Vectors and Direction Investigation Key Question: How do you give directions in physics?

Vectors and Direction A scalar is a quantity that can be completely described by one value: the magnitude. You can think of magnitude as size or amount, including units.

Vectors and Direction A vector is a quantity that includes both magnitude and direction. Vectors require more than one number. –The information “1 kilometer, 40 degrees east of north” is an example of a vector.

Vectors and Direction In drawing a vector as an arrow you must choose a scale. If you walk five meters east, your displacement can be represented by a 5 cm arrow pointing to the east. *use a ruler, not the boxes

Vectors and Direction Suppose you walk 5 meters east, turn, go 8 meters north, then turn and go 3 meters west. Your position is now 8 meters north and 2 meters east of where you started. The diagonal vector that connects the starting position with the final position is called the resultant.

Vectors and Direction The resultant is the sum of two or more vectors added together. You could have walked a shorter distance by going 2 m east and 8 m north, and still ended up in the same place. The resultant shows the most direct line between the starting position and the final position. A B C R R = A+B+C

*Use a ruler not the boxes on graph paper!

Representing vectors with components Every displacement vector in two dimensions can be represented by its two perpendicular component vectors. The process of describing a vector in terms of two perpendicular directions is called resolution.

Representing vectors with components Cartesian coordinates are also known as x-y coordinates. –The vector in the east-west direction is called the x-component. –The vector in the north-south direction is called the y-component. The degrees on a compass are an example of a polar coordinates system. Vectors in polar coordinates are usually converted first to Cartesian coordinates.

Adding Vectors Writing vectors in components make it easy to add them.

Subtracting Vectors To subtract one vector from another vector, you subtract the components.

1.You are asked for the resultant vector. 2.You are given 3 displacement vectors. 3.Sketch, then add the displacement vectors by components. 4.Add the x and y coordinates for each vector: –X 1 = (-2, 0) m + X 2 = (0, 3) m + X 3 = (6, 0) m –= ( , ) m = (4, 3) m –The final displacement is 4 meters east and 3 meters north from where the ant started. Calculating the resultant vector by adding components An ant walks 2 meters West, 3 meters North, and 6 meters East. What is the displacement of the ant?

Calculating Vector Components Finding components graphically makes use of a protractor. Draw a displacement vector as an arrow of appropriate length at the specified angle. Mark the angle and use a ruler to draw the arrow.

Finding components mathematically Finding components using trigonometry is quicker and more accurate than the graphical method. The triangle is a right triangle since the sides are parallel to the x- and y-axes. The ratios of the sides of a right triangle are determined by the angle and are called sine and cosine.

Y X

Finding the Magnitude of a Vector When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem.

Adding Vectors Algebraically 1.Make a chart 2.Find the x- and y- components of all the vectors 3.Add all of the numbers in the X column 4.Add all of the numbers in the Y column 5.This is your resultant in rectangular coordinates. VectorXY A = (r, Θ)= rcosΘ= rsinΘ B = (r, Θ)= rcosΘ= rsinΘ R = A + BA x + B x A y + B y

What Quadrant? Your answer for Θ is not necessarily complete! –If you have any negatives on your R x or R y, you need to check your quadrant. (+,+) = 1 st = 0-90 o (-,+) = 2 nd = 90 o – 180 o (-,-) = 3 rd = 180 o -270 o (+,-) = 4 th = 270 o -360 o

Equilibriant Like “equilibrium” The vector that is equal in magnitude, but opposite in direction to the resultant. Ex. R = (30m, -50 o ) E = (30m, 130 o )

Forces in Two Dimensions Investigation Key Question: How do forces balance in two dimensions?

Force Vectors If an object is in equilibrium, all of the forces acting on it are balanced and the net force is zero. If the forces act in two dimensions, then all of the forces in the x- direction and y-direction balance separately.

Equilibrium and Forces It is much more difficult for a gymnast to hold his arms out at a 45- degree angle. Why?

Forces in Two Dimensions 2) Use the y-component to find the force in the gymnast’s arms. 1) Resolve the force supported by the left arm into the x and y components.

Forces in Two Dimensions The force in the right arm must also be 495 newtons because it also has a vertical component of 350 N. The vertical force supported by the left arm must be 350 N because each arm supports half the weight. (F y = 350) Resultant

Forces in Two Dimensions When the gymnast’s arms are at an angle, only part of the force from each arm is vertical. (350 N) The resultant force must be larger (495 N) because the vertical component in each arm is only part of the resultant.

The inclined plane An inclined plane is a straight surface, usually with a slope. Consider a block sliding down a ramp. There are four forces that act on the block: –gravity (weight). –Normal force –friction –the reaction force acting on the block. FnFn FgFg FaFa FfFf

Forces on an inclined plane The friction force is equal to the coefficient of friction times the normal force in the y direction: F f = -  Fn cosθ F n = mg F f = -  mg cosθ.

Motion on an inclined plane Newton’s second law can be used to calculate the acceleration once you know the components of all the forces on an incline. According to the second law: a = F m Force (kg. m/sec 2 ) Mass (kg) Acceleration (m/sec 2 )

Motion on an inclined plane Since the block can only accelerate along the ramp, the force that matters is the net force in the x direction, parallel to the ramp. If we ignore friction, and substitute Newton's' 2nd Law and divide by m, the net force in the x direction is: F x = a msin θg Fm = a = g sin θ

Motion on an inclined plane To account for friction, the acceleration is reduced by the opposing force of friction: F x = mg sin θ -  mg cos θ

F x = (50 kg)(9.8 m/s 2 ) (sin 20 o ) = N F net = F x – F f = N – 30 N = N Calculate the acceleration: a = F/m a = N ÷ 50 kg = 2.75 m/s 2 Calculating acceleration A skier with a mass of 50 kg is on a hill making an angle of 20 degrees. The friction force is 30 N. What is the skier’s acceleration?

A Global Positioning System (GPS) receiver determines position to within a few meters anywhere on Earth’s surface. The receiver works by comparing signals from three different GPS satellites. About twenty-four satellites orbit Earth and transmit radio signals as part of this positioning or navigation system. Robot Navigation