Physics 30
Unit One: Kinematics and Dynamics
Text: Merrill Physics: Principles and Problems
Chapter 6: Vectors (Dynamics)
Graphical Method of Vector Addition
 A vector quantity can be represented by an arrow-tipped line segment
 Length of the line (drawn to scale) represents he magnitude of the quantity
 Direction of the arrow represents the direction of the vector
 We can add these vectors together graphically (drawn to scale)
1) Vector Addition in One Dimension
Notes:
 Vectors are considered to be in one dimension if the vector only travels in one direction (example: east) and/or
back in the exact OPPOSITE direction (example: west)
 To add these together graphically, we must draw the first vector to scale facing the proper direction. We then
add the second vector to it by drawing the tail of the second vector onto the head of the first vector and drawing
it to scale.
 Example:


o In this case, vector A & vector B get added together to make vector AB (or the resultant vector R)
If the vectors are in different directions, we follow the same procedure as listed above:
Example:
o In this case, vector A and B get added together to make vector AB (or R) which is shorter than the
original vector A

REMEMBER: VECTORS ARE ALWAYS ADDED HEAD TO TAIL
2) Vector Addition in Two Dimensions
Notes:
 Vectors can also represent motion in more than one dimension (motion east AS WELL as motion north, etc.)
 We add vectors the same way in two dimensions as we do in one dimension: VECTORS ARE ALWAYS
ADDED HEAD TO TAIL
 Draw the first vector to scale, then draw the tail of the second vector on the head of the first vector;
continue as needed with any other vectors
 The RESULTANT vector is drawn from the TAIL of the initial vector to the HEAD of the final vector

Example: Two students walk 95 m [E] then turn and walk 55 m [N]
The resultant vector would be 110 m [E30oN]

We can measure the resultant vector and measure the angle using a protractor

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We can also add vectors that are not perpendicular (whether they be force, position, or velocity, etc.)
Example: A force A of 45 N and a force B of 65 N are exerted on an object at a point P. Force A acts in the
direction of [E60oN], while force B acts in the direction [E].

The two vectors are drawn to scale (head to tail), then the resultant vector is drawn by connecting the tail of the
initial vector with the head of the final vector
 Magnitude is determined by measuring the resultant vector (using the scale)
 Direction is determined by using a protractor to measure the angle
3) Addition of Several Vectors
Notes:
 There are often times when there are more than two vectors acting upon an object
 To handle this situation, follow the same procedure as the before – add the tail of the next vector to the head of
the previous vector – once all the vectors are added together, connect the tail of the initial vector to the head of
the final vector
 NOTE: it does not matter what order all the vectors are combined together – the resultant vector will be the
same regardless
 REMEMBER: when placing vectors head-to-tail, the direction and length of each vector must not be changed
4) Independence of Vector Quantities
Notes:
 Perpendicular vectors are INDEPENDENT of one another
 Example: If a boat travels east at 8.0 m/s across a river that flows north at 5.0 m/s, what will be the resultant
velocity?
 In the above question, the velocity north does not change the velocity east – the boat travels at both
simultaneously
o This only applies for perpendicular vectors
 In the above example, if it takes the boat 10 seconds to get to the other side of the river, in that time it will travel
50 m north as well. This DOES NOT change that fact that the boat traveled 80 m east across the river in that
time as well.
Analytical Method of Vector Addition
 In the graphical method, vectors are represented by arrows. In the analytical method, a vector is represented by
TWO NUMBERS: the first number gives the magnitude, the second give the direction (an angle with refereces)
 When we add two vectors together we use TRIGONOMETRY to find the resultant vector
o Trigonometry deals with the relationships among angles and side of triangles
 There are three trigonometric functions that we use:
o sine:
sin θ = opposite side/hypotenuse
o cosine:
cos θ = adjacent side/hypotenuse
o tangent:
tan θ = opposite side/adjacent side

Example:
o sin θ = B/hyp
o cos θ = A/hyp
o tan θ = B/A

Using these formulas and given the lengths of the sides of the triangle, we can solve for the angles of the triangle –
which will give us DIRECTION (using a calculator, or, heaven-forbid, a trigonometric ratio table – yechh)
5) Adding Perpendicular Vectors
Notes:
 When we have perpendicular vectors, they meet each other at a right angle – this means that the Resultant Vector
will form the hypotenuse of the triangle
 To find the MAGNITUDE of the vector, we can use the Pythagorean theorem (c2 = a2 + b2) so long as the triangle
has a right angle.
 The interior angles can be found by using the trigonometric ratios to give us DIRECTION (usually, because the
adjacent and opposites sides will be KNOWN, we use the tangent function to get the angle for direction)
 These types of problems involve two calculations – one to find magnitude of the resultant vector, and one to find the
direction of the resultant vector
 Example:
An airplane flying due east at 90.0 km/h is being blown north at 50.0 km/h. What is the resultant velocity of
the plane?
MAGNITUDE:
c2 = a2 + b2
c2 = (90.0 km/h)2 + (50.0 km/h)2
c = 103 km/h
DIRECTION:
tan θ = opp / adj
tan θ = 50.0 km/h / 90.0 km/h
θ = 29o
θ
6) Components of Vectors
Notes:
 Just as two vectors can be added together to make a resultant vector, any vector can be broken down into 2 (or
more parts)
 When we break a vector into two or more parts, these parts are called COMPONENT vectors
 When we find the magnitude and direction of the component vectors, this process is called VECTOR
RESOLUTION
 To do this, first draw the original vector, then draw in an x & y axis (to represent the directions)
 Once these are drawn we can then sketch in the vector that would make up the ‘x’ component of the vector and
the ‘y’ component of the vector
 We can determine their magnitudes by using the given value of the initial vector, the given direction of the
original vector (something in degrees) and the trigonometric ratios (sin/cos/tan) to solve for our desired sides
7) Adding Vectors at Any Angle
Notes:
There are two methods to find resultant vectors when we add vectors together that are not perpendicular
Method #1: Vector Resolution
 We can use vector resolution to add vectors that do not meet at a 90 degree angle
 We resolve both vectors into their ‘x’ and ‘y’ components
 From there, add the ‘x’ components together – do the same for the ‘y’ components
 The resulting ‘x’ and ‘y’ components can be drawn out and, like any other two vectors, we can find the resultant
vector
Method #2: Law of Cosines/Law of Sines
Law of Cosines can be used on ANY triangle (not just right triangles)
 The law works as follows, applies to ALL triangles, and is used to find the MAGNITUDE of a side of a triangle:
c2 = a2 + b2 – 2ab cos C

NOTE:
o small letters = sides
o Large letters = angles
Law of Sines can also be used on ANY triangle
 The law works as follows, applies to ALL triangles, and is used to find the DIRECTION (or angle) in a triangle:


To use this, we have to know two sides and one angle
NOTE: we only have to use 2 out of the 3 parts of the equation – whatever works for the question
Applications of Vectors
 All forces and acceleration are vector quantities, so we can use vectors to understand Newton’s Laws, and to
establish how an object will react that has two or more forces acting upon it
8) Equilibrium
Notes:
 Equilibrium is established when all the forces acting upon an object sum to zero
 If there is no net force acting upon an object, it will NOT be accelerated (according to Newton’s Laws)
 Example: if two equal forces are acting upon an object in opposite directions
 An EQUILIBRANT FORCE is a force that can be applied to an object that will cause the object to attain equilibrium
 To find the equilibrant force, first you must find the net force acting upon an object – the equilibrant force is equal in
magnitude, but in the exact opposite direction of the net force
 NOTE: the equilibrant force can be attained by 2 or more forces acting togther
 Example: p. 122 text book
9) Gravitational Force and Inclined Planes
Notes:
 The gravitational force of an object is always directed towards the centre of the earth
 To understand the forces acting upon an object on an inclined plane, we must resolve the force of gravity into the
two components: one parallel to the surface, and one perpendicular to the surface.
 If we know the force of gravity, and the angle of the incline, we can use this to determine the perpendicular surface
force, and the force parallel to the surface
 These can be calculated using either graphical or trigonometric methods
 NOTE: the object will not be accelerating perpendicular to the surface, because the surface will be holding it up –
the force that holds the object up is the normal force – it will balance the perpendicular force of gravity
 The only other force acting upon the object will be the parallel force (due to gravity)
 This force will be moving it down the incline – if there is no friction opposing the movement then Newton’s
second law can be used to solve problems
 Note: if there is no friction, then a = F/m: in the case of inclined planes, this means that
a = W sin θ / m
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Since W / m = g (acceleration due to gravity) this formula becomes:
a = g sin θ
Most times there is FRICTION involved – this will oppose the parallel component of the force of gravity on an
inclined plane
The resolution of weight on an inclined plane (into parallel and perpendicular components) can be used to measure
the force of friction or the coefficient of static friction between the two surfaces
Example: We can determine the coefficient of static friction
 If a coin is placed on the cover of a book and just begins to move when the over angle is 38o with the
horizontal. What is the coefficient of static friction?
 KNOWN: angle of inclined plane = 38o
 UNKNOWN: μ
 EQUATIONS: Ff = μFN
Fper = W cos θ
Fpar = W sin θ
SOLUTION
 b/c coin stays on the book cover, perpendicular force must equal normal force:
FN = Fper = W cos θ
 when coin is not moving, the forces parallel to the book cover are equal in magnitude
Ff = Fpar = W sin θ
 when motion is just ready to start, friction is at its maximum
Ff = μFN
THUS W sin θ = μ W cos θ
 this means that
μ = sin θ / cos θ = tan θ
μ = tan 38o
μ = 0.78