vectors and direction investigation key question: how do you give directions in physics?

Vectors and Direction Vectors and Direction

Investigation Key Question:

How do you give directions in physics?

Vectors and DirectionVectors 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.


• 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.


• 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


• 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.


• 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 Representing vectors with componentscomponents

• 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 Representing vectors with componentscomponents

• 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 VectorsAdding Vectors

• Writing vectors in components make it easy to add them.

Subtracting VectorsSubtracting 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:

– X1 = (-2, 0) m + X2 = (0, 3) m + X3 = (6, 0) m– = (-2 + 0 + 6, 0 + 3 + 0) m = (4, 3) m– The final displacement is 4 meters east and 3 meters

north from where the ant started.

Calculating the resultant vector Calculating the resultant vector by adding componentsby adding components

An ant walks 2 meters West, 3 meters North, and 6 meters East. What is the displacement of the ant?

Calculating Vector ComponentsCalculating 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 Finding components mathematicallymathematically

• 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.

Finding the Magnitude of a Finding the Magnitude of a VectorVector

• 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 Adding Vectors AlgebraicallyAlgebraically

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.

Vector X Y

A = (r, Θ) = rcosΘ = rsinΘ

B = (r, Θ) = rcosΘ = rsinΘ

R = A + B Ax + Bx Ay + By

What Quadrant?What Quadrant?

• Your answer for Θ is not necessarily complete!– If you have any negatives on your Rx or Ry,

you need to check your quadrant.

(+,+) = 1st = 0-90o

(-,+) = 2nd = 90o – 180o

(-,-) = 3rd = 180o -270o

(+,-) = 4th = 270o -360o

EquilibriantEquilibriant

• Like “equilibrium”

• The vector that is equal in magnitude, but opposite in direction to the resultant.

• Ex. R = (30m, -50o)E = (30m, 130o)

Forces in Two DimensionsForces in Two Dimensions

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

Force VectorsForce 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 ForcesEquilibrium and Forces

• It is much more difficult for a gymnast to hold his arms out at a 45-degree angle.

• Why?


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.


• 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. (Fy

= 350)

Resultant


• 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 planeThe 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.

Fn

Fg

Fa

Ff

Forces on an inclined planeForces on an inclined plane

• The friction force is equal to the coefficient of friction times the normal force in the y direction:

Ff = -Fn cosθ

Fn = mg

Ff = -mg cosθ.

Motion on an inclined planeMotion 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/sec2)

Mass (kg)

Acceleration (m/sec2)


• 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:

Fx =

a

m sin θ

g

F m =

a = g sin θ


• To account for friction, the acceleration is reduced by the opposing force of friction:

Fx = mg sin θ - mg cos θ

Fx = (50 kg)(9.8 m/s2) (sin 20o) = 167.6 N

Fnet = Fx – Ff = 167.6 N – 30 N = 137.6 N

Calculate the acceleration: a = F/m

a = 137.6 N ÷ 50 kg = 2.75 m/s2

Calculating accelerationCalculating 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

vectors and direction investigation key question: how do you give directions in physics?

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