1. Chapter 4 Revisited Forces in two dimensions
Analyzing force problems is only slightly more complicated when forces act in more than one dimension.

2. Newton’s 2nd Law in 2-D
You must still identify all forces and draw your free body diagram.
Choose a coordinate system.
You then resolve your problem into an x-problem and a y-problem (like projectile motion).

3. Equilibrium
An object either at rest or moving with a constant velocity is said to be in equilibrium
The net force acting on the object is zero so
In two dimensions this becomes,

4. Accelerated motion
Similar to equilibrium except
Use components

5. Multiple objects
If more than one object is present, draw free body diagram for each object
Apply equations, keeping track of signs

6. Sample problem
Larry pushes a 200 kg block on a frictionless floor at a 45o angle below the horizontal with a force of 150 N while Moe pulls the same block horizontally with a force of 120 N.
a) Draw a free body diagram.
b) What is the acceleration of the block?
c) What is the normal force exerted on the block?

8. Inclined Planes
Choose the coordinate system with x along the incline and y perpendicular to the incline
Replace the force of gravity with its components
The normal force is perpendicular to inclined surfaces. It’s always equal to the component of weight perpendicular to the surface. x

9. Sample problem
Find the normal force exerted on a 2.5-kg book resting on a surface inclined at 28 above the horizontal.
If the angle of the incline is reduced, do you expect the normal force to increase, decrease, or stay the same?

11. Sample Problem
How long will it take a 1.0 kg block initially at rest to slide down a frictionless 20.0 m long ramp that is at a 15o angle with the horizontal?

13. Static friction on a ramp
surface fs N
Without friction, the book will slide down the ramp. If it stays in place, there is sufficient static friction holding it there.
Physics
W = mg q
Wx = mgsinq and N = mgcosq
At maximum angle before the book slides, we can prove that ms = tanq.

14. Static friction on a rampx
surface fs
Assume q is maximum angle for which book stays in place. N
SF = 0
Wx = fs
mgsinq = msmgcosq
ms = sinq/cosq = tanq
Physics q Wx
W = mg q
fs = mgsinq and N = mgcosq
At maximum angle before the book slides, we can prove that ms = tanq.

15. Problem
A 900kg polar bear slides down a snow bank at an angle of 25 degrees. The coefficient of friction between the bear and the snow is .05; find the force of friction.

16. Problem
A 10-kg wooden box slides on a wooden ramp. The coefficient of kinetic friction is 0.30.
What is the friction force between the box and ramp if the ramp is at a 45o angle?
What is the acceleration of the box?