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<<< Displacement and Force in Two Dimensions >>>

Vectors


If we apply two forces on an object and these forces are along one of the axes [X Y], then we say the forces are in one dimension.
If we apply two forces on an object and each force is on a different axis [X,Y], then we say the forces are in two dimensions.

Physical Quantities

Scalar Quantities

Vector Quantities

Time - Mass - Distance

Force - Velocity - Displacement - Acceleration



Resultant of Vectors in Two Dimensions
Finding the resultant of two or more vectors graphically
Move one vector until its tail coincides with the head of the first vector (maintaining magnitude and direction)
Then draw a line from the tail of the first vector to the head of the second vector - this is the required resultant

This method is used to find the resultant of more than two vectors
The order of moving vectors doesn't affect the final result


Resultant of Vectors in Two Dimensions
Calculating the resultant of two vectors mathematically
If the two vectors are perpendicular, we follow this method as shown in the diagram

Move one vector until its tail coincides with the head of the other
Draw a line from the tail of the first to the head of the last vector. Note the resultant is the hypotenuse of a right triangle, so by Pythagoras' theorem:
\[ 𝐶^2 = 𝐴^2 + 𝐵^2 \] \[c=10\]
The direction of the resultant is determined by the angle it makes with one of the vectors
\[𝜃 = tan^{−1}\frac{B}{A}\] This is the angle the resultant makes with the first vector \[𝜃 = tan^{−1}\frac{A}{B}\] This is the angle the resultant makes with the second vector


Test Yourself
F1 = 10 N
(Eastward)

F2 = 5 N
(Northward)
Find the resultant of the two vectors and determine its direction
  • Click here to show solution



    • Test Yourself

    Ahmed moves eastward a distance of \[30 m\] then changes direction and moves southward a distance of \[40 m\] Find the resultant displacement
  • Click here to show solution


    • If the angle between two vectors is not 90 degrees
    To find the resultant we use the cosine rule \[\vec R=\vec A+\vec B\] \[R^2=A^2+B^2-2.A.B.Cos 𝜃\] For direction: \[\frac {R}{sin(𝜃)}=\frac {A}{sin(a)}=\frac {B}{sin(b)}\]





    Test Yourself
    From the figure below we have two displacements \[A=3 m , B=5 m\] as shown Find the resultant displacement \[\vec R=\vec A +\vec B\] and determine the angle between the resultant and vector A

  • Click here to show solution

    • Vector Components (Vector Resolution)
    The process of projecting a vector onto two perpendicular axes and converting one vector into two components
    The value of this vector on each axis
    \[𝐴_𝑋= A . Cos 𝜃 \]
    \[𝐴_y= A . Sin 𝜃\]





    In this simulation, when resolving vectors there are positive and negative signs depending on the vector's position and which quadrant it's in
    Note the signs of the components in each quadrant
    Test Yourself
    A force vector \[F=60 N \] makes an angle of \[𝜃=30^0\] southwest. The components of the force on the perpendicular axes equal

  • Click here to show solution

    • What is the purpose of vector resolution?
    When we have vectors at an angle and need to find their resultant
    We resolve each vector into components and sum the components on each axis, considering the signs We end up with two perpendicular vectors Then we apply the resultant of two perpendicular vectors And find the final resultant


    Test Yourself
    Majid moved 60 meters in the direction \[30^0\] northeast
    Then changed direction and moved 40 meters in the direction \[40^0\] northwest
    Calculate the total displacement

  • Click here to show solution

    • Results of Vector Resultants in Two Dimensions

    Visual Representation:

    Diagram showing two vectors \[A and B\] and their resultant \[\vec A+\vec B\]

    Mathematical Equations:

    \[R = A + B\]
    \[ R_x = A_x + B_x\]
    \[ R_Y = A_Y + B_Y\]

    Magnitude Calculation:
    \[ |R| = \sqrt {(R_x^2 + R_Y^2)}\]

    \[𝜃 = tan^{−1}\frac{A}{B}\] This is the angle the resultant makes with the second vector

    Influencing Factors:

    • Magnitude of each vector
    • Direction of each vector
    • Angle between vectors (θ)

    Practical Applications:

    • Calculating forces in mechanical systems
    • Aircraft navigation considering wind effects
    • Stress analysis in engineering structures

    Kinetic and Static Friction Forces and Factors Affecting Friction

    Friction Force

    Friction force is the resistance of an object to motion as it moves over the surface of another object.
    Friction force is not considered a fundamental force like gravity or electromagnetic force;
    scientists view friction as the result of electromagnetic attraction between charged particles of two contacting surfaces


    Its direction is always opposite to the motion and parallel to the surface the object moves on
    There are two types of friction force:
    Static friction which appears when we apply a pulling force and the object remains stationary
    Kinetic friction which appears when we apply a pulling force and the object moves
    In this simulation, apply a horizontal pulling force and observe what happens. At what pulling force does the object move? If we change the object's mass, do we need more force to move it? What is the effect of increasing the normal force on friction force?
    Apply a horizontal pulling force and observe what happens when changing the nature of the contacting surfaces. Do static and kinetic friction forces remain the same?







    Example 2)"From previous experiments, what are the factors that change friction force?




    Example 3)"Identify the forces acting on the following objects"


    A pulling force was applied to the right parallel to the rough horizontal surface and the object remained stationary

    A pulling force was applied to the right at an angle to the horizontal causing the object to move on a rough horizontal plane
    A pulling force was applied parallel to the surface to lift the object to the top of the rough inclined plane

    An object was left to slide down a rough inclined plane
  • Click here to show solution

    • Static and Kinetic Friction

    Friction in Physics

    Static Friction

    The force that resists the initiation of motion between two stationary surfaces.

    Equation \[F_s = μ_s× N\]

    Factors affecting:

    • Coefficient of static friction (μs)
    • Normal force (N)
    • Nature of surfaces (roughness/smoothness)

    Practical uses:

    • Preventing furniture from sliding
    • Car brakes when stopped
    • Climbing inclined surfaces

    Kinetic Friction

    The force that resists motion between two relatively moving surfaces.

    Equation \[ F_k = μ_k × N\]

    Factors affecting:

    • Coefficient of kinetic friction (μk)
    • Normal force (N)
    • Object's velocity (in some cases)

    Practical uses:

    • Heat generation in car brakes
    • Writing with pen on paper
    • Movement of conveyor belts in factories

    Coefficient of Static Friction and Coefficient of Kinetic Friction

    The coefficient of static friction is the ratio between static friction force and normal force \[μ_S =\frac{F_S}{F_n}\] The coefficient of kinetic friction is the ratio between kinetic friction force and normal force \[μ_k =\frac{F_k}{F_n}\]
    In this simulation we'll apply a horizontal or angled pulling force on an object placed on a rough horizontal surface and determine the coefficients of static and kinetic friction
    Do they match the values shown in the bottom right icons?
    Consider \[g=9.75 \frac{m}{s^2} \]


    Friction in Physics

    Results of Static and Kinetic Friction Coefficients in Physics

    Static Friction

    The force that resists motion between two surfaces at relative rest.

    Equation \[Fs ≤ μ_s.N\]

    • μs: Coefficient of static friction
    • N: Normal force

    Kinetic Friction

    The force that resists motion between two surfaces in relative motion.

    Equation \[F_k = μ_k.N\]

    • μk: Coefficient of kinetic friction
    • N: Normal force

    Factors affecting friction:

    • Nature of contacting surfaces (roughness/smoothness)
    • Force pressing the objects together
    • Materials composing the surfaces
    • Contact area (in some cases)

    Practical Applications:

    1. Vehicle braking systems
    2. Walking on surfaces
    3. Design of vehicle tires
    4. Industrial machinery and power transmission
    5. Writing with pen and paper
    6. Securing nails in walls

    Important Notes:

    • Coefficient of static friction is usually greater than kinetic
    • Friction coefficient doesn't depend on surface area
    • Kinetic friction sometimes decreases with increasing velocity

    Equilibrium of an Object Under Multiple Concurrent Forces


    We say an object is in equilibrium when the net force acting on it is zero \[\sum F_X=0\;\;\;\;\;\sum F_Y=0\] In this case the object may be at rest \[v=0 \;\;\;\;\;\; a=0\] or moving with constant velocity \[v=constan\;\;\;\;\;\;a=0\]



    Solved Example
    Three concurrent forces acted on an object as shown below putting the object in equilibrium Determine the magnitude and direction of force \[F_2=?\]
    Since the object is in equilibrium \[\sum F_X=0\;\;\;\;\;\sum F_Y=0\] \[F_X=50.cos36.8+F_2cos𝜃+0=0\Rightarrow \;\;F_2cos𝜃= -40 \] \[F_Y=50.sin36.8+F_2sin𝜃-40 =0\Rightarrow \;\;F_2sin 𝜃= 10 \] \[𝜃 = tan^{−1}\frac{f_y}{f_x}=tan^{−1}\frac{10}{-40}=-14^0\] This is the angle between the vector and the negative \[X\] axis \[𝜃=180-14 =166^0\] This is the angle between the vector and the positive \[X\] axis \[F_2cos 166= -40\Rightarrow F_2=\frac {-40}{cos 166}=41.2\;N\]

    Motion of an Object on a Smooth Inclined Plane


    When an object moves on a smooth inclined plane it's affected by two forces: the normal force and the weight \[F_N\;\;\;\;\;F_g\]
    We identify the acting forces and draw perpendicular coordinate axes where the horizontal axis is parallel to the plane of motion and the weight makes an angle with the vertical axis equal to the plane's inclination We resolve the weight into two components \[Fg_ X = m . g .sin 𝜃 \;\;\;\;\;\;\;Fg_ Y = m . g .cos 𝜃\] We calculate acceleration using Newton's second law \[Fg_ X = m . a\Rightarrow m .g .sin 𝜃=ma\Rightarrow a=g .sin 𝜃\] Note that mass doesn't affect the object's acceleration





    Motion of an Object on a Rough Inclined Plane


    Motion on an inclined plane - if the surface is rough there's friction force
    The object won't move until the horizontal component of weight exceeds the friction force which opposes motion Note in the figure that the weight has been resolved into two components \[Fg_ X = m . g .sin 𝜃 \;\;\;\;\;\;\;Fg_ Y = m . g .cos 𝜃\]
    We calculate acceleration using Newton's second law \[Fg_ X -F_K= m . a\] Friction force is calculated using the friction coefficient \[𝜇_K=\frac{F_K}{F_N}\Rightarrow F_K=𝜇_K.F.N=𝜇_K.m . g .cos 𝜃\] \[m . g .sin 𝜃 -𝜇_K.m . g .cos 𝜃= m . a\] \[ a=g .sin 𝜃 -𝜇_K . g .cos 𝜃\]


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