Aim: How can we explain forces at an angle? Do Now: Solve for the x and y components: 10 N x y 30° x = 5 N x = 8.7 N

Why do crazy football coaches say to get low?  Demo  rfogIU&feature=related rfogIU&feature=related  If force is at an angle, some component of the force is in the x direction and some is in the y

Forces at an Angle  A 12 kg box is pulled across a table with a force of 50 N at an angle of 40 o above the horizontal. Draw a free body diagram Write a net force equation in the x and y direction What is the normal force? If the box moves at a constant velocity, what is the force of friction? What is the coefficient of friction? 40 o 50 N 12 kg

Question 2  A small child pulls a 25 kg wagon with 30 N of force over a frictionless surface. The angle that the handle makes with the ground is 27 o What is the horizontal component of the force? What is the vertical component of the force? Write a net force equation in the x and y direction What is the acceleration of the wagon If the wagon starts from rest, how far does the child go in 4 seconds?

Normal Force Do you remember where the normal force is directed? What if the plane is inclined?

On an incline:

The problem with an incline: FgFg x y The object moves along the plane Everything would need to be resolved into x and y components That’s a lot of sins and cosines Solution – rotate the x and y axis x y

This way, only F g needs to be resolved into x and y components - F ║ and F ┴

FNFN F g = mg F F II θ θ How can we solve for F ║ and F ┴ Mathematically, these θ’s are equal

FNFN FgFg F F II θ θ F F N = ? F N = F ┴ F N = F g cosθ F N = mgcosθ At rest or moving with a constant velocity: F F = ? F F = F ║ F F = F g sinθ F F = mgsinθ

FNFN FgFg F F II θ θ F On a frictionless incline: F Net = ma F ║ = ma F g sinθ = ma mgsinθ = ma gsinθ = a As θ increases, F II increases, so F net increases, and the object accelerates faster

On a surface with friction…  The force of friction decreases as the angle increases since: Remember: cos(90 o ) = 0 and cos(0 o ) = 1 The higher the angle, the lower the value of cosine

Ex: A 50 kg object rests on a table that is inclined 25 o from the horizontal. (a) Determine the components of gravity acting on the object. (b) What is the Normal force? F II = F g sinθ F II = mgsinθ F II = (50 kg)(9.8 m/s 2 )sin(25 o ) F II = 207 N F = F g cosθ F = mgcosθ F = (50 kg)(9.8 m/s 2 )cos(25 o ) F = 444 N F N = F ┴ = 444 N

What is the force of friction if the object is at rest? F F = F ║ F F = 207 N Assume the incline is now frictionless What is the acceleration down the incline? F Net = ma F ║ = ma 207 N = (50 kg)a a = 4.14 m/s 2

What is the coefficient of static friction? F F = µF N 207 N = µ(444 N) µ = 0.47 How far will the object travel in 5 s? d = v i t + ½at 2 d = (0 m/s)(5 s) + ½(4.14 m/s 2 )(5 s) 2 d = m