Forces in Two Dimensions Chapter 5 Forces in Two Dimensions

5.1 Vectors Vector problem from Chapter 4: If you pushed on a table with 40 N of force and your friend pushed with 40 N of force in the same direction, the resultant force would be: 80 N

If you pushed on a table with 40 N of force and your friend pushed with 60 N of force in the opposite direction, the resultant force would be: 20 N towards you

Vectors in Multiple Directions: To add vectors that are not at right angles to each other: Create a vector diagram If they are at right angles: Create a vector diagram OR Resolve algebraically using Pythagorean’s Theorem and SOH, CAH, TOA Pythagorean’s Theorem: R2 = A2 + B2

Practice Problem #1 A person walks 50 km east and then turns down a street that is 75o south of east and travels another 50 km. What is the person’s total distance walked? What is the person’s resulting displacement from the starting point?

Practice Problem #2 An airplane travels east at 200 m/s. A wind blows towards the north at 50 m/s. What is the resulting velocity of the plane?

Components of Vectors A single vector may be thought of as a resultant of 2 vectors which are called perpendicular components There is one horizontal and one vertical component for every vector Vector resolution – breaking a vector down into its components

Adding Vectors at Any Angle Resolve each vector into its horizontal and vertical components Vx=V cos q Vy=V sin q Sum the results for each Find the magnitude using the Pythagorean Theorem Find angle by the following formula: tan q = Vy (sum) Vx (sum)

Boat Problem! A boat heads east across a river that is 2.8 km wide with a velocity of 25 km/h. The river flows south with a velocity of 7.2 km/h. What is the resultant velocity of the boat? How long does it take the boat to cross the river? How far upstream is the boat when it reaches the opposite side?

5.2 Friction Static – starting friction; works against the start of motion Kinetic – sliding friction; works against keeping an object in motion Ff = m FN If the object is moving with a constant velocity, then the applied force (often called horizontal force) is equal to the frictional force so… FH = Ff

m – the coefficient of friction Greater for rougher surfaces Lesser for smoother surfaces Has no units! Is a number between 0 and 1

Try This! A 64-N box is pulled across a rough horizontal surface. What is the force necessary to keep the box moving at a constant speed if the coefficient of friction between the box and the floor is 0.81?

…and this… A 9.0 kg crate is resting on a floor. A 61-N force is required to just start motion of the crate across the floor. What is the coefficient of friction between the floor and the crate?

5.3 Force in Two Dimensions Equilibrium When the sum of all forces acting on an object is zero Equilibrant Force The force that will put all other forces in equilibrium To calculate: Find the resultant The equilibrant is equal in magnitude and opposite in direction Use the same force and add or subtract 180o to the direction

Try This: Two forces act on an object. One is 125 N pulling toward 57o. The other is 182 N pulling toward 124o. Find the one force that would put the other two in equilibrium. You may use any method that you want.

Motion Along An Inclined Plane A skier has several forces working on him as he moves down a hill: Gravitational force toward center of the earth Normal force perpendicular to the hill Frictional force parallel to the hill

Calculating the Components of Weight on an Inclined Plane: Fgx = Fg sin q (Parallel to the inclined plane) Fgy = Fg cos q (Perpendicular to the inclined plane)