Introduction of Mechanics

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1. Scalar and vector quantities

⇒ Scalar Quantity

• A scalar quantity is a physical quantity that has only magnitude (amount) but no direction.
• Scalars are used to describe quantities that are invariant under coordinate transformations, meaning their value remains the same regardless of the coordinate system used to measure them.
• Examples of scalar quantities are
  • Temperature (represented by T, (SI unit is Kelvin (, Base quantity)
  • Time (represented by t, (SI unit is second (s), Base quantity)
  • Mass (represented by m, (SI unit is kilogram (kg), Base quantity)
  • Length (represented by l (SI unit is meter (m), Base quantity)
  • Electric current (represented by I (SI unit is ampere (A), Base quantity)
  • Intensity of light (represented by L (SI unit is candela (cd), Base quantity)
  • Amount of a substance (represented by n (SI unit is mole (mol), Base quantity)
  • Density (represented by , (SI unit is kilogram per meter cube ([math]\text{kgm}^{-3} [/math]), Drive Quantity)
  • Pressure (represented by V (SI unit is meter cube ([math] \text{m}^3 [/math]), Drive quantity)
  • Charge (represented by Q (SI unit is coulomb (C or As), Drive quantity)
• Scalar quantities are often represented by a single number or a simple value, unlike vectors, which have both magnitude and direction.
• Some key properties of scalar quantities are

• Commutativity: Scalars can be added or multiplied in any order.
• Associativity: Scalars can be grouped in any way when adding or multiplying.
• Distribution: Scalars can be distributed over addition and multiplication.

⇒ Vector Quantity

• A vector quantity is a physical quantity that has both magnitude (amount) and direction.
• Vectors are used to describe quantities that have both size and direction, and are often represented graphically as arrows in a coordinate system.

Representation

Figure 1 A vector representation

• Examples of vector quantities include:
  • Displacement (represented by d (SI unit is meter (m), Base Quantity)
  • Velocity (illustrated by v (SI unit is meter per second ([math]\text{m.s}^{-1}[/math]), Drive quantity)
  • Acceleration (determined by a (SI unit is meter per second per second ([math] \text{m.s}^{-2} [/math]), Drive quantity)
  • Force, Weight (showed by F / W (same SI unit is newton (N (, Drive quantity)
• Vector quantities are often represented mathematically using boldface letters.
• Some key properties of vector quantities:
  • Magnitude (length): The size of the vector.
  • Direction: The direction in which the vector points.
  • Addition: Vectors can be added graphically or mathematically.
  • Scalar multiplication: Vectors can be multiplied by a number, which changes their magnitude.
  • Dot product (scalar product): The product of two vectors that results in a scalar.
  • Cross product (vector product): The product of two vectors that results in another vector.

⇒ Resultant Vector

• The resultant vector, also known as the net vector or total vector, is the vector that results from the combination of two or more vectors.
• It is the vector that represents the overall effect of all the vectors acting together.

Figure 2 R represent the resultant vector

• Resultant vectors are used in various applications, such as:
  • Force vectors: To find the net force acting on an object.
  • Velocity vectors: To find the net velocity of an object.
  • Acceleration vectors: To find the net acceleration of an object.

2. The Addition of vectors

• The addition of vectors is a fundamental operation in vector mathematics.
• It involves combining two or more vectors to produce a resultant vector. Here are the key aspects of vector addition:

Figure 3 Addition of vectors

Figure 4 graphical representation of rambler walk.

[math] |AC|^2 = |AB|^2 + |BC|^2 \\
|AC|^2 = |8 \, \text{km}|^2 + |6 \, \text{km}|^2 \\
|AC|^2 = |64 + 36| * \text{km}^2 \\
|AC|^2 = 100 \, \text{km}^2 \\
\text{Taking the square root on both sides:} \\
\sqrt{|AC|^2} = \sqrt{100 \, \text{km}^2} \\
\text{Then,} \, |AC| = 10 \, \text{km}[/math]

Directions can be calculated on a bearing from due north; this is the angle θ, in Figure 4, which is also the angle θ, in the triangle.

The displacement is 10km on a bearing of 53°

3.The resolution of vectors

• Resolving a vector Split it into two mutually perpendicular components that add up to the original vector.

  
  Figure 5 a passenger pulling his wheelie bag

• Figure 5 shows a passenger pulling his wheelie bag at the airport. He pulls the bag with a force, F, part of which helps to pull the bag forward and part of which pulls the bag upwards, which is useful when the bag hits a step.
• The force, F, can be resolved into two components:
• A horizontal component F_h. 
• A vertical component F_v.
• A vector can be resolved into any two components that are perpendicular, but resolving a vector into vertical and horizontal components is often useful, due to the action of gravity.
• Using the laws of trigonometry:
• 
• The vertical component of the force is
• [math]F_v= F sinθ [/math]
• The horizontal component of the force is
• [math] F_h = F cosθ [/math]

⇒ An inclined plane

• An inclined plane is a flat surface that is tilted at an angle to the horizontal. It is a fundamental concept in physics.
• An inclined plane is used to describe a variety of phenomena, such as
  • Motion on an inclined plane: Objects can move up or down an inclined plane, and the force of gravity acts along the surface of the plane.
  • Forces on an inclined plane: The force of gravity, normal force, and frictional force act on an object on an inclined plane.
  • Inclined plane in mechanics: Inclined planes are used to change the direction of motion, increase or decrease the force of gravity, and to create a mechanical advantage.
  • Real-world applications: Inclined planes are used in various real-world applications, such as:
  • Ramps: Used to load and unload heavy objects.
  • Stairs: A series of inclined planes that connect different levels.
  • Roofs: Inclined planes that provide protection from the elements.
  • Highways: Inclined planes that connect different elevations.
  • Figure 6 shows a car at rest on a sloping road. The weight of the car acts vertically downwards, but here it is useful to resolve the weight in directions parallel (||) and perpendicular (⊥) to the road.

    
    Figure 6 a car at rest on a sloping road.

  • The component of the weight parallel to the road provides a force to accelerate the car downhill.
  • The component of the weight acting along the slope is
  • The component of the weight acting perpendicular to the slope is
  • W _||=W sinθ
  • The component of the weight acting perpendicular to the slope is
  • W _⊥=W cosθ

 ⇒ Forces in equilibrium 

• Forces in equilibrium are forces that are balanced, meaning they cancel each other out. When the net force on an object is zero, the object is said to be in equilibrium.
• The Principle of Moments states that when a body is balanced, the total clockwise moment about a point equals the total anticlockwise moment about the same point.
• Here are some key points about forces in equilibrium (condition for equilibrium):
  • First Condition: The sum of all forces acting on an object must be zero.
  • Second Condition: The sum of all torques (rotational forces) acting on an object must be zero.
  • Types of Equilibrium:
    – Static Equilibrium: The object is at rest and remains at rest.
    – Dynamic Equilibrium: The object is moving, but the net force is zero.
  • Examples:
    • A book on a table (static equilibrium)
    • A car moving at constant velocity (dynamic equilibrium)
    • In Figure 8 the motorcyclist has a weight of 800 N and the bike has a weight of 2500 N – a total of 3300 N. Those forces are balanced by the two contact forces exerted by the road, so the bike remains at rest. This is an example of the application of Newton’s first law of motion.

      
      Figure 8 the motorcyclist on road

    • This example is easy to solve as all of the forces act in one direction.
    • Shows all the forces acting on the stationary car first shown in Figure 6. Newton’s first law of motion applies to this situation too, although the forces do not all act in a straight line.
    • However, the vector sum of the forces must still be zero.
    • Three forces act on the car: (1) its weight, (2) a normal reaction perpendicular to the road, and (3) a frictional force parallel to the road.
    • Three weights are set up in equilibrium. A central weight, W3, is suspended by two light strings that pass over two frictionless pulleys. These strings are attached to the weights W1 and W2. Since point O is stationary, the three forces acting on that point, T1, T2 and W3, must balance. The tensions T1 and T2 are equal to W1 and W2 respectively.

Figure 9 In this example θ₁ = 60° and θ₂ = 30°

• We can demonstrate the balance of forces by resolving the forces acting on point O vertically and horizontally.
• The forces must balance in each direction.
• So, resolving horizontally:
• T₁  sin θ₁ = T₂  sin θ₂
• And resolving vertically:
• T₁  cos θ₁ =T₂  cos θ₂ = W₃
• We can check using the example numbers in the diagram:
• T₁  sin θ = 3N sin 60 = 2.6N
• T₂  sin θ = 5N sin 30 = 2.5N
• So, the horizontal components balance to within a reasonable experimental error. Resolving vertically:
• 2.5N + 2.6N = 5.8N

4.Turning moments

⇒Introducing moments

• Turning moment, also known as torque, is a measure of the rotational force that causes an object to rotate or turn. (Also known as moment of a force about a point)
• It is a vector quantity.
• It’s unit is typically measured in newton-meters (N·m)
• Formula:
  Turning momentum=Forced appiled * distance prependicular from the piovt
  
  •  is the turning moment (torque)
  •  is the distance from the axis of rotation to the point where the force is applied
  •  is the force applied
• A pivot is a point or axis around which something rotates or turns.
• It is a fixed point that remains stationary while other parts move or rotate around it.
• An object is in equilibrium, when the sum of the forces = 0 and the sum of the turning moments = 0.
• Newton’s first law of motion that an object will remain at rest if the forces on it balance.
• However, if the body is to remain at rest without translational movement, or rotation, then the sum of the forces on it must balance, and the sum of the moments on the object must also balance.
• Another way of expressing Newton’s first law is to say that when the vector sum of the forces adds to zero, a body will remain at rest or move at a constant velocity.

⇒ Centre of mass

• The center of mass (COM) is the point where the total mass of an object or system is concentrated.
• It is the point where the object or system would balance if it were suspended from that point.
• Properties of Center of Mass:
  • Unique point: The COM is a unique point for any object or system.
  • Mass concentration: The COM is where the total mass of the object or system is concentrated.
  • Balance point: The object or system would balance if suspended from the COM.
  • Invariance: The COM remains unchanged under rigid body transformations (translation and rotation).
• Importance of Center of Mass:
  Motion analysis: COM is used to describe the motion of objects and systems.
  Stability analysis: COM is used to determine the stability of objects and systems.
  Collision analysis: COM is used to determine the outcome of collisions.
  Robotics and computer graphics: COM is used to simulate and animate objects.
• Figure 10a shows a rod in equilibrium on top of a pivot.
• Gravity acts equally on both sides of the rod, so that the clockwise and anticlockwise turning moments balance.
• This rod is equivalent (mathematically) to another rod that has all of the mass concentrated into the midpoint, this is known as the centre of mass.
• In Figure 10b the weight acts down through the pivot, and the turning moment is zero.
• The centre of mass is the point in a body around which the resultant torque due to the pull of gravity is zero.
• This means that you can always balance a body by supporting it under its centre of mass. In figure 10c the tapered block of wood lies nearer to the thicker end.

Figure 10 Shows centre point of a rod where total weight of the body acting downward

⇒ Moments in action

• Moments in action refer to the turning effect of a force around a pivot or fulcrum.
• Here are some key aspects of moments in action.
  Clockwise and counterclockwise moments: Depending on the direction of rotation.
  Moment arm: The distance from the pivot to the point where the force is applied.
  Force and moment: The product of the force and moment arm.
  Torque and rotation: The relationship between moments and rotation.
• Moments in action are used in various applications, such as:
  Mechanical advantage: Moments are used to gain a mechanical advantage in tools and machines.
  Rotation and torque: Moments are used to understand and calculate rotation and torque in mechanical systems.
  Stability and balance: Moments are used to analyze and determine the stability and balance of objects and systems.
• Figure 11 shows the action of a force to turn a spanner, but the line of action lies at an angle of 45° to the spanner.
• How do we determine the turning moment now?
• This can be done in two ways.
  • First, a scale drawing shows that the perpendicular distance between the line of the force and the pivot is 0.21m.

    
    Figure 11 The action of a force to turn a spanner

  • So, the turning moment is:
  • 100N * 0.21m = 21N.m
  • Secondly, the turning moment can be calculated using trigonometry, because the perpendicular distance is (the length of the spanner) × sin θ.
  • Moment=F * l * sin θ
    Moment = 100N * 0.3m * sin 45 = 21Nm

 ⇒ Couples

• A couple is a pair of forces that are equal in magnitude, opposite in direction, and act on different points in an object.
• Couples tend to produce rotation or torque, and are commonly used in mechanics and engineering to simplify complex systems.
• Some key aspects of couples:
  • Equal and opposite forces: The two forces that make up a couple are equal in magnitude and opposite in direction.
  • Different points of application: The forces act on different points in the object.
  • Moment of a couple: The product of the force and the distance between the points of application.
• Couple calculated by using:
• Couple = Force * prependicular distance between line of action
• 
• Examples of couples include:

  Scissors: The two blades of scissors form a couple, with each blade exerting an equal and opposite force.
  Wheel and axle: The forces exerted on the wheel and axle form a couple, allowing the wheel to rotate.
  Gears: The teeth of gears form couples, transmitting torque and rotation.

• A couple when you turn a steering wheel,
• The two forces turn the wheel but exert no translational force.
• If you take one hand off the wheel (which you should do only to change gear), you can still exert a couple, but the second force is applied by the reaction from the steering wheel (see Figure 12).

Figure 12 Turn a steering wheel