1. Forces in Two Dimensions
Chapter 5
Forces in Two Dimensions

2. 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

3. 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

4. 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

5. 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?

6. 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?

8. 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

9. 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)

10. 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?

11. 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

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

13. 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?

14. …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?

15. 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

16. 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.

17. 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

18. 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)