Inclined Planes

Lecture and Lab!!

Inclined Planes and Gravitational Force

To analyze the forces acting on an object on an inclined plane (a tilted surface), the weight of the object, Fg, is resolved into 2 perpendicular components. One component is parallel to the incline, F-parallel, and the other is perpendicular to the surface, F-perpendicular

Continued!!

F-Parallel θ

F-perpendicular θ

Fg

** Due to similar triangles the two angles θ are equal**Also note that W is the hypotenuse!

Example # 1A book weighing 55N is resting on an inclined plane tilted at 30° from the horizontal. Find the F-parallel and F-perpendicular for this weight.

Solution

F-parallel 30° F-perpendicular

Fg= 55N

F-parallel = opposite sideF-perpendicular = adjacent sideFg = hypotenuse

Angle = 30°Therefore: sin 30° = F-parallel = F-parallel = 27.5N (F-parallel) Fg 55N

Solution Part 2cos 30 ° = F-perpendicular = F-perpendicular = 47.6N (F-

perpendicular) Fg 55N

The perpendicular component is 47.6N, and that the parallel component is 27.5N

Inclined PlanesWhat other forces are acting on the block besides gravity?

FN (normal force) is pushing upward perpendicular to the surface.

Since the book is not moving downwards(through the surface), Fnet = 0.

                                            

Inclined PlanesIf there is no friction, any motion down the plane is caused by F-parallel (note that F-parallel is Fnet)Therefore:a = F-parallel = Fg sin θ

m mSince Fg = mg , then a = g sin θ

Example # 1 ContinuesFind the acceleration and direction of the book’s motion.

Fg = 55N

Angle = 30°Force causing motion is F-parallel (assume the surface is frictionless)

Example # 1 Continueda = g sin θ = (-9.8m/s2)(sin 30°) = -4.9m/s2 (downward)

How does friction fit into this?Static friction is overcome when an angle is reached that causes an object to move down the plane.

Example # 2A key is placed on a book cover and begins to move when the cover is tilted at 40° from the horizontal. What is μ (coefficient of friction) between the key and the book?

SolutionAngle = 40°Ff = μFN

FN = F-perpendicular

Ff = F- parallel (since friction is parallel to the surface)

F-perpendicular = Fg cos θ

F-parallel = Fg sin θ

Solution continued..Therefore:

Ff = μFN

F-parallel = μ F-perpendicular

Fg sin θ = μ Fg cos θ

Sin θ = μ cos θTherefore μ = tan θ = tan 40° = 0.84

Inclined Planes

Get into your lab groups and make your way to the back!!!!