Chapter 2: Forces
Scalars and Vectors
A scalar quantity can have magnitude, algebraic sign (positive or negative), and units, but not a direction in
space.
When adding vectors, consider the directions of the vectors being added.
Vector Notation:
• An arrow over a boldface symbol indicates a vector quantity: F
• When written without the arrow and in italics ( F ), it represents the magnitude of the vector (which is
scaler).
• Absolute value bars also indicate magnitude F = | F |
• The magnitude of a vector may have units but it never negative, only positive or zero.
Vector representation:
• Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector.
• By convention, the length of a vector arrow is proportional to the magnitude of the vector.
• The negative of a vector is simply in the opposite direction to the original vector.
Graphical Vector Addition:
Components of a Vector:
• Any vector can be expressed as the sum of vectors parallel to the x-, y-, and (if needed) z-axes.
• The x-, y-, and z- components of a vector indicate the magnitude and direction of the three vectors along
the axes.
• A component has magnitude, units, and an algebraic sign ( + or − ).
• The sign of a component indicates the direction along that axis.
• The process of finding the components of a vector is called resolving the vector into its components.
Strategy for finding the x and y components of a vector from magnitude and direction.
1. Draw a right triangle with the vector as the hypotenuse and the other two sides parallel to the x- and y-
axes.
2. Determine one of the angles in the triangle.
3. Use trigo functions to find the magnitudes of the components. Make sure your calculator is in “degree
mode” to evaluate trigonometric functions of angles in degrees and “radian mode” for angles in radians.
4. Determine the correct algebraic sign for each component.
Note: the components are scalars, the hypotenuse is the vector.
,Strategy for finding the Magnitude and Direction of a Vector from Its x and y-components.
1. Sketch the vector on a set of x- and y-axes in the correct quadrant, according to the signs of the
components.
2. Draw a right triangle with the vector as the hypotenuse and the other two sides parallel to the x and y-
axes.
3. In the right triangle, choose which of the unknown angles you want to determine.
4. Use the inverse tangent function to find the angle.
The lengths of the sides of the triangle represent Fx and Fy.
If θ is opposite the side parallel to the x-axis, then tanθ = opposite/adjacent = Fx/ Fy.
If θ is opposite the side parallel to the y-axis, then tanθ = opposite/adjacent = Fy/ Fx.
5. Interpret the angle: specify whether it is the angle below the horizontal, or the angle west of south, or the
angle clockwise from the negative y-axis, etc.
6. Use the Pythagorean theorem to find the magnitude of the vector.
Strategy for Adding Vectors Using Components
1. Find the x- and y-components of each vector to be added.
2. Add the x-components (with their algebraic signs) of the vectors to find the x-component of the sum.
3. Add the y-components (with their algebraic signs) of the vectors to find the y-component of the sum.
4. If necessary, take the x- and y-components of the sum to find the magnitude and direction of the sum
using pythag and trig for the angle.
Free-body diagrams
A free-body diagram (FBD) is a simplified sketch of a single object with force vectors drawn to represent
every force acting on that object.
• Include only forces acting on the object, not forces the object exerts on its environment or any forces that
act on other objects.
• A coordinate system and labels for angles are often added.
Vector Multiplication
Vectors can be multiplied in two ways:
• Dot product, indicated as follows: 𝐴 ∙ 𝐵
• Cross product, indicated as follows: 𝐴 × 𝐵
We will not be using the cross product in this course.
The dot product is defined as follows: 𝐴 ∙ 𝐵 = |𝐴| |𝐵| cos 𝜃 = 𝐴 𝐵 cos 𝜃
• 𝜃 is the angle BETWEEN the two vectors, 𝐴 and 𝐵
• The dot product of two vectors is a SCALAR quantity.
• To calculate the dot product of two vectors, if the components of the vectors are known, simply multiply
the components and add together:
𝐴 ∙ 𝐵 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵z
Dot Product in physics
• The dot product is a mathematical tool that has many useful applications in Physics.
• In this course, it is used in defining the work done by a constant force: 𝑊 = 𝐹 ∙ 𝑟
• 𝐹 is the force vector and 𝑟 the displacement vector.
,Newton’s 1st and 3rd Law Gravitational Forces:
Interactions and Forces:
Long-range/fundamental forces are contact free forces and
there are four of them identified in Physics:
• Gravitational force (gravity)
• Electromagnetic force
• Strong nuclear force
• Weak nuclear force
All non-fundamental (convenience) forces are contact forces:
• Normal Forces
• Forces of Friction (Kinetic and static frictional forces)
• Tension
Note that all forces are vector quantities, i.e. they have both magnitude and direction. In the SI system, the
unit of forces is Newton (N).
Net Force
When more than one force acts on an object, the subsequent motion of the object is determined by the net
force acting on the object.
The net force is the vector sum of all the forces acting on an object.
Newtons 1st Law of Motion
An object’s velocity vector remains constant if and only if the net force acting on the object is zero.
Inertia:
Newton’s first law is also called the law of inertia.
In physics, inertia means resistance to changes in velocity.
Inertia and Equilibrium
For an object to be in equilibrium, the sum of all forces = 0
Definition of Equilibrium: An object is in equilibrium when it has zero acceleration.
Newton’s 3rd Law of Motion
In an interaction between two objects, each object exerts a force on the other.
These two forces are equal in magnitude and opposite in direction.
Equivalently, we can write F (on B by A) = -F (on A by B)
Force of Gravity
Newton’s law of Universal Gravitation:
There exists an attractive force between 2 objects that is proportional to the product of their masses and
inversely proportional to the square of the distance between them.
𝑚1 𝑚2
𝐹=𝐺 , 𝐺 = 6.67 × 10−11 N.m2/kg2
𝑟2
m1 and m2 = masses of the object (kg)
r = the distance between them (m)
G = the universal gravitational constant (Nm2/kg2)
F = the attractive force (N)
, Weight
For an object on the Earth’s surface, g, free fall acceleration, is approximately constant everywhere on the
planet.
As you rise above the Earth’s surface the gravitational field weakens. The gravitational field is proportional
to the inverse square of the distance from Earth’s centre.
Contact Forces:
Normal Forces
The normal force is one component of the force that a surface exerts on an object with which it is in contact
namely, the component that is perpendicular to the surface.
Frictional Force
The component of the surface force that acts parallel to the
contact surface. The force of friction acts between two
objects while their surfaces are in contact, opposing the
motion of one object slipping along the other.
The force of friction is caused by irregularities in the
surfaces that are in contact with each other.
Static Friction
Static friction is the force that prevent objects from moving
with respect to a contacting surface and allows them to
maintain their stationary status.
𝑓𝑠𝑚𝑎𝑥 = 𝜇𝑠 𝑁
𝜇𝑠 is the coefficient of static friction, and N is the normal
force.
The magnitude of the static frictional force can have any
value from zero up to a maximum value.
Scalars and Vectors
A scalar quantity can have magnitude, algebraic sign (positive or negative), and units, but not a direction in
space.
When adding vectors, consider the directions of the vectors being added.
Vector Notation:
• An arrow over a boldface symbol indicates a vector quantity: F
• When written without the arrow and in italics ( F ), it represents the magnitude of the vector (which is
scaler).
• Absolute value bars also indicate magnitude F = | F |
• The magnitude of a vector may have units but it never negative, only positive or zero.
Vector representation:
• Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector.
• By convention, the length of a vector arrow is proportional to the magnitude of the vector.
• The negative of a vector is simply in the opposite direction to the original vector.
Graphical Vector Addition:
Components of a Vector:
• Any vector can be expressed as the sum of vectors parallel to the x-, y-, and (if needed) z-axes.
• The x-, y-, and z- components of a vector indicate the magnitude and direction of the three vectors along
the axes.
• A component has magnitude, units, and an algebraic sign ( + or − ).
• The sign of a component indicates the direction along that axis.
• The process of finding the components of a vector is called resolving the vector into its components.
Strategy for finding the x and y components of a vector from magnitude and direction.
1. Draw a right triangle with the vector as the hypotenuse and the other two sides parallel to the x- and y-
axes.
2. Determine one of the angles in the triangle.
3. Use trigo functions to find the magnitudes of the components. Make sure your calculator is in “degree
mode” to evaluate trigonometric functions of angles in degrees and “radian mode” for angles in radians.
4. Determine the correct algebraic sign for each component.
Note: the components are scalars, the hypotenuse is the vector.
,Strategy for finding the Magnitude and Direction of a Vector from Its x and y-components.
1. Sketch the vector on a set of x- and y-axes in the correct quadrant, according to the signs of the
components.
2. Draw a right triangle with the vector as the hypotenuse and the other two sides parallel to the x and y-
axes.
3. In the right triangle, choose which of the unknown angles you want to determine.
4. Use the inverse tangent function to find the angle.
The lengths of the sides of the triangle represent Fx and Fy.
If θ is opposite the side parallel to the x-axis, then tanθ = opposite/adjacent = Fx/ Fy.
If θ is opposite the side parallel to the y-axis, then tanθ = opposite/adjacent = Fy/ Fx.
5. Interpret the angle: specify whether it is the angle below the horizontal, or the angle west of south, or the
angle clockwise from the negative y-axis, etc.
6. Use the Pythagorean theorem to find the magnitude of the vector.
Strategy for Adding Vectors Using Components
1. Find the x- and y-components of each vector to be added.
2. Add the x-components (with their algebraic signs) of the vectors to find the x-component of the sum.
3. Add the y-components (with their algebraic signs) of the vectors to find the y-component of the sum.
4. If necessary, take the x- and y-components of the sum to find the magnitude and direction of the sum
using pythag and trig for the angle.
Free-body diagrams
A free-body diagram (FBD) is a simplified sketch of a single object with force vectors drawn to represent
every force acting on that object.
• Include only forces acting on the object, not forces the object exerts on its environment or any forces that
act on other objects.
• A coordinate system and labels for angles are often added.
Vector Multiplication
Vectors can be multiplied in two ways:
• Dot product, indicated as follows: 𝐴 ∙ 𝐵
• Cross product, indicated as follows: 𝐴 × 𝐵
We will not be using the cross product in this course.
The dot product is defined as follows: 𝐴 ∙ 𝐵 = |𝐴| |𝐵| cos 𝜃 = 𝐴 𝐵 cos 𝜃
• 𝜃 is the angle BETWEEN the two vectors, 𝐴 and 𝐵
• The dot product of two vectors is a SCALAR quantity.
• To calculate the dot product of two vectors, if the components of the vectors are known, simply multiply
the components and add together:
𝐴 ∙ 𝐵 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵z
Dot Product in physics
• The dot product is a mathematical tool that has many useful applications in Physics.
• In this course, it is used in defining the work done by a constant force: 𝑊 = 𝐹 ∙ 𝑟
• 𝐹 is the force vector and 𝑟 the displacement vector.
,Newton’s 1st and 3rd Law Gravitational Forces:
Interactions and Forces:
Long-range/fundamental forces are contact free forces and
there are four of them identified in Physics:
• Gravitational force (gravity)
• Electromagnetic force
• Strong nuclear force
• Weak nuclear force
All non-fundamental (convenience) forces are contact forces:
• Normal Forces
• Forces of Friction (Kinetic and static frictional forces)
• Tension
Note that all forces are vector quantities, i.e. they have both magnitude and direction. In the SI system, the
unit of forces is Newton (N).
Net Force
When more than one force acts on an object, the subsequent motion of the object is determined by the net
force acting on the object.
The net force is the vector sum of all the forces acting on an object.
Newtons 1st Law of Motion
An object’s velocity vector remains constant if and only if the net force acting on the object is zero.
Inertia:
Newton’s first law is also called the law of inertia.
In physics, inertia means resistance to changes in velocity.
Inertia and Equilibrium
For an object to be in equilibrium, the sum of all forces = 0
Definition of Equilibrium: An object is in equilibrium when it has zero acceleration.
Newton’s 3rd Law of Motion
In an interaction between two objects, each object exerts a force on the other.
These two forces are equal in magnitude and opposite in direction.
Equivalently, we can write F (on B by A) = -F (on A by B)
Force of Gravity
Newton’s law of Universal Gravitation:
There exists an attractive force between 2 objects that is proportional to the product of their masses and
inversely proportional to the square of the distance between them.
𝑚1 𝑚2
𝐹=𝐺 , 𝐺 = 6.67 × 10−11 N.m2/kg2
𝑟2
m1 and m2 = masses of the object (kg)
r = the distance between them (m)
G = the universal gravitational constant (Nm2/kg2)
F = the attractive force (N)
, Weight
For an object on the Earth’s surface, g, free fall acceleration, is approximately constant everywhere on the
planet.
As you rise above the Earth’s surface the gravitational field weakens. The gravitational field is proportional
to the inverse square of the distance from Earth’s centre.
Contact Forces:
Normal Forces
The normal force is one component of the force that a surface exerts on an object with which it is in contact
namely, the component that is perpendicular to the surface.
Frictional Force
The component of the surface force that acts parallel to the
contact surface. The force of friction acts between two
objects while their surfaces are in contact, opposing the
motion of one object slipping along the other.
The force of friction is caused by irregularities in the
surfaces that are in contact with each other.
Static Friction
Static friction is the force that prevent objects from moving
with respect to a contacting surface and allows them to
maintain their stationary status.
𝑓𝑠𝑚𝑎𝑥 = 𝜇𝑠 𝑁
𝜇𝑠 is the coefficient of static friction, and N is the normal
force.
The magnitude of the static frictional force can have any
value from zero up to a maximum value.
