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# Mechanics Maths ## Want to get better grades?

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Mechanics Maths
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics Mechanics is the area of study in physics and mathematics that examines how forces affect a body and its motion. It deals with the movement of physical objects and the relationship between force, mass, and motion. So mechanics studies stationary objects, where the forces acting over them are in equilibrium.

There are two main subsections of mechanics that deal with objects depending on if they are in equilibrium (statics) or in movement (dynamics) . For objects in motion, it is divided into the study of the forces and their effects (dynamics) or the variables of motion (kinematics) .

## What is kinematics?

Kinematics deals with displacement, time, velocity, and acceleration without considering the forces that cause the objects to move.

A simple example of this is the study of a car in motion. We can observe time, displacement, velocity, and acceleration.

A moving car will have a certain displacement . Recording two different moments when moving introduces the concept of time . When we combine the two, displacement over time, we have velocity . If the car isn't moving at a constant rate, the concept of acceleration (the change of velocity) comes in.

## What is dynamics?

Dynamics is the area of mechanics that studies the forces that cause or modify the movement of an object. Dynamics is divided into linear dynamics and rotational dynamics. The first studies an object moving in linear motion, and the second studies objects that rotate around a fixed center, such as a chair in a carousel.

Dynamics works with concepts such as forces, the mass of the object in motion, its momentum (defined as the velocity multiplied by the object's mass), and energy.

Engineering requires you to apply the principles of mechanics from the point of view of kinematics or dynamics. Multiple applications range from the design of airplanes, bridges, cars, and buildings to the development of rockets for space exploration.

## Quantities, units, and assumptions in mechanics

The study of mechanics is linked to quantities, which are the properties you can measure of an object. In an object in motion, the most important properties are the distance an object covers, the time it takes to cover this distance, the speed it has, how the speed changes, and the forces affecting the object.

The quantities to measure use units. Units are standards used for each property we are measuring. Mechanics specifically uses the units for velocity (meters per second or m / s) and forces (Newtons), amongst others.

Another important aspect when dealing with mechanics is the simplification of the systems analyzed. These assumptions allow you to study mechanics by reducing its complexity.

### Physical quantities and units

In trying to understand what laws govern specific systems, we will need to quantify the physical elements that are going to be involved in the system.

Anything we can measure is known as a physical quantity . For example, if I say I weigh 80kg or the ruler is 30cm, you can assume 80kg is my mass, and 30cm is the length of the ruler. Every physical quantity must have two things:

For example, if you say 20 kg of salt, 20 is the numerical value of the salt you have. This is not enough to conclude how much salt you have until the unit kg is added. The kilograms, or kg, is a SI unit - an international standard.

Units are necessary to specify the specific amount of what property of the substance we are measuring.

### assumptions

Applying mathematics to real-life events can be complicated. There are so many variables it can be hard to know where to begin. You start by making the problem as simple as you possibly can.

There are certain things you can ignore, including:

• Air resistance.

• friction

• Energy dissipation.

• mass distribution.

It's helpful to know some keywords that are used for these assumptions. For example, 'smooth surface' means there is no friction present on the surface, or if a particle has a ' negligible mass', it means you can assume its weight is zero.

## Acceleration in kinematics

Remember, kinematics is an area of study that focuses on the movement of objects, disregarding the forces that cause the movements. This part of mechanics explores the concept of motion, and its relationship with time, velocity, and acceleration. The movements of the objects in kinematics can have a constant acceleration or a variable acceleration .

### Constant acceleration and SUVAT equations

Constant acceleration can also be called one-dimensional equations for motion for constant acceleration. This employs the use of SUVAT equations to find the values of any of the variables. SUVAT is an acronym of the variables to study. They are:

s, displacement in meters [m].

u, initial velocity in meters over seconds [m / s].

v, final velocity in meters over seconds [m/s].

a, acceleration in meters over seconds squared [m / s 2 ] .

t, time in seconds [s].

### Variable acceleration

In contrast to constant acceleration, variable acceleration primarily explores motion in objects where acceleration keeps changing. A variable acceleration means a variable velocity.

In mathematics, the formulations found to model the movement of an object are related to a mathematical area of study - differentiation .

A typical example is to use the classical SUVAT formulation to calculate the acceleration from the displacement. The first derivation of the displacement will give you the velocity, and if you derive the velocity, you will obtain the acceleration.

If you are given the SUVAT formulation for the acceleration and want to find the displacement, you apply the inverse operation named integration . Integrating the acceleration will give you the velocity, and if you integrate the velocity, you will obtain the displacement. Here are the equations:  ### Projectiles and parabolic motion

Projectiles and parabolic motion deal with objects projected through the air, describing a parabola during their movement. An example is throwing a ball.

This part of kinematics employs concepts of mathematics such as trigonometry because of the angles involved in the movements of the objects.

## Forces and Newton's laws

Force can change the motion of an object. A straightforward way to describe force is as a pull or a push against an object. Newton's laws of motion and its mathematical expressions are central to how we describe forces every day.

These laws cover three significant ideas: the reciprocity of forces, the forces altering the state of movement of an object, and how mass, acceleration, and force relate to each other.

Another important aspect of the study of forces is how we use them to move objects and the mechanisms you can create to produce or affect them. Two examples of these mechanisms are pulleys and moments produced by a bar.

Forces can also be present when an object has no movement; one example is the force of gravity on you as you remain standing. The study of forces when an object does not move ( in equilibrium ) or change its movement is called statics .

### Newton's laws

Newton came up with three specific laws to describe the motion of an object.

• Newton's first law of motion states that an object continues to be in a state of rest or a state of motion at a constant speed along a straight line unless a force acting over the object changes this.

A ball will roll indefinitely if nothing stops it from moving. In this case, the friction against the air and the ground will cause it to stop.

• Newton's second law of motion states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it. It can be modeled in an equation as: Where f is the force in Newtons, m is the mass in kg, and a is the acceleration in m / s 2 .

• Newton's third law of motion is also called the action and reaction forces law. It states that when a body exerts a force over another, the other body will exert a force equal in magnitude and opposite in direction.

An example is when you push against a hard wall, you will feel a push in the other direction.

### pulleys

A pulley comprises a wheel and a fixed axle, with a groove along the edges to guide a rope or cable. It is not easy to lift heavy objects, so that is where pulleys come in. Put two or more wheels together and run a wheel around them, and there you have an excellent lifting machine. The more pulleys you add to your machine, the more mechanical advantage you have at lifting a load easily. Pulley system lifting a weight, the system has two pulleys and allows a force F to lift a weight against the gravity force mg

### statics

Statics deal with objects at rest and ones that are moving with constant velocity. In this object, forces are in equilibrium, so there is no change to its movement. One example of this is the forces over a building. The building structure is affected by gravity pulling it down, the force is distributed along the building, and the structure reacts to create an equilibrium.

### friction

Friction is the force that resists the rolling and sliding of an object over a surface. Friction is a dissipative force, meaning that it can decrease the velocity of the objects in motion.

### moments

A moment is a force you apply to something multiplied by the distance between the pivot and the force.

When a force is not enough to turn something around, you will need a pivot, too. Pivots and forces have a special relationship - if you push with the same force further away from the pivot, you can turn the item more easily due to a larger moment.

moment = force distance

In a moment, the distance is the perpendicular distance to the point where you apply the force. Force F1 will produce Force F2 thanks to the pivot, and the moment will be equal to force F2 per its distance to the pivot

## Mechanics Maths - Key takeaways

• Mechanics is the area of study of physics and mathematics that deals with how forces affect a body in motion or repose.

• Kinematics is an area of study that focuses on the movement of objects, disregarding the forces that cause the movements.

• Any property we can measure in an object is known as a physical quantity.

• Assumptions help reduce the complexities of real-life applications of mechanics by ignoring certain variables.

• The influence that can change the state of an object (motion or repose) is referred to as force.

• Mass is one significant variable to be considered when exploring the effects of motion in objects, and mass is a central variable in Newton's second law.

• Statics deal with objects at rest and ones that are moving with constant velocity. In this case, the forces acting over the objects are at equilibrium.

• Dynamics, in contrast, is the section that deals with the forces that put the objects in motion.

• Projectiles and parabolic motion study with objects that describe a parabola while moving.

Statics and dynamics.

Mechanics is the area of study of physics and mathematics that deals with how force affects a body. It is concerned with the relationship between force, matter, and motion.

Mathematics is the tool required to solve mechanical problems since you are dealing with specific systems with specific rules that are modelled mathematically.

Mechanics is studied in both maths and physics. It studies the relationship between force, matter and motion (which are topics in physics), and uses mathematical modelling to do this.

## Final Mechanics Maths Quiz

Question

What is the equation for the coefficient of friction?

μ = F / L

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What are pulleys?

A pulley comprises a wheel and a fixed axle, with a groove along the edges to guide a rope or a cable

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Why are Newton's laws important?

They tell us why objects move or sit still, why we are not floating, and informs the mechanism in guns and cars

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Why is force considered a vector quantity?

They are considered vector quantities because they have a magnitude and a direction.

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What is a resultant force?

A resultant force is a single force which is a representation of the vector sum of more than one force

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What are Newton's three laws of motion?

• Newton's first law of motion states that an object continues to be in a state of rest or a state of motion at a constant speed along a straight line unless it is compelled to change that state by a net force acting on it. Net force is the vector sum of all the forces acting on an object.

• Newton's second law of motion states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it.

• Newton's third law states that when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction.

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What is the formula for solving moments?

moment = force distance (from pivot perpendicular to force's line of action).

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What do the letters in SUVAT stand for?

• s = Displacement
• u = initial velocity
• v = final velocity
• a = acceleration
• t = time.

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What is free fall?

It is when an object experiences acceleration due to gravity.

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Give two examples of projectiles.

• A bullet’s movement at the instant it is fired from a gun.

• A car driven off a cliff.

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What is constant acceleration?

It is also called one-dimensional equations of motion for constant acceleration. It deals with all kinematic problems where the acceleration is stable and constant.

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What is a particle in equilibrium?

A particle is said to be in equilibrium when its net force is zero.

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What is variable acceleration?

When acceleration is different between points along its path, it is considered variable acceleration.

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What is an example of variable acceleration?

A car moving in a crowded environment will slow down till it has enough clear space to move forward.

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What are projectiles?

Projectiles deal with objects projected through the air either by being thrown, hit, or fired.

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The path of a projectile.

Trajectory.

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What are kinematic equations?

They are the equations used to describe the motion of objects with constant acceleration.

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What is mechanics?

Mechanics is the area of study in physics and mathematics that examines how forces affect a body and its motion.

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What are the two subsections of mechanics?

Kinematics and dynamics

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How are units relevant to quantities in mechanics?

Units give meaning to the numerical values of measurements. They describe the measured property of the object.

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What assumptions can you make?

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Is friction a force?

Yes, it is a force.

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If the object is static can it be considered in equilibrium?

Yes, it is in equilibrium.

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What is kinematics?

Kinematics is an area of study that focuses on the movement of objects, disregarding the forces that cause the movements.

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What is the acronym for the equations of motion with constant acceleration?

SUVAT

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What is an example of a projectile in parabolic motion?

A ball launched through the air.

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What is the purpose of Newton's laws?

Newton's laws explain how forces affect the movement of objects.

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Which Newton's law describes the force exerted over an object and the change of speed of the object?

Newton's second law.

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What is the equation for Newton's second law of motion?

f = ma

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How do you calculate the moment produced by a force?

The distance multiplied by the force.

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True or false, statics studies objects where velocity changes and the forces are not in equilibrium?

False

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Why do we use modelling assumptions?

Modelling assumptions allow us to simplify real world problems and analyze them using known mathematical techniques.

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State whether the following statement is true or false : A model is often a simplified abstraction of reality.

True

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State whether the following statement is true or false : Modelling assumptions do not affect the validity of the model itself.

False

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What are the common assumptions used when we model gravity in mechanics?

• All objects with mass are attracted towards the Earth

• Earth’s gravity is uniform and acts vertically downwards

• g (gravitational constant) is constant and is taken as 9.8 m/s², unless otherwise stated

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What are the common assumptions used when we model a particle in mechanics?

• Mass of the particle is concentrated at a single point.

• Rotational forces and air resistance can be ignored

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What are the common assumptions used when we model a wire in mechanics?

The wire is treated as a one-dimensional object

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What is/are the common assumption(s) used when we model an inextensible string in mechanics?

Acceleration is the same in objects connected by a taut inextensible string

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In mechanics, what is a bead?

A particle with a hole in it for threading on a wire or string

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In mechanics, what is a lamina?

Object with area but negligible thickness, like a sheet of paper.

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Which of the following is a common assumption when we mention a light object in mechanics?

Mass of the object is zero or negligible.

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Which of the following is a common assumption when we mention a smooth surface in mechanics?

There is no friction between the surface and any object on it.

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In mechanics, what is a peg?

A support from which a body can be suspended or rested.

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What are the common assumptions used when we model a rod in mechanics?

• mass is concentrated along a line

• no thickness

• rigid (does not bend or buckle)

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An ant falls off the top of a building. Is the following an appropriate assumption to model the motion of the ant: the effect of air resistance is negligible.

No.

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An ice puck is hit and it slides across the ice. The ice is assumed to be a smooth surface. What will be the impact of this assumption on the calculation of the motion of the puck?

No effect of friction between the puck and the ice will be taken into account.

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What are the five SUVAT variables ?

s= Displacement, u= Initial velocity, v= Final velocity, a= Acceleration, t= Time taken

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You are provided s= Displacement, u= Initial velocity, v= Final velocity, and asked to find acceleration. Which equation should you use?

v² = u² + 2 as

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You are provided s= Displacement, u= Initial velocity, a= Acceleration, and asked to find the time taken. Which equation should you use?

s = ut + ½at²

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You are provided u= Initial velocity, v= Final velocity, t= Time taken and asked to find displacement. Which equation should you use?

s = ½ (u + v) t

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