Vectors and scalars
Vectors and scalars
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Physics
Vectors and scalars
A Scalar Quantity is one which has magnitude only. Examples: length, area, energy, time.
A Vector Quantity is one which has both magnitude and direction. Examples: displacement, acceleration, force.
Vectors can be represented on a diagram by an arrow, where the vector is in the same direction as the quantity it is representing.
Composition (addition) of two perpendicular vectors
- When adding two vectors, they should be arranged tail-to-tail (the arrow represents the head) and the rectangle should then be completed.
- The resultant is the line joining the two tails to the opposite corner.
- The direction is from the tails to the opposite corner.
- Mathematically, the length of the vector can be found by using Pythagoras’ Theorem.
- Mathematically, the angle can be found by using Tan q = Opp/Adj.
Experiment: To find the Resultant of Two Forces
Attach three Newton Balances to a knot in a piece of thread.
- Adjust the size and direction of the three forces until the
knot in the thread remains at rest.
- Read the forces and note the angles.
- The resultant of any two of the forces can now be shown to be
equal to the magnitude and direction of the third force.
Resolving a vector into two perpendicular Components
You have just seen that two perpendicular vectors can be added together to form a resultant.
Well let’s say we started off with the resultant. Would we be able to get back the two original vectors?
First we need to remember that for a right-angled triangle:
Sin q = Opposite/Hypothenuse, therefore Opposite = Hypothenuse x Cos q {Opp = H Sin q}
Cos q = Adjacent/Hypothenuse, therefore Adjacent = Hypothenuse x Cos q {Adj = H Cos q}
Example
Consider a velocity vector representing a velocity of 50 ms-1, travelling at an angle of 600 to the horizontal:
The Opposite is equal to H Sin q, which in this case = 50 Cos 600 = 43 ms-1.
The Adjacent is equal to H Cos q, which in this case = 50 Sin 600 = 25 ms-1.
Now look over problem 4 and 5, page 86. Then try questions 1 – 4, page 87, followed by questions 7 and 8, page 88.
Leaving Cert Physics Syllabus: Vectors and Scalars
Content |
Depth of Treatment |
Activities |
STS |
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Vectors and Scalars |
Distinction between vector and scalar quantities. |
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Vector nature of physical quantities: everyday examples. |
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Composition of perpendicular vectors. |
Find resultant using newton balances or pulleys. |
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Resolution of co-planar vectors. |
Appropriate calculations. |
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What do you get if you cross a mosquito with a rock climber?
You can't cross a vector with a scalar!
Boom Boom!
Exam questions
- [2003]
Give the difference between vector quantities and scalar quantities and give one example of each.
- [2006 OL]
Force is a vector quantity. Explain what this means.
[2003]
A cyclist travels from A to B along the arc of a circle of radius 25 m as shown.
- Calculate the distance travelled by the cyclist.
- Calculate the displacement undergone by the cyclist.
- [2004]
Two forces are applied to a body, as shown. What is the magnitude of the resultant force acting on the body?
- [2003]
Describe an experiment to find the resultant of two vectors.
Exam solutions
- A vector has both magnitude and direction whereas a scalar has magnitude only.
- A vector is a quantity which has magnitude and direction.
- The displacement is equivalent to one quarter of the circumference of a circle = 2πr/4 = 25π/2
= 12.5π = 39.3 m.
- Using Pythagoras: x2 = 252 + 252 Þ x = 35.3 m. Direction is NW
- R2 = F12 + F22 Þ R2 = 52 +122
R = 13 N
The marking scheme didn’t look for direction, but it should have, particularly since this is a vectors question and force is a vector. q = tan-1 (5/12).
- Attach three Newton Balances to a knot in a piece of thread.
- Adjust the size and direction of the three forces until the
- knot in the thread remains at rest.
- Read the forces and note the angles.
- The resultant of any two of the forces can now be shown to be
equal to the magnitude and direction of the third force.
Source : http://www.thephysicsteacher.ie/LC%20Physics/Student%20Notes/8.%20Vectors%20and%20Scalars.doc
Web site link: http://www.thephysicsteacher.ie
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Vectors and scalars
Vectors and scalars
Vectors and scalars
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