Chapter 5 Forces in Two Dimensions 1

Vectors: Vectors have both magnitude and direction. Vectors must be added using vector addition. You will have to treat vertical and horizontal vectors separately. You can add vectors in any order as long as you do not change there length or direction. 2 2

MATHEMTATICAL: 140 degrees counterclockwise from +x axis Measuring Angles GEOGRAPHICAL : 40 degrees North of West 50 degrees West of North MATHEMTATICAL: 140 degrees counterclockwise from +x axis 3

Vector Direction Examples Vector 35 m/s, due South Observe in the first example that the vector is said to have a direction of 40 degrees. This means that the tail of the vector was pinned down, and the vector was rotated an angle of 40 degrees in the counterclockwise direction beginning from due east. Observe in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down, and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east; a rotation of 240 degrees is equivalent to rotating the vector through two quadrants (180 degrees) and then an additional 60 degrees into the third quadrant. 4 4 4

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Resultant Vector The vector that results from the addition of 2 or more vectors. Always drawn from the “tail” of the first vector to the “tip” of the last vector. Direction should always be measured between the first vector and the resultant. PH Ch 4 Vector 6 6

Adding Vectors and Finding the Resultant: Method 1 Scaled Vector Diagram/Graphically Decide on a scale (EX: 1 km = 1 cm) Use a ruler to measure the vectors and a protractor to measure angle direction. Draw the vectors tip to tail. Draw the resultant vector from the tail of the first to the tip of the last vector. Use a ruler to measure the magnitude of the resultant vector Use a protractor to measure the angle of direction (angle between the 1st vector and the resultant vector). 7 7

Adding Vectors and Finding the Resultant: Method 2 Mathematical Method If the two vectors being added are at right angles, the magnitude can be found using the Pythagorean Theorem and the direction can be found using trig ratios (SOH CAH TOA). If the two vectors being added are at some angle other than 90, the magnitude and direction can be found by using the Law of Cosines and the Law of Sines. 8 8

Adding Vectors PH Ch 4 Vector 9 9

Law of Cosines a2 = b2 + c2 - 2bc (cos A) b2 = a2 + c2 - 2ac (cos B)                                      Law of Cosines a2 = b2 + c2 - 2bc (cos A) b2 = a2 + c2 - 2ac (cos B) c2 = a2 + b2 - 2ab (cos C) 10 10

                                     Law of Sines a = b = c__ sin A sin B sin C 11 11

Adding Vectors - perpendicular Add these vectors -determine the resultant. 2.0 m/s, 90 deg 7.0 m/s, 0 deg Resultant: Red 12 12 12

PH Ch 4 Vector

Adding Vectors Add the following vectors - determine the resultant. 3.0 m/s, 45 deg 5.0 m/s, 135 deg Parallelogram method PH Ch 4 Vector 14 14 14

Examples: A person walks 100m N and loses all sense if direction. Without knowing the direction, she walks 100m again. Draw a vector representation and determine the range of her displacement. 2. You are traveling from SMCC to Jackson for the cross country meet. You travel 30 km west, 20 km north, and 10 km west. Find your displacement (magnitude and direction) both graphically and mathematically. PH Ch 4 Vector 15 15

PH Ch 4 Vector

Examples: In your basketball game, you start at half court and run straight down the sideline 20 m. You then make a sharp 90 degree cut towards the lane. You run 15 m before the ball is thrown to you and you catch it with a jump stop. What is the magnitude and direction of your displacement? A person jogs 15 km and then turns to the right at a 45 degree angle and continues to run 25 more kilometers. Find the resultant vector (magnitude and direction) for the jogger. PH Ch 4 Vector 17 17

PH Ch 4 Vector

PH Ch 4 Vector

Components of a Vector – the horizontal and vertical vectors that make up the resultant 20

Components You can use trig to find the components. *Be careful if the angle is bigger than 90 degrees. You may have to use a reference angle. 21 21

Method 3: Vector Resolution/Components Two or more vectors can be added by: Resolving each vector into its x and y components. Add all the x-components to form the x- component of the resultant: Rx = Ax + Bx + Cx… Add all the y-components to form the y- component of the resultant: Ry = Ay + By + Cy… 22 22

Use the Pythagorean Theorem to find the magnitude of the resultant R. R2 = Rx2 + Ry2 Use tangent to find the direction of R. PH Ch 4 Vector 23 23

PH Ch 4 Vector 24 24

Examples: A bus travels 23 km on a straight road that is 30º N of E. What are east and north components of its displacement? A hammer slides down a roof that makes a 40 angle with the horizontal. What are the magnitudes of the components of the hammer’s velocity at the edge of the roof if it is moving at a speed of 4.25 m/s? 25

PH Ch 4 Vector

EX: Add the following three vectors using the component method: A is 4 m south, B is 7.3 m northwest, C is 6 m 30⁰ south of west. 27

PH Ch 4 Vector

Example: 2. A GPS receiver told you that your home was 15 km at a direction of 40º north of west, but the only path led directly north. If you took that path and walked 10 km, how far and in what direction would you then have to walk to reach your home? 29

PH Ch 4 Vector

FRICTION µ = “mu” = coefficient of friction Kinetic friction force: the force exerted on one surface by another surface when the objects are in motion EX: Sliding your book across your desk Ff = µkFN µ = “mu” = coefficient of friction Ff is proportional to the force pushing one surface against the other (FN) 31

FRICTION Static friction force: the force exerted on one surface by another surface when there is no motion between the two surfaces. EX: Pushing on a car or an extremely heavy crate… 32

FRICTION Eventually there is a limit to this static friction force – once the applied force is greater than the maximum static friction force, the object will begin to move. If the applied force increases, the static friction force will increase up to a maximum value. Ff ≤ μsFN At the instant before motion: Ff = μsFN 33

FRICTION Besides the normal force, friction also depends on the types of surfaces that are in contact. Different surfaces have different coefficients of friction (for static and kinetic) Table 5-1 p.129 34

EX: You push a 25 kg wooden box across a wooden floor at a constant speed of 1 m/s. How much force do you exert on the box? 35

PH Ch 4 Vector

EX: A small child is dragging a heavy, rubber- soled shoe by its laces across a sidewalk at a constant speed of 0.35 m/s. If the shoe has a mass of 1.56 kg, what is the horizontal component of the force exerted by the child? PH Ch 4 Vector

PH Ch 4 Vector

EX: If the child pulls with an extra 2 N in the horizontal direction, what will be the acceleration of the shoe? PH Ch 4 Vector

PH Ch 4 Vector

Inclined Planes A tilted surface is an inclined plane. Objects accelerate down inclined planes because of an unbalanced force. The force of gravity acts in the downward direction. The normal force acts in a direction perpendicular to the surface. PH Ch 4 Vector

Components of vectors Analyzing forces on inclined planes will involve resolving the weight vector (Fgrav) into two perpendicular components. - one parallel to the surface - one perpendicular to the inclined surface. The parallel force causes acceleration PH Ch 4 Vector

Components of vectors F11   Wt F11 The angle of incline always equals the angle between the weight vector and its perpendicular component. Use sine and cosine to find the components. PH Ch 4 Vector

Practice PH Ch 4 Vector

PH Ch 4 Vector

Example A trunk weighing 562 N is resting on a plane inclined 30º above the horizontal. Find the components of the weight force parallel and perpendicular to the plane. PH Ch 4 Vector

PH Ch 4 Vector

Example A 62 kg person on skis is going down a hill sloped at 37º. The coefficient of kinetic friction between the skis and the snow is 0.15. How fast is the skier going 5 s after starting from rest? PH Ch 4 Vector

PH Ch 4 Vector

Equilibrant Equilibrant – a force that puts an object in equilibrium. To find the equilibrant: Find the resultant of all the forces on the object. The equilibrant is the same in magnitude but opposite in direction. Equilibrant PH Ch 4 Vector

EX: What is the equilibrant for an 8 N force applied at 0º,a 6 N force applied at 90º, and a 7 N force applied at 60º? PH Ch 4 Vector

PH Ch 4 Vector

EX: What is the tension in each cable? PH Ch 4 Vector

PH Ch 4 Vector