Composition and Resolution of Forces

Contents

**Define and classify the term Mechanics** 1

**State different Laws of Mechanics** 4

**Describe the Fundamental and Derived Units** 5

**Recall the Dimensional Homogeneity** 6

**Recall the Scalar and Vector quantities** 7

**State various Idealization in Mechanics** 8

**List and recall various Forces with examples** 10

**Describe the composition of two force systems** 11

**Recall the method of following i. Resolution of Forces ii. Composition of number of forces** 12

**Define applied and Non-applied Forces** 13

**Explain the concept of Free Body Diagram** 14

**Describe equilibrium of connected bodies** 16

**State and prove Varignon’s Theorem** 18

**Define Couple and list its properties** 19

**Recall the Resultant of non-concurrent Force system** 20

**Define and classify the term Mechanics**

Mechanics is a branch of physics that deals with the study of motion and its causes. It is concerned with the behavior of objects under the influence of various forces and their interactions with each other. The term “mechanics” can be defined as the study of how objects move and the forces that cause them to do so.

There are two main branches of mechanics: classical mechanics and quantum mechanics.

Classical Mechanics: This branch of mechanics deals with the motion of macroscopic objects that are large enough to be observed directly. It includes the study of motion of objects under the influence of forces, such as gravity, friction, and tension. This branch of mechanics is used to explain the motion of objects in everyday life, such as the motion of a pendulum or the movement of a car.

Quantum Mechanics: This branch of mechanics deals with the behavior of objects at the atomic and subatomic scale. It is concerned with the study of the behavior of particles, such as electrons and photons, and how they interact with each other. Quantum mechanics uses mathematical models to explain the behavior of particles, such as the wave-particle duality and the Heisenberg uncertainty principle.

In summary, mechanics is a branch of physics that deals with the study of motion and its causes, and it can be divided into two main branches: classical mechanics and quantum mechanics.

**Define the following terms i. Mass and weight ii. Space and time iii. Velocity and acceleration iv. Momentum and continuum v. Rigid body of particles**

i. Mass and weight: Mass is a measure of the amount of matter in an object, and it is expressed in units of kilograms (kg) or grams (g). Mass is an intrinsic property of an object, meaning that it is determined by the properties of the object itself, and does not depend on the location or other external factors.

Weight, on the other hand, is a force that acts on an object due to gravity. It is equal to the mass of an object multiplied by the acceleration due to gravity (g). The weight of an object is proportional to its mass, but it is also affected by the strength of the gravitational field in which it is located. Weight is expressed in units of newtons (N).

ii. Space and time: Space refers to the three-dimensional region in which objects can exist and move. Time, on the other hand, is the dimension in which events occur in a linear sequence. Space and time are interrelated and form a four-dimensional space-time continuum.

iii. Velocity and acceleration: Velocity is a vector quantity that represents the rate of change of an object’s position in space. It is expressed in units of meters per second (m/s) and is a measure of the speed and direction of an object’s motion.

Acceleration, on the other hand, is the rate of change of velocity. It represents the change in the speed or direction of an object’s motion over time and is expressed in units of meters per second squared (m/s^{2}). An object is said to be accelerating when its velocity changes, either in magnitude (speed) or direction.

iv. Momentum and continuum: Momentum is a vector quantity that represents the movement of an object. It is equal to the product of an object’s mass and velocity and is expressed in units of kilograms-meters per second (kg-m/s). Momentum is conserved, meaning that the total momentum of a system of objects remains constant if there are no external forces acting on it.

Continuum refers to the idea that a material or a physical system is considered to be a continuous entity, with no breaks or gaps between its parts. In mechanics, the continuum idea is applied to materials, such as solids, liquids, and gases, and it is used to explain the behavior of objects under different conditions.

v. Rigid body of particles: A rigid body is a solid object that does not deform or change its shape when subjected to external forces. It can be modelled as a collection of particles that are connected to each other and move as a single unit. The motion of a rigid body is described in terms of its position, orientation, and angular velocity. The concept of a rigid body is important in mechanics, as it is used to explain the behavior of objects in a variety of real-world applications, such as the motion of vehicles, machines, and other structures.

**State different Laws of Mechanics**

There are several laws of mechanics that form the foundation of the study of motion and its causes. These laws are:

- Newton’s First Law of Motion: Also known as the law of inertia, this law states that an object at rest tends to stay at rest, and an object in motion tends to stay in motion, with a constant velocity, unless acted upon by an external force.
- Newton’s Second Law of Motion: This law states that the acceleration of an object is directly proportional to the net force acting on the object, and inversely proportional to its mass. Mathematically, it can be expressed as F = ma, where F is the net force, m is the mass of the object, and a is its acceleration.
- Newton’s Third Law of Motion: This law states that for every action, there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B will exert an equal and opposite force on object A.
- Law of transmissivity of forces : The law of transmissibility of forces is a principle in physics that states that the effect of a force on a body is independent of where the force is applied. In other words, if a force is applied to a rigid body at a certain point, it will produce the same effect as if it were applied to any other point along the same line of action.
- Law of Gravitation: This law states that every object in the universe is attracted to every other object by a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This law was first stated by Sir Isaac Newton and is a fundamental principle of classical mechanics.

These laws of mechanics form the foundation of the study of motion and its causes, and they are used to explain a wide range of physical phenomena, from the motion of objects in everyday life to the behaviour of objects in space.

**Describe the Fundamental and Derived Units**

In physics, units are used to quantify physical quantities and make measurements. There are two types of units: fundamental units and derived units.

- Fundamental Units: Fundamental units are the basic units used to measure physical quantities that cannot be expressed in terms of other units. There are seven fundamental units in the International System of Units (SI): meter (m) for length, kilogram (kg) for mass, second (s) for time, mole (mol) for amount of substance, kelvin (K) for temperature, ampere (A) for electric current, and candela (cd) for luminous intensity.
- Derived Units: Derived units are units that can be expressed in terms of fundamental units. For example, velocity is a derived unit, and it is expressed in terms of meters per second (m/s). Other examples of derived units include force (newton, N), energy (joule, J), power (watt, W), and pressure (pascal, Pa).

Derived units are important because they allow us to express more complex physical quantities in terms of the fundamental units. This makes it easier to compare and manipulate physical data, and to relate different physical quantities to each other.

In addition to the SI units, there are other systems of units used in physics, such as the cgs (centimeter-gram-second) system and the imperial system. However, the SI system is the most widely used and internationally recognized system of units in physics, and it is used for all scientific and engineering calculations.

**Recall the Dimensional Homogeneity**

Dimensional homogeneity is a property of physical quantities in physics that states that each term in an equation must have the same dimensions, or units. This is an important concept in physics because it allows us to check the consistency of equations and to make sure that the units in an equation are consistent with each other.

For example, if we have an equation that relates distance, time, and velocity, such as v = d/t, each term in the equation must have the same dimensions. The dimension of velocity is length per time, so distance must have the dimension of length, and time must have the dimension of time. If this condition is not satisfied, the equation is not dimensional homogeneous and is therefore not physically meaningful.

Dimensional homogeneity is also used in the development of physical theories and models. It helps to ensure that the equations used in a theory are consistent with each other, and that the units used in the equations are consistent with the units used to describe the physical quantities being modelled.

In summary, dimensional homogeneity is an important concept in physics because it helps to ensure that equations and theories are consistent and physically meaningful. By checking the dimensional homogeneity of equations, we can make sure that the units used in an equation are consistent with each other, and that the equations accurately describe the physical quantities they are meant to represent.

**Recall the Scalar and Vector quantities**

In physics, physical quantities can be classified into two types: scalar quantities and vector quantities.

- Scalar Quantities: Scalar quantities are physical quantities that have magnitude, but not direction. Examples of scalar quantities include mass, temperature, time, and distance. The magnitude of a scalar quantity can be expressed in terms of a numerical value and a unit, but it does not have a direction associated with it.
- Vector Quantities: Vector quantities are physical quantities that have both magnitude and direction. Examples of vector quantities include velocity, acceleration, force, and displacement. The magnitude of a vector quantity can be expressed in terms of a numerical value and a unit, and it also has a direction associated with it. This direction can be specified using a coordinate system or using angle and magnitude.

It is important to understand the difference between scalar and vector quantities in physics because different mathematical operations are used to manipulate these different types of physical quantities. For example, scalar quantities can be added and subtracted using simple arithmetic, while vector quantities require more complex vector operations, such as vector addition and vector subtraction.

In summary, scalar quantities are physical quantities that have magnitude, but not direction, while vector quantities have both magnitude and direction. Understanding the difference between these two types of physical quantities is important in physics because it affects how these quantities are mathematically manipulated and how they are used to describe physical systems.

**State various Idealization in Mechanics**

Idealization is a concept in mechanics that refers to the process of simplifying a physical system in order to make it easier to analyze and understand. In mechanics, idealisation is often used to simplify real-world systems into simpler, more idealised models.

There are several types of idealisation that are commonly used in mechanics:

- Idealised shapes: In mechanics, objects are often idealised as simple shapes, such as spheres, cylinders, and blocks. This simplifies the calculations and analysis of the physical system, and allows for more straightforward solutions to be obtained.
- Idealised material properties: Material properties such as elasticity and strength are often idealised in mechanics. For example, a material may be idealised as a perfect spring, which has a constant elasticity, or as a perfectly rigid body, which has no elasticity at all.
- Idealised forces: Forces such as friction, air resistance, and other forms of drag are often idealised in mechanics. For example, friction may be idealised as a constant force that acts in the opposite direction to the velocity of an object.
- Idealised systems: Complex physical systems, such as machines and vehicles, are often idealised into simpler models, such as systems of masses and springs, or systems of rigid bodies connected by hinges. This simplifies the calculations and analysis of the system and allows for more straightforward solutions to be obtained.

Idealization is an important concept in mechanics because it allows us to simplify complex physical systems into more manageable models. By idealising a physical system, we can make it easier to analyze, understand, and predict the behavior of the system. However, it is important to remember that idealisation is not always accurate and that real-world systems may behave differently than their idealised models.

**Define the term Force**

Force is a concept in physics that describes an influence that changes or has the potential to change the motion of an object. In other words, a force is something that can cause an object to start moving, change direction, or change speed. Forces can be either contact forces, which require physical contact between two objects, or non-contact forces, which act at a distance and do not require physical contact.

Forces are often described in terms of their magnitude, direction, and point of application. The magnitude of a force is a scalar quantity that describes the strength of the force, while the direction of the force is a vector quantity that describes the direction in which the force is acting. The point of application is the specific location on the object where the force is being applied.

There are several types of forces in physics, including gravitational forces, electromagnetic forces, and the forces associated with the strong and weak nuclear forces. In mechanics, the most commonly studied forces are gravitational forces and contact forces, such as tension, friction, and normal force.

In summary, force is a concept in physics that describes an influence that has the potential to change the motion of an object. Forces are characterized by their magnitude, direction, and point of application, and can be either contact forces or non-contact forces. Understanding the concept of force is fundamental to the study of mechanics and is essential for understanding the motion of objects and the behavior of physical systems.

**List and recall various Forces with examples**

There are several types of forces in physics, including:

- Gravitational force: This is a non-contact force that acts between all objects in the universe. It is proportional to the mass of the objects and the distance between them, and always acts in the direction of the center of mass of the objects. For example, the force of gravity between the Earth and the Moon is what keeps the Moon in orbit around the Earth.
- Tension force: This is a contact force that acts in a rope, cable, or string when it is stretched or pulled. Tension acts along the length of the rope, cable, or string and pulls in opposite directions at the ends. For example, when you pull a rope, the tension force acts along the length of the rope and pulls in opposite directions at the ends.
- Friction force: This is a contact force that opposes motion between two objects that are in contact. Friction is proportional to the normal force and the coefficient of friction, which is a material property that depends on the roughness of the surfaces in contact. For example, when you try to slide a book across a table, friction opposes the motion and makes it harder to move the book.
- Normal force: This is a contact force that acts perpendicular to the surface of an object and supports the object against gravitational force or any other external forces. The normal force is proportional to the weight of the object and acts in the upward direction. For example, the normal force supports a book resting on a table against the gravitational force acting on it.
- Electrostatic force: This is a non-contact force that acts between electrically charged objects. It is proportional to the product of the charges and the inverse square of the distance between them, and acts along the line connecting the charges. For example, the electrostatic force between two positively charged objects repels each other, while the force between a positively charged object and a negatively charged object attracts each other.
- Magnetic force: This is a non-contact force that acts between magnets and moving charges. It is proportional to the product of the current and the magnetic field, and acts perpendicular to both the current and the magnetic field. For example, the magnetic force acts between a bar magnet and a wire carrying an electric current, and can be used to generate electricity.

These are some of the most commonly studied forces in physics. The study of forces and their interactions is a fundamental aspect of mechanics and is essential for understanding the motion of objects and the behavior of physical systems.

**Describe the composition of two force systems**

A force system is a collection of forces that act on an object. The composition of two force systems refers to the process of adding two or more force systems together to determine the total effect on an object.

There are two main methods for composing force systems: graphical and analytical.

- Graphical method: The graphical method involves representing each force as a vector and drawing the vectors on a coordinate system. The vectors are then added together, either by the head-to-tail method or by the parallelogram method, to determine the total effect of the force system. The result is a vector that represents the net force, or the vector sum of the individual forces.
- Analytical method: The analytical method involves representing each force as an equation or a numerical value, and adding the equations or values together to determine the net force. The result is a scalar or a vector quantity that represents the net force.

Regardless of the method used, the composition of force systems is an important concept in mechanics because it allows us to determine the total effect of multiple forces acting on an object. This is important for understanding the motion of objects and the behavior of physical systems, and is a key aspect of the study of mechanics.

**Recall the method of following i. Resolution of Forces ii. Composition of number of forces**

- Resolution of forces: The resolution of forces is a method for breaking down a single force into its component parts. This is done by representing the force as a vector and resolving it into its components along two or more mutually perpendicular axes. The components can then be used to understand the effect of the force on an object.

For example, if a force is acting on an object at an angle, it can be resolved into its horizontal and vertical components. This allows us to understand how the force is affecting the object in different directions, and can help us determine the net effect of the force on the object’s motion.

- Composition of a number of forces: The composition of a number of forces refers to the process of adding two or more forces together to determine the net effect of the forces on an object. There are two main methods for composing forces: graphical and analytical.

The graphical method involves representing each force as a vector and adding the vectors together, either by the head-to-tail method or the parallelogram method. The result is a vector that represents the net force, or the vector sum of the individual forces.

The analytical method involves representing each force as an equation or a numerical value and adding the equations or values together to determine the net force. The result is a scalar or vector quantity that represents the net force.

Regardless of the method used, the composition of forces is an important concept in mechanics because it allows us to determine the total effect of multiple forces acting on an object. This is important for understanding the motion of objects and the behavior of physical systems, and is a key aspect of the study of mechanics.

**Define applied and Non-applied Forces**

Applied forces and non-applied forces are two types of forces that can act on an object in mechanics.

- Applied forces: Applied forces are forces that are deliberately imposed on an object by an external source. Examples of applied forces include pushing, pulling, and lifting. These forces are often easy to observe and measure, and play a major role in determining the motion of an object.
- Non-applied forces: Non-applied forces, also known as reactive forces, are forces that result from the interaction of an object with its environment. Examples of non-applied forces include gravitational force, friction, and normal force. These forces are often more difficult to observe and measure than applied forces, but are no less important in determining the motion of an object.

Both applied and non-applied forces play a crucial role in determining the motion of an object, and understanding the relationship between these forces is an important aspect of the study of mechanics. Understanding the distinction between applied and non-applied forces is also important in many practical applications, such as in the design of machines, vehicles, and structures, where the effects of forces must be taken into account to ensure the safety and performance of these systems.

**Explain the concept of Free Body Diagram**

A free body diagram (FBD) is a graphical representation of the forces acting on a body in a given situation. It is a powerful tool for analyzing the motion of objects and for understanding the behavior of physical systems.

An FBD is a simple diagram that shows the body in question and the forces acting on it. The forces are represented as arrows pointing in the direction of their effect on the body, and are labelled with their magnitude and direction.

The main purpose of an FBD is to simplify the problem by considering only the forces that are directly acting on the body. All other forces and influences are ignored, allowing the analyst to focus on the key factors that are affecting the body’s motion.

To create an FBD, it is important to first identify all of the forces that are acting on the body, including both applied and non-applied forces. The forces can then be represented as vectors, with their direction and magnitude shown on the diagram.

Once the FBD has been created, it can be used to determine the net force acting on the body, which is the vector sum of all of the forces acting on it. This information can then be used to determine the body’s acceleration, velocity, and position over time.

The use of FBDs is an important aspect of the study of mechanics, and is widely used in engineering, physics, and other fields where the motion of objects and the behavior of physical systems is of interest. It is a powerful tool for visualizing and analyzing the forces acting on an object, and for understanding the complex interactions between forces, motion, and energy.

**State Lami’s Theorem**

Lami’s theorem is a fundamental concept in mechanics that deals with the forces acting on a system of particles. It states that the total force acting on a system of particles is equal to the sum of the forces acting on each particle, assuming that the forces are applied at the same point in space.

The theorem is based on the idea that the forces acting on a system of particles can be decomposed into their components, and that the components can be treated as separate forces acting on each particle in the system. The total force acting on the system is then equal to the vector sum of the forces acting on each individual particle.

Lami’s theorem is particularly useful in the study of rigid bodies, where the forces acting on the body can be complex and difficult to understand. By using the theorem, the forces can be decomposed into their components, making it easier to analyze the motion of the body and to determine the net force acting on it.

The theorem is also useful in the study of particle dynamics, where it can be used to determine the forces acting on individual particles in a system and to study the interactions between particles.

In summary, Lami’s theorem is a fundamental concept in mechanics that states that the total force acting on a system of particles is equal to the sum of the forces acting on each particle, assuming that the forces are applied at the same point in space. It is a useful tool for understanding the behavior of physical systems and for analyzing the motion of objects.

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**Describe equilibrium of connected bodies**

Equilibrium of connected bodies refers to a state in which the forces acting on a system of interconnected bodies are balanced, resulting in no net movement of the system as a whole. In other words, the net force acting on the system is equal to zero, and the net torque (or rotational force) is also equal to zero.

For a system of connected bodies to be in equilibrium, the following two conditions must be satisfied:

- Force balance: The net force acting on the system must be equal to zero. This means that the sum of all the forces acting on the system must add up to zero, so that the bodies remain at rest or in uniform motion.
- Torque balance: The net torque acting on the system must also be equal to zero. This means that the sum of all the torques acting on the system must add up to zero, so that the bodies do not rotate about any axis.

To determine the conditions for equilibrium, one must first identify the forces acting on the system, and then perform a force balance and torque balance analysis. This involves calculating the net force and net torque acting on the system, and checking if both are equal to zero.

It’s worth noting that a system of connected bodies can be in equilibrium even if some of the bodies are accelerating, as long as the net force acting on the system is zero. This type of equilibrium is called dynamic equilibrium.

In summary, equilibrium of connected bodies refers to a state in which the forces and torques acting on a system of interconnected bodies are balanced, resulting in no net movement or rotation of the system. A system is in equilibrium if the net force and net torque acting on the system are both equal to zero.

**Define the term Moment**

The term “moment” in mechanics refers to the rotational force or torque applied to an object, which causes it to rotate around an axis. A moment is a measure of the ability of a force to rotate an object, and it is defined as the product of the force and its perpendicular distance from the axis of rotation.

The unit of moment is the Newton-meter (Nm), which is equivalent to the Joule (J), the standard unit of energy. A moment of 1 Nm is equivalent to a force of 1 N acting at a distance of 1 meter from the axis of rotation.

Moments can be caused by a variety of forces, such as gravitational forces, electromagnetic forces, and frictional forces. In mechanics, moments play a crucial role in the analysis of rotating systems, as they are used to describe the rotational motion and stability of objects.

Moments can also be analyzed mathematically, using the concepts of vector calculus and matrix algebra. The mathematical treatment of moments enables engineers to predict the behavior of rotating systems under different conditions, and to design and optimize components for various applications.

In summary, the term “moment” in mechanics refers to the rotational force or torque applied to an object, which causes it to rotate around an axis. A moment is a measure of the ability of a force to rotate an object, and it is defined as the product of the force and its perpendicular distance from the axis of rotation.

**State and prove Varignon’s Theorem**

Varignon’s theorem, also known as Varignon’s principle, states that the moment of a force about a point is equal to the vector sum of the moments of its components about the same point. In other words, the theorem states that the moment of a force about a point is independent of the choice of the point of application of the force.

The theorem is named after Pierre Varignon, a French mathematician who first stated the theorem in the late 17th century. The theorem is widely used in mechanics, engineering, and physics, as it provides a convenient method for finding the moment of a force about a point, without having to calculate the moment of each component of the force individually.

The theorem can be stated mathematically as follows:

Let F be a force, and let A and B be two points. Then the moment of F about A is equal to the sum of the moments of its components about B. That is:

M_{A }= M_{B1} + M_{B2 }

where M A is the moment of F about A, M_{B1} is the moment of the x-component of F about B, and M_{B2 } is the moment of the y-component of F about B.

The theorem can be proved by considering the moments of the components of the force F about A and B. The moment of a force about a point is defined as the product of the force and its perpendicular distance from the point. Since the x and y components of a force are perpendicular, their moments about A and B are equal. Hence, the moment of F about A is equal to the sum of the moments of its components about B.

In summary, Varignon’s theorem states that the moment of a force about a point is equal to the vector sum of the moments of its components about the same point. The theorem provides a convenient method for finding the moment of a force about a point, without having to calculate the moment of each component of the force individually, and it is widely used in mechanics, engineering, and physics.

**Define Couple and list its properties**

A couple is a special type of force system that consists of two equal and opposite forces acting on a body. These two forces are separated by a perpendicular distance, also known as the moment arm. The moment arm creates a rotational effect on the body, which is known as the moment of a couple.

Properties of a couple include:

- Magnitude: The magnitude of a couple is equal to the product of one of the forces and the moment arm.
- Direction: The direction of a couple is perpendicular to the plane defined by the two forces and the moment arm.
- Sense: The sense of a couple can be determined by the right-hand rule, which states that if the thumb of your right hand points in the direction of one of the forces, then the fingers will curl in the direction of the moment.
- Resultant: A couple is a non-collinear force system, meaning that the forces do not lie along the same line. As a result, a couple has no resultant force, meaning that it does not produce any linear motion of the body.
- Equilibrium: A body is said to be in equilibrium under the action of a couple if the net moment about any point is zero.

In conclusion, a couple is a fundamental concept in Mechanics, and its properties and theorems are essential for analyzing rotational motion and equilibrium of bodies.

**Recall the Resultant of non-concurrent Force system**

The resultant of a non-concurrent force system is the single force that can replace the original system, and produce the same effect on a body as the original system. In other words, it is the sum of all the forces in the system and represents the net force acting on the body.

The calculation of the resultant of a non-concurrent force system involves vector addition of all the forces in the system. This can be done graphically using a vector diagram, where the vectors representing the individual forces are added head-to-tail to produce the resultant vector. Alternatively, the calculation can be done mathematically by determining the components of each force in a coordinate system and then adding the components.

It’s important to note that the resultant of a non-concurrent force system can also be used to determine the conditions for equilibrium. If a body is in equilibrium, it means that the net force acting on the body is zero, and the body is either at rest or moving at a constant velocity. In other words, the resultant force must be zero for a body to be in equilibrium.

In conclusion, the concept of the resultant of a non-concurrent force system is crucial for analyzing the motion and stability of a body under the influence of multiple forces. Understanding the methods for determining the resultant and using it to analyze equilibrium conditions is essential for understanding the principles of Mechanics.