Physics and Physical Measurement Topic 1.3 Scalars and Vectors

Scalars Quantities Scalars can be completely described by magnitude (size) Scalars can be added algebraically They are expressed as positive or negative numbers and a unit examples include :- mass, electric charge, distance, speed, energy

Vector Quantities Vectors need both a magnitude and a direction to describe them (also a point of application) When expressing vectors as a symbol, you need to adopt a recognized notation e.g. They need to be added, subtracted and multiplied in a special way Examples :- velocity, weight, acceleration, displacement, momentum, force

Addition and Subtraction The Resultant (Net) is the result vector that comes from adding or subtracting a number of vectors If vectors have the same or opposite directions the addition can be done simply same direction : add opposite direction : subtract

Co-planar vectors The addition of co-planar vectors that do not have the same or opposite direction can be solved by using scale drawings to get an accurate resultant Or if an estimation is required, they can be drawn roughly or by Pythagoras’ theorem and trigonometry Vectors can be represented by a straight line segment with an arrow at the end

Triangle of Vectors Two vectors are added by drawing to scale and with the correct direction the two vectors with the tail of one at the tip of the other. The resultant vector is the third side of the triangle and the arrow head points in the direction from the ‘free’ tail to the ‘free’ tip

Example a b+= R = a + b

Parallelogram of Vectors Place the two vectors tail to tail, to scale and with the correct directions Then complete the parallelogram The diagonal starting where the two tails meet and finishing where the two arrows meet becomes the resultant vector

Example a b+= R = a + b

More than 2 If there are more than 2 co-planar vectors to be added, place them all head to tail to form polygon when the resultant is drawn from the ‘free’ tail to the ‘free’ tip. Notice that the order doesn’t matter!

Subtraction of Vectors To subtract a vector, you reverse the direction of that vector to get the negative of it Then you simply add that vector

Example a b- = R = a + (- b) -b

Multiplying Scalars Scalars are multiplied and divided in the normal algebraic manner Do not forget units! 5m / 2s = 2.5 ms -1 2kW x 3h = 6 kWh (kilowatt-hours)

Multiplying Vectors A vector multiplied by a scalar gives a vector with the same direction as the vector and magnitude equal to the product of the scalar and a vector magnitude A vector divided by a scalar gives a vector with same direction as the vector and magnitude equal to the vector magnitude divided by the scalar You don’t need to be able to multiply a vector by another vector

Resolving Vectors The process of finding the Components of vectors is called Resolving vectors Just as 2 vectors can be added to give a resultant, a single vector can be split into 2 components or parts

The Rule A vector can be split into two perpendicular components These could be the vertical and horizontal components Vertical component Horizontal component

Or parallel to and perpendicular to an inclined plane

These vertical and horizontal components could be the vertical and horizontal components of velocity for projectile motion Or the forces perpendicular to and along an inclined plane

Doing the Trigonometry  Sin  = opp/hyp = y/  V  Cos  = adj/hyp = x/  V  V y x Therefore y =  V  sin  In this case this is the vertical component Therefore x =  V  cos  In this case this is the horizontal component  V cos  V sin 

Quick Way If you resolve through the angle it is cos If you resolve ‘not’ through the angle it is sin

Adding 2 or More Vectors by Components First resolve into components (making sure that all are in the same 2 directions) Then add the components in each of the 2 directions Recombine them into a resultant vector This will involve using Pythagoras´ theorem

Question Three strings are attached to a small metal ring. 2 of the strings make an angle of 70 o and each is pulled with a force of 7N. What force must be applied to the 3rd string to keep the ring stationary?

Answer Draw a diagram 7N F 70 o 7 sin 35 o 7 cos 35 o + 7 cos 35 o

Horizontally 7 sin 35 o - 7 sin 35 o = 0 Vertically 7 cos 35 o + 7 cos 35 o = F F = 11.5N And at what angle? 145 o to one of the strings.