Download 4-8 inclined planes

4-8 Applications Involving Friction, Inclines
An object sliding down an incline has three forces acting
on it: the normal force, gravity, and the frictional force.
• The normal force is always perpendicular to the surface.
• The friction force is parallel to it.
• The gravitational force points down.
If the object is at rest,
the forces are the same
except that we use the
static frictional force,
and the sum of the
forces is zero.
Inclined Planes
• Label the direction of N and mg.
N
θ
mg
Use slide show mode (press F5) for animations.
Inclined Planes
• Mark the direction of acceleration a.
N
a
θ
mg
Inclined Planes
• Choose the coordinate system with x in the same
or opposite direction of acceleration and y
perpendicular to x.
y
N
x
a
θ
mg
Inclined Planes
• Now some trigonometry
y
N
x
a
θ
90- θ
θ
mg
Inclined Planes
• Replace the force of gravity with its components.
y
N
x
a
θ
θ
mg
Inclined Planes
• Use Newton’s second law for both the x and y
directions
Fx  max  ma
y
N
x
a
 mg sin   ma
Fy  may  0
θ
θ
mg
 N  mg cos   0
The force and acceleration in the x-direction have a negative sign
because they point in the negative x-direction.
Inclined Planes
• Why is the component of mg along the x-axis –mgsinθ
• Why is the component of mg along the y-axis –mgcosθ
y
N
x
a
θ
θ
mg
Inclined Planes
• Why is the component of mg along the x-axis: –mgsinθ
• Why is the component of mg along the y-axis: –mgcosθ
Inclined Planes
• Why is the component of mg along the x-axis: –mgsinθ
• Why is the component of mg along the y-axis: –mgcosθ
Inclined Planes
• Why is the component of mg along the x-axis: –mgsinθ
• Why is the component of mg along the y-axis: –mgcosθ
sinθ =
opposite
hypotenuse
cosθ =
adjacent
hypotenuse
Inclines

Ff
FN
mg cos 


mg 
mg sin 


Tips
•Rotate Axis
•Break weight into components
•Write equations of motion or
equilibrium
•Solve
Friction & Inclines
A person pushes a 30-kg shopping cart up a 10 degree incline with a force of 85
N. Calculate the coefficient of friction if the cart is pushed at a constant
speed.
Fa  Ff  mg sin 
Fa
Fn
Ff  k FN
Fa  k FN  mg sin  FN  mg cos 
Fa  k mg cos   mg sin 
Fa  mg sin   k mg cos 
mg cos 

Ff
mg
mg sin 

Fa  mg sin 
k 
mg cos 
85  (30)(9.8)(sin10)
k 

(30)(9.8)(cos10)
0.117
Example
A 5-kg block sits on a 30 degree incline. It is attached to string that is thread
over a pulley mounted at the top of the incline. A 7.5-kg block hangs
from the string.
 a) Calculate the tension in the string if the acceleration of the system is 1.2
m/s/s
 b) Calculate the coefficient of kinetic friction.
T
FN
FNET  ma
m1 g  T  m1a
m2gcos30
30
T
m2g
m1
30
m2gsin30
m1g
T  ( Ff  m2 g sin  )  m2 a
Ff
FN  m2 g cos 
Example
FNET  ma
m1 g  T  m1a
m1 g  m1a  T
T  ( Ff  m2 g sin  )  m2 a
T  F f  m2 g sin   m2 a
T  k FN  m2 g sin   m2 a
T  m2 a  m2 g sin   k FN
(7.5)(9.8)  (7.5)(1.2)  T T  m a  m g sin 
2
2
 k
T  64.5 N
FN
FN  m2 g cos 
T  m2 a  m2 g sin 
 k
m2 g cos 
64.5  (5)(1.2)  (5)(9.8)(sin 30)
 k
(5)(9.8)(cos 30)
k  0.80 N
Practice Problems

43. (a) A box sits at rest on a
rough 30 degrees inclined
plane. Draw a free body
diagram. (b) How would the
diagram change if the box
were sliding down the plane?
(c)How would it change if the
box were sliding up the plane
after an initial shove?

40. The coefficient of static
friction between hard rubber
and normal street pavement is
0.8. On how steep a hill
(maximum angle) can you
leave a car parked?
4-9 Problem Solving – A General Approach
1. Read the problem carefully; then read it again.
2. Draw a sketch, and then a free-body diagram.
3. Choose a convenient coordinate system.
4. List the known and unknown quantities; find
relationships between the knowns and the
unknowns.
5. Estimate the answer.
6. Solve the problem without putting in any numbers
(algebraically); once you are satisfied, put the
numbers in.
7. Keep track of dimensions.
8. Make sure your answer is reasonable.
Summary of Chapter 4
• Newton’s first law: If the net force on an object
is zero, it will remain either at rest or moving in a
straight line at constant speed.
• Newton’s second law:
• Newton’s third law:
• Weight is the gravitational force on an object.
• The frictional force can be written:
(kinetic friction) or
(static friction)
• Free-body diagrams are essential for problemsolving
Homework:
Chapter 4 Problems 41, 53, 55, 56
Kahoot 4-8