-Concurrent force systems.

Coplanar Concurrent Force System: Since all forces pass through a single point, they cannot produce a moment about that point. Thus, only two equations of translational equilibrium are needed:
$\sum F_x = 0$
$\sum F_y = 0$
Coplanar Non-Concurrent Force System: Forces can produce translation as well as rotation. Therefore, three equations are required:
$\sum F_x = 0$
$\sum F_y = 0$
$\sum M = 0$ (Sum of moments about any point in the plane is zero).

Explain the concept of an Equilibrant Force.

An Equilibrant Force is a single force which, when applied to a system of forces, brings the entire system into a state of equilibrium.

Magnitude: The magnitude of the equilibrant is exactly equal to the magnitude of the resultant force of the system.
Direction: Its line of action is exactly opposite (180 degrees) to the resultant force.
Collinearity: It acts along the same line as the resultant but in the opposite sense.
If Resultant $= R$ acting at angle $\theta$ , Equilibrant $= -R$ acting at angle $\theta + 180^\circ$ .

Discuss how the resolution of a force into non-perpendicular components is done.

While usually resolved into rectangular (perpendicular) components, a force can be resolved into two non-perpendicular components using the sine rule of triangles.

If a force $F$ is to be resolved into two components $F_1$ and $F_2$ making angles $\alpha$ and $\beta$ with $F$ respectively on opposite sides:
By forming a force triangle and applying the sine rule:
$\frac{F}{\sin(180^\circ - (\alpha + \beta))} = \frac{F_1}{\sin \beta} = \frac{F_2}{\sin \alpha}$
Since $\sin(180^\circ - \theta) = \sin \theta$ , we get:

$F_1 = \frac{F \sin \beta}{\sin(\alpha + \beta)}$
$F_2 = \frac{F \sin \alpha}{\sin(\alpha + \beta)}$

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