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5' Using Newtons Laws

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Determine the object, or objects, of interest. Determine real ... that a roller-coaster car stays on the track at. the top of the loop? Identify forces on car ... – PowerPoint PPT presentation

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Title: 5' Using Newtons Laws


1
5. Using Newtons Laws
2
Using Newtons Laws
  • General Method
  • Determine the object, or objects, of interest.
  • Determine real forces are acting on each object.
  • For each object, find the net force.
  • Insert the net force into the 2nd law and solve

3
Multiple Objects
4
Example Saving a Climber
Newtons 2nd law applies to each climber For
this example, we assume no friction rope does
not stretch rope of negligible mass
5
Example (2)
  • Forces on climber Steve
  • Gravity
  • Normal force
  • Tension in rope
  • Forces on climber Paul
  • Gravity
  • Tension in rope

6
Example (3)
Steve
Paul
1. Choose coordinate system for
each climber. 2. Sum forces for each and apply
2nd law.
7
Example (4) Steve
Steve
8
Example (5) Paul
Paul
Note for Paul, we have chosen a frame of
reference with x pointing down.
9
Example (6)
As usual, we equate components. But for this
problem only the x components are relevant
Steve Paul
10
Example (7)
By assumption
Therefore,
Steve Paul
11
Example (8)
Acceleration
Rope tension
12
Circular Motion
13
Circular Motion
is a unit vector that points from the center of
the circle to the object.
is the velocity of the object. As the
object moves around the circle, the direction of
the velocity changes as does the direction of the
unit vector .
14
Circular Motion
Since the path is a circle, the velocity is
always perpendicular to the unit vector . We
can therefore write
where the unit vector is in the direction of
the velocity and perpendicular to the unit vector
.
15
Circular Motion
The unit vectors and are related by a
clockwise rotation by 90o. If we represent this
rotation by the symbol R we can write one unit
vector in terms of the other
and the velocity as
16
Circular Motion
By definition, the acceleration is
since the speed v and rotation R are constant.
17
Circular Motion
The change in the unit vector is in the same
direction as the velocity. Therefore, we can
write
where c is a constant to be determined.
18
Circular Motion
The constant c is the speed with which the tip
of the unit vector moves around the circle of
unit radius
Therefore,
19
Circular Motion
When
is substituted into
we get
since
20
Circular Motion
We see that the acceleration is directed towards
the center of the circle, that is, it is
centripetal, and its magnitude is
21
Example 5.7 Loop-the-Loop!
What is the minimum speed needed to
guarantee that a roller-coaster car stays on the
track at the top of the loop?
  • Identify forces on car
  • Gravity
  • Normal force from track

22
Example 5.7 (2)
We have just seen that to move in a circle,
an object must have a centripetal
acceleration. According to the 2nd law, the
acceleration is caused by a net force.
23
Example 5.7 (3)
Since the acceleration is in the same direction
as the net force, it follows that the net force
must be centripetal, that is, directed towards
the center of the loop. What are these forces?
24
Example 5.7 (4)
Presumably, they must be the two forces we have
identified the weight and the normal force.
As usual, we need to set up a coordinate system.
25
Example 5.7 (5)
Coordinate system Take y to be downwards
Take x to the right
26
Example 5.7 (6)
At the top of the loop, the normal force and the
gravitational force point downwards and towards
the center of the circle. Therefore, in the y
direction
27
Example 5.7 (7)
Solving for v we get
At the minimum speed the car is on the verge of
leaving the tracks at the top of the loop. This
occurs when the normal force, n, is zero!
28
Friction
29
The Nature of Friction
  • Friction is an electrical force between the
    molecules of surfaces in contact.
  • Unlike gravity, however, friction is a very
    complicated force to describe accurately.

30
The Nature of Friction
  • But, for many everyday situations, such as
    dragging an object along a floor, we can describe
    frictional forces using simple, approximate,
  • expressions.

31
Frictional Forces
  • Static Friction This is the frictional force
    between surfaces that are at rest relative to
    each other. The maximum static frictional force
    is found to be
  • fs ms n
  • where n is the magnitude of the normal force.
    ms is called the coefficient of static friction.

32
Frictional Forces
  • Kinetic Friction This is the frictional force
    between surfaces that are moving relative to each
    other. Its value is found to be
  • fk mk n
  • where n is the magnitude of the normal force.
    mk is called the coefficient of kinetic friction.

33
Frictional Forces
  • It is found that as the applied force increases
    so does the opposing frictional force until a
    maximum value is reached. When the applied force
    exceeds the maximum frictional force the object
    accelerates.
  • During acceleration the frictional force
    decreases and remains constant when the motion is
    constant.

34
Frictional Forces
Maximum frictional force Accelerating
ms n
mk n
Frictional force
Constant speed
At rest
Time
Frictional force remains equal to and opposite
the applied force.
35
Friction in Action
Without friction it would be impossible to walk
or make a vehicle move.
As you push against the ground, the ground pushes
you forwards!
36
Example Dragging a Box
What rope tension is needed to move the box
at constant velocity, assuming a coefficient of
kinetic friction mk between box and floor?
37
Example Dragging a Box
Draw free-body diagram for box. The magnitude of
the kinetic friction force is fk mk n
38
Example Dragging a Box
The motion is constant, so the forces cancel
fk T cosq 0 (x-dir.) mg n T sinq
0 (y-dir.)
39
Example Dragging a Box
The magnitude of the tension is therefore
40
Summary
  • Big Idea A net force causes changes in motion.
  • How to Apply
  • Find all real forces on a body, sum them, and
    apply Newtons 2nd and 3rd laws.
  • Frictional force
  • Increases until object moves, then reduces and
    remains constant when motion is constant.
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