Download Chapter 6 - Applying Newton`s Laws

Chapter 5 - Applying Newton's Laws
I.
In solving dynamics problems, the following procedure can be used to set up and solve dynamic
problems:
A.
Pick a Particle:
-
B.
Draw a Free-body Diagram:
-
C.
the particle can be an object or a point on the object
you should know something about the particle and it should be related directly to what you
want to find
this is a diagram with vectors representing the forces acting on the particle
the vectors start at the particle and point away from the particle in the directions the forces act
remember the forces are noncontact (gravitational force) and contact forces
Draw a Coordinate System:
-
for static situations, try to draw the axes to make the analysis simple
for kinetic situations, draw one axis parallel to the motion, with the positive direction being the
direction in which the object is moving; the other axis is perpendicular (the advantage of
choosing the axes in this way is that the sum of the forces in one direction will be zero)
D.
Resolve the Forces into Components:
E.
Apply the Laws of Motion:
-
F.
sum the forces in the x-direction and set them equal to 0 or ma, depending on the type of
motion the object undergoes in that direction
sum the forces in the y-direction and set them equal to 0 or ma, depending on the type of
motion the object undergoes in that direction
Examples:
1.
A 1.50 kg book is at rest on a horizontal table. Find the force the table exerts on the book.
(Define: normal force, N)
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2) A 10 kg block is suspended from the ceiling by a rope of negligible mass. Find the "tension in the
rope." (What if the rope does have mass?)
3) A 100 N weight is being supported by three ropes. Find
the tensions in the ropes.
30
o
37
A
o
B
C
100 N
4) A 40 N force acting at an angle of 30 o above the horizontal pulls a 12 kg block over a smooth
horizontal surface. Find the acceleration of the block.
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5) What horizontal force is necessary to push a 100 kg block at a constant velocity up a smooth, 37o
inclined plane? (What about down the plane?)
F
37o
6)
Elevator problem: A 60 kg person standing on a bathroom scale in the elevator.
a) What is the reading on the bathroom scale?
b) What is the reading when accelerating upward at 2 m/s2?
c) What is the reading when decelerating 2 m/s2 at while traveling upward?
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7) A 5 kg block is at rest at the top of a smooth, 30 o inclined plane. The height of the block above the
ground is 2 m. How fast is it traveling when it reaches the bottom of the incline?
30o
Multiple Body Problems:
1) A 10 kg block is at rest on a smooth horizontal surface. It is connected to a 5 kg block by a light
string passing over a massless, frictionless pulley. Find the acceleration of the system and the
tension in the string.
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2) If the surfaces are frictionless, which way will the
system move and what is the acceleration of the
system?
100 kg
50 kg
53
o
37
Frictional Forces: the resistive force an object feels when moving on a surface
-
the frictional force is denoted by the letter f
-
the force is always parallel to the surfaces of contact and is directed opposite to the direction of
motion of the object relative to the surface, or in a direction opposite to the intended direction of
motion of the object relative to the surface.
-
the mechanics of the frictional force - exert an applied force of increasing magnitude on an object,
and graph the applied force against the frictional force
f
F
f
F
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o
On what does the friction force depend?
-
depends on the types of surfaces in contact; the differences are indicated by the different
values of the coefficient of friction
-
depends on the normal force, N, i.e., f is proportional to N (good approximation at slow
speeds)
Define a proportionality constant,  , called the coefficient of friction
-
coefficient of static friction, s
s =
f
maximu m static frictional force
 s max
normal force
N
or, fs max = sN and 0  f s  f s max  s N
-
coefficient of kinetic (sliding) friction, k
k =
f
kinetic frictional force
 k
normal force
N
or, fk = kN
-
in general, s > k
From where does the frictional force arise?
-
irregularities in the surfaces
- molecular bonding
Examples:
1) An 80 N force pulls horizontally on a 20 kg block on a horizontal surface. The coefficients of
friction are s = 0.30 and k = 0.25. Will the block move? If so, then what type of motion?
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2) Change the direction of the force to 37o above the horizontal.
3) A 150 N force pulls a 20 kg block up a 30 o inclined plane where the coefficient of kinetic friction is
0.20. Find the acceleration.
4) A block is given a kick on a horizontal surface and is traveling at 8 m/s immediately after the
kick. The coefficient of kinetic friction between the block and the surface is 0.30. How far will the
block travel before coming to rest?
5-7
5) What is the magnitude of a horizontally directed force that will cause a 20 kg block to accelerate
up a 37o incline at 4 m/s2. Take the coefficient of kinetic friction to be 0.25.
6) A 10 kg block is at rest on a horizontal surface. It is connected to a 5 kg block by a light string
passing over a massless, frictionless pulley. The coefficient of kinetic friction between the 10 kg
block and the surface is 0.20. Find the acceleration of the system and the tension in the string.
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7) If the coefficient of friction between all surfaces in contact is 0.25, find the force F that will
produce an acceleration of 2m/s2.
a)
4 kg
8 kg
F
b)
4 kg
F
8 kg
c)
4 kg
F
8kg
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II. Newton’s Laws and Curvilinear Motion
A. Anytime there is an acceleration, there must be a net, external force causing it, i.e.,

 F  ma ,

where the force and accelerations are in the same directions.
B. In curvilinear motion there are radial accelerations, ar , and tangential accelerations, at .
Therefore, there must be net forces in the radial direction and tangential directions that cause the
accelerations:
F
r
 mar and
 F  ma
t
t
Look at the motion of a pendulum:
v2
and no tangential acceleration.
r
Therefore, there must be a net, external force causing it. This force is called the radial force of the
"centripetal" force. That is,
v2
.
Fr  centripeta l force = m
r
C. In uniform circular motion, there is a radial acceleration ar =

Examples of horizontal circular motion:
1.
A 1.5 kg block at the end of a 1.2 m long string is whirled in a horizontal circle on a smooth
table. The time for one revolution is 2 sec. Find the tension in the string. What is the
centripetal force? What is the centrifugal force?
2.
A car traveling at 25 m/s travels on a flat, circular road. The coefficient of friction between
the tires and the road is 0.50. What is the smallest radius circular road the car can travel in
without sliding outward? Is the coefficient of friction the static or kinetic coefficient? What is
the centripetal force? What is the centrifugal force?
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3.
A car travels around a 60o banked track whose radius is 40 m. If the coefficient of friction
between the tires and the road is 0.60, then what are the maximum and minimum velocities
at which the car can travel without sliding up or down the banked track?
4.
A small mass m is at the end of a string of length L. The mass travels in a horizontal circular
path with the string tracing out a cone where the angle between the string and a vertical is .
Find the speed of the mass.
Examples of vertical circular motion:
1) What are the tensions in the string at the top and bottom of the circular path?
a.
at the top
b. at the bottom
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2.
A 60 kg person sits on the seat of a Ferris wheel. The radius of the Ferris wheel is 10 m and it
makes one revolution every 15 seconds. Find the magnitude of the force the person exerts on
the seat at the top and bottom of the circular path?
3.
A bucket filled with water is whirled in a vertical circle with a radius R. What is the
minimum speed at which the bucket can be traveling at the top so that the water does not
spill out?
4.
A car travels over the top of a hill of radius R. What is the maximum speed at which the car
can travel and not lose contact with the road?
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D. Resistive forces - air resistance: at low velocities the air resistance force is proportional to the
velocity of the object and acts opposite to the direction of travel of the object. Find its velocity as
a function of time.
E. Resistive forces – air resistance: at higher velocities the air resistance is proportional to the square
of the velocity, that is, the “drag force” is (1/2)CAv2, where C is the drag coefficient,  is the
density of the fluid, and A is the cross-sectional area of the object.
F. Motion in Accelerated Frames of Reference (noninertial frames): introduce fictitious, inertial, or
pseudo forces equal to ma.
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