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Vector
components
and motion
Objectives
- Distinguish between vectors and scalars and give examples of each.
- Use vector diagrams to interpret the relationships among vector quantities such as force and acceleration.
- Use mathematical and graphical vector addition to make predictions and solve problems involving vectors.
- You are given force vectors A and B : Which of these represents vector C in the vector equation? A. 70 N east B. 70 N west C. 10 N east D. 10 N west
- Which of the quantities below are vectors? Choose all that apply. A. speed B. velocity C. acceleration D. mass E. distance
Assessment Assessment
- Two component forces acting on an object are depicted in this vector diagram. What is the magnitude of the force F?
- Which vector diagram correctly depicts the vector equation: ?
Assessment Physics terms
- vector
- vector diagram
- magnitude
- scalar
- resultant
- component force
- net force
Equations
There are many different variables that are important in physics.
These variables are either vectors or scalars.
What’s the difference? Why is it important?
Two types of variables
A scalar is a type of quantity that can be fully described using only information about its magnitude. Temperature is a scalar quantity. We can fully describe the temperature with a magnitude (number value).
Scalars Brainstorm with a partner
Can you name other scalar quantities? Mass is a scalar. Time is also a scalar. We don’t say that an event lasted 15 seconds due north!
Mass is a scalar quantity
A vector is a type of variable that can only be fully described by providing information about its magnitude AND its direction. Force is a vector. This box is being pulled by a 50 N force acting at 40° above the horizontal.
Vectors
- 50 N is the magnitude or amount of the force.
- 40° above the horizontal is the direction.
F = 50 N
40°"
Coordinate axes indicate which direction is positive.
Coordinate axes Finding the resultant
Along the y -axis:
Finding the resultant
Along the y -axis: Along the x -axis: The resultant force is:
Finding the resultant
Along the y -axis: Along the x -axis: The resultant force is: 20 N
The resultant tells you that it is as if
a single force of 20 N pushed the box to the right.
The resultant force
m
The box will accelerate in the direction of the net force: 20 N Adding vectors is not the same as just using arithmetic. 3 N plus 4 N may not give you 7 N!
Adding perpendicular vectors
When two forces act at right angles, you can use the Pythagorean theorem to determine the resultant force.
Adding perpendicular vectors Use the Pythagorean theorem
In this case, adding 3 N and 4 N gives you 5 N. Do you see why?
The resultant force is 5 N
The resultant vector is a single vector that represents the sum of two or more individual vectors. This object will accelerate in the direction of the resultant (net) force vector.
The resultant
When adding vectors, it is often helpful to use vector components. Any vector can be resolved into x - and y -components.
Vector components
What are the components of vector F?
Vector components
Adding vectors with components
The resultant force is 8 N to the right. You can check your work by adding the vectors graphically. (^) F 1 F 2
Check your work
The resultant force is 8 N to the right. Click on this interactive calculator on page 173
Vector components Vector components
10 60 If you know a vector’s magnitude and direction you can find its x and y components. What are the components of this vector? F = 10 N, +60°
Vector components
If you know a vector’s magnitude and direction you can find its x and y components. What are the components of this vector? F = 10 N, +60° 10
60 F = (+5.0, +8.7) N
Vector components
10 If you know the x - and y - components of a vector, you can find its magnitude and direction. What is the magnitude and direction of this vector? F = (-10, +10 ) N
If you know the x - and y - components of a vector, you can find its magnitude and direction. What is the magnitude and direction of this vector? F = (-10, +10 ) N F = 14 N, +135° (or 45° above the –x axis)
Vector components
- 10 14 135 1. Which of the quantities below are vectors? Choose all that apply. A. speed B. velocity C. acceleration D. mass E. distance
Assessment
- You are given force vectors A and B : Which of these represents vector C in the vector equation? A. 70 N east B. 70 N west C. 10 N east D. 10 N west
- Which of the quantities below are vectors? Choose all that apply. A. speed B. velocity C. acceleration D. mass E. distance
Assessment
- You are given force vectors A and B : Which of these represents vector C in the vector equation? A. 70 N east B. 70 N west C. 10 N east D. 10 N west
Assessment
- Which of the quantities below are vectors? Choose all that apply. A. speed B. velocity C. acceleration D. mass E. distance Subtracting a vector is the same as adding its opposite.
Assessment
- Two component forces acting on an object are depicted in this vector diagram. What is the magnitude of the force F?
Assessment
- Two component forces acting on an object are depicted in this vector diagram. What is the magnitude of the force F?
Use the sine function to determine the y -component
Determine the y -component
Use the sine function to determine the y -component
Determine the y -component
Specify the vector using these components: F = (Fx , Fy )
Polar vectors
Vectors can also be specified in polar form, using magnitude and direction.
Polar vectors
Vectors can also be specified in polar form, using magnitude and direction. If you know the components, you can use them to find the direction (angle). Determine the angle using the inverse tangent function.
Determine the angle
The resultant is: F = 5 N, - 36.9o One warning: scientific calculators can use either degrees or radians for angles.
Radians and degrees
- The default is radians on many calculators, in many computer programming languages, and in many applications!!!
- If you want to use degrees, you may need to press the button on your calculator to toggle between degrees and radians.
