PHYS 100 Chapter Notes - Chapter 3: Sin 34, Protractor, Pythagorean Theorem
10 May 2018
School
Department
Course
Professor
Chapter 3
Kinematics in Two Dimensions
- The independence of Perpendicular motions
o Independence of motion
▪ THe horizontal and vertical componenets of two-dimensional motion are
independent of each other
• Any motion in the horizontal direction does not affect motion in the
vertical direction
Vector addition and SUbtration
- Vector sin two dimensions
o A vector is a quantity that has magnitude and direction
▪ Displacement, velocity acceleration and force
• D stands for a vector
• Direction represented by 0
• F stands for Force
- Vector Addition: Head-to-Tail Method
o Graphical way to add vectors
o The tail of the vector is the starting point of the vector
o The Head is the final pointed end of the arrow
▪ Step 1: Draw an arrow to represent the first vector
▪ Step 2: Draw an arrow to represent the second vector
▪ Step 3: Draw an arrow from the tail of the first vector to the head of the last
• This becomes the resultant
▪ Step 4: To get the magnitude, measure its length with a ruler
▪ Step 5: Get the direction of the resultant, measure the angle it makes with the
reference frame using a protractor
- Vector Subtraction
o To define subtraction, the negative of a vector B is defined to be -B
▪ The negative of any vector has same magnitude but the opposite direction
▪ The subtraction of vector B from A is defined as -B to A.
• Vector subtraction is the addition of a negative vector
- Multiplication of Vectors and Scalars
o When multiplying the vector by a positive scalar, the magnitude changes but direction
stays the same
o When vector A is multiplied by scalar C
▪ The magnitude becomes a value of cA
▪ If C is positive, the direction does not change
▪ If C is negative, the direction is reversed
- Resolving a vector into components
o This is used when finding out the answer to this example
▪ A person is walking 10.3 blocks in a direction 29.0º north of east and want to
find out how many blocks east and north had to be walked
Vector Addition and Subtration: Analytic Methods
- Analytic Methods
o Concise, accurate and precise than graphical methods
o Only limited by the accuracy and precision with which physical quantities are known
- Resolving a vector in Perpendicular components
find more resources at oneclass.com
find more resources at oneclass.com
o Similar to Pythagoreon Theorum
▪ A+B = C
▪ Ax+Ay =A
o If the vector A is known, then its magnitude A and angle θ, to find Ax or Ay use the
following
▪ Ax = A Cos θ
▪ Ay = A Sin θ
o Example: A = 10.3, θ = 29.1
▪ Find Ax & Ay
• Ax = A cos θ
o 10.3(cos 29.1) = 9
• Ay = A Sin θ
o 10.3 (Sin 29.1) = 5
- Calculating a resultant Vector
o If perpendicular Ax and Ay of a Vector A are known, then A can be found analytically
▪ To find Magnitude A & Direction θ of a vector from its components Ax & Ay we
use the relationships
• A = √Ax2+Ay2
• Θ = tan-1 (Ay/Ax)
- Adding Vectors Using analytic Methods
o Consider where Vectors A & B are added to produce resultant R
▪ If A & B represent two parts of a walk, then R is total displacement
▪ We can find R & θ by using the equation
• A = √Ax2+Ay2
• Θ = tan-1 (Ay/Ax)
▪ Step 1: Identify the X & Y-axes that will be used in the problem
• Find the components of each vector to be added along with the chosen
axes
• Use: Ax = A Cos θ
o Ay = A Sin θ
▪ Step 2: Find the components of the resultant along each axis by adding the
components of the vectors along that axis
• Rx = Ax + Bx
• Ry = Ay + By
▪ Step 3: To get the Magnitude R, use Pythagorean Theorem
• R = √R2x + R2y
▪ Step 3: To get the direction
• Θ = tan-1 (Ry/Rx)
▪ Example:
▪ Add the vector A to Vector B, using
perpendicular components along the X-y
axes.
▪ Vector A represents first leg of a walk in
which a person walks 53m in direction 20
degree north of east
▪ Vector B represents the second leg, a
displacement of 34m in a direction 63
degree north of east
• Given: A = 53m
o ΘA = 20
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Independence of motion: the horizontal and vertical componenets of two-dimensional motion are independent of each other, any motion in the horizontal direction does not affect motion in the vertical direction. Vector sin two dimensions: a vector is a quantity that has magnitude and direction, displacement, velocity acceleration and force, d stands for a vector, direction represented by 0, f stands for force. Multiplication of vectors and scalars: when multiplying the vector by a positive scalar, the magnitude changes but direction stays the same, when vector a is multiplied by scalar c, the magnitude becomes a value of ca. If c is positive, the direction does not change. If c is negative, the direction is reversed. Analytic methods: concise, accurate and precise than graphical methods, only limited by the accuracy and precision with which physical quantities are known. Resolving a vector in perpendicular components: similar to pythagoreon theorum.
