Inclined Planes

•Rotated coordinate system

•Motion on inclines

•Motion with friction on inclines

Inclined Planes

• Since many real life situations occur either on a ramp, hill, roof, or some other form of an incline, inclined plane problems are very common.

• Thus far we have looked at only horizontal (x) and/or vertical (y) motion.

• Using this xy coordinate system for inclined plane situations can be problematic.– This is because objects moving up (or

down) an incline are moving in both the x and y direction at the same time.

• Therefore, a rotated coordinate system is used to solve incline plane problems.

• x-direction is rotated to be parallel to the plane’s surface (called F|| in text)

• y-direction is rotated to be perpendicular to the plane’s surface (called in text)F

Rotated Coordinate System

Inclined Plane

y

x

NOTE: Positive is taken to be in the direction of the motion

Show ALL of the Forces acting on the block below.

Fg

FN

f

What force makes the block slide down the incline?

Fg

FN

f

Gravity

• In fact it is the component of the force of gravity that acts parallel to the incline that makes the block slide down the ramp.

• This component is called Fx

• Is the normal force equal to the force of gravity?

• It is less than the force of gravity. The normal force is the component of the gravity (Fy) that acts perpendicular to the incline.

• FN = Fy

• Thus FN and Fx are based on the mass of the block and the angle of the incline.

What is the formula for FN and Fx?

Fg

FN

FxFy

Fx

Fy = FN

q

In this triangle where does q go?

WHY?

Fx = Fgsinq=mgsinq

Fy =FN= Fgcosq

=mgcosq

Examples• Without friction:

1. Assuming a frictionless incline, find the acceleration of the 10. kg block below with incline angles:

A) 0o B) 30.o C) 45o D) 60.o E) 90.o

netFa

m

xFm

sinmgm

q sing q

• Thus, acceleration is independent of mass in problems without friction

• Same as for free fall!

A) 0o B) 30.o C) 45o

D) 60.o E) 90.o

Page 196. #2

Examples with Friction

Determine the acceleration on a block with mass m, on an incline with angle q, with a coefficient of sliding friction mk.

netFa

m xF f

m

sin cos kmg mga

mq qm

cosN k kf F mgm qm

sin cos kg gq qm

• Thus, acceleration is independent of mass in problems with friction!

EX. 3. Determine the acceleration on a block with mass 25.0kg, on an incline with angle 25.0o, with a coefficient of sliding friction mk0.100

sin cos ka g gq qm

2 29.8 sin25 9.8 cos25 (0.100)o om m

s s

23.25

ms

In the previous example, find the time it would take for the block slide 30. cm if it starts sliding from rest.

21

1

2d vt at

22

10.30 0 (3.25 )

2

m mm t

s s

0.43t s

Lets do #1,3,4 on page 196

1.A copy of Physics: Concepts and Connections leaves the printing pressand slides down a 4.0-m-long ramp into the arms of an eager physicsstudent. If the ramp is inclined at an angle of 25° to the horizontaland has a coefficient of kinetic friction of 0.10, how long will it takethe 2.0-kg textbook to reach the student?

t =1.6s

3.• A skateboarder slides down a frictionless ramp inclined at an

angle of 30° to the horizontal. He then slides across a frictionless horizontal floor and begins to slide up a second incline at an angle of 25 ° to the horizontal. The skateboarder starts at a distance of 10 m from the bottom of the first incline. How far up the second incline will he go if the coefficient of kinetic friction on the second incline is 0.10?

4.Batman is driving the Batmobile down a hill coming from the Bat Cave.The hill is inclined at an angle of 30.° to the horizontal and has a coefficient of kinetic friction of 0.28. What force must the Batmobile'sengine apply to cause the Batmobile to accelerate at 10.0g?

Page 224#7, 8, 9, 10

Inclined planes can be used to determine coefficient of static friction

• The force of static friction is equal to the force of gravity acting along the plane, if the block is at rest.

Also,

And,

sins xf F mg q

cosN yF F mg q

sin

coss

sN

mgfF mg

qm

q tansm q

• The tan of the angle of the incline gives the coefficient of static friction.

• To find the maximum static friction, raise the incline until the object moves.

Page 224 #6