Mechanics 1 Scalars and Vectors Monday, 10 December 2018 Leeds City College

Vectors and Scalars Scalars are quantities that have a value (or magnitude). Vectors have a magnitude and direction. Vectors and scalars can have the same units, for example velocity has units m s-1 … …and speed has units m s-1. Vectors can be used as scalars. You will sometimes be asked to find the magnitude of the vector. This means that you just put down the value. In some situations, direction is absolutely critical. Monday, 10 December 2018 Leeds City College

Note The product of a vector × scalar is a vector: Momentum = mass × velocity. Momentum is vector, mass is scalar, velocity is a vector. The product of two vectors is a scalar. Potential energy = mass × gravity × displacement Energy is a scalar, mass is a scalar, while gravity and displacement are vectors. The square of a vector is a scalar. Displacement squared (= area) is always positive. Monday, 10 December 2018 Leeds City College

Resultants of a Vectors For vectors in the same direction and the same plane, they simply add up. 3N 4N 7N For vectors in the opposite direction and the same plane, they simply take away. The resultant is in the direction of the bigger force 3N 1N 4N The resultant force is 5 N at an angle of 36.9o to the horizontal. These forces are separated to make them clearer. They are in the same plane. For vectors that are perpendicular, we use Pythagoras’ theorem. Resultant2 = 32 + 42 = 9 + 16 = 25.  Resultant = (25) = 5 N. 4N 3N 5N To work out the angle we can use the tan function:   tan q = ¾ = 0.75  q = tan-1(0.75) = 36.9o Monday, 10 December 2018 Leeds City College

Vectors not at right-angles 1 F2 F1 FR The cosine rule can be use to find the resultant. a2=b2+c2- 2bccosA Monday, 10 December 2018 Leeds City College

Vectors not at right angles 2 F2 FR F1 q a2=b2+c2- 2bccosA F22=F12+FR2- 2F1FRcosq Monday, 10 December 2018 Leeds City College

To Do Worked examples of vector sums. Answer Question 2. Monday, 10 December 2018 Leeds City College

Resolving Vectors Any vector in any direction can be resolved into horizontal and vertical components. These can be calculated by accurate drawing or trigonometry. F F sin q Vertical component q F cos q Horizontal component Monday, 10 December 2018 Leeds City College

Vectors in 3 D Resultant y R2 = x2 + y2 + z2 z You need to be aware of this, but it won’t be in the exam. x Monday, 10 December 2018 Leeds City College