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Simplifying Problems

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Simplifying Problems. Free-Body Diagrams. used to isolate a system of interest and to identify and analyze the external forces that act directly upon it. Free-Body Diagrams. common forces in free-body diagrams include: tension forces gravity (weight) normal force friction. Ideal Strings.

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slide2

Free-Body Diagrams

  • used to isolate a system of interest and to identify and analyze the external forces that act directly upon it
slide3

Free-Body Diagrams

  • common forces in free-body diagrams include:
    • tension forces
    • gravity (weight)
    • normal force
    • friction
slide4

Ideal Strings

  • ideal strings...
    • have no mass; therefore, do not affect acceleration
    • do not stretch
slide5

Ideal Strings

  • ideal strings...
    • exert only pulling forces—you can’t push on a string!
    • exert forces only in line with the string
slide6

Ideal Strings

  • hold objects at fixed distances
  • all objects connected by the string are pulled with the same speed and acceleration
slide7

Connected Objects

There is no single, uniform way to solve every problem involving mechanics and connected objects.

Free-body diagrams can be very helpful in the analysis of these problems.

slide8

Connected Objects

Drawing a “world diagram” is a good way to start.

It should include all connections and include an arrow showing the direction of motion (if known).

slide9

Connected Objects

When drawing individual free-body diagrams for each object, include force vectors showing all forces acting on the object.

slide10

Connected Objects

Select a coordinate system for each object.

It is not necessary for all objects to use the same coordinate system.

slide11

Example 8-1

  • Draw the world diagram.
  • Draw a free-body diagram of block 2.
  • Calculate the acceleration of the system.
  • Calculate the tension force for block 2.
slide13

Ideal Pulleys

  • used to change the direction of tension in a string
  • has the following characteristics:
slide14

Ideal Pulleys

  • It consists of a grooved wheel and an axle. It can be mounted to a structure outside the system or attached directly to the system.
slide15

Ideal Pulleys

  • Its axle is frictionless.
  • The motion of the string around the pulley is frictionless.
slide16

Ideal Pulleys

  • It changes the direction of the tension in the string without diminishing its magnitude.
slide17

Example 8-2

The free-body diagrams are drawn first.

Take special note of the coordinate system used for each block!

Check all directions when you have finished.

slide18

Example 8-3

The free-body diagrams are drawn first.

Be especially careful with the components this time!

Are the pulleys moving in the direction you calculated?

slide19

Inclines

  • Since coordinate systems are chosen, it is usually wisest to make the x-axis parallel to the incline.
  • Of course, the x-axis and y-axis must be perpendicular.
slide20

Normal Force

  • This is the force exerted by a surface on the object upon it.
  • It is always exerted perpendicular to the surface (hence, “normal”).
  • It is notated N.
slide21

Normal Force

  • On a flat surface, the normal force has a magnitude equal to the object’s weight, but with the opposite direction.
  • N = -Fw norm = -Fwy
slide22

Normal Force

  • On an inclined surface, the normal force has a magnitude smaller than the magnitude of the object’s weight.
  • Trigonometry is needed to find N’s components.
slide23

Normal Force

  • If an object is not moving, the normal force can be used to measure the object’s weight.
  • This is simplest with an unaccelerated reference frame.
slide24

Normal Force

But what if the object and scale are accelerating??

  • If they are accelerating upward, the apparent weight on the scale will be greater than the actual weight (see Ex. 8-6).
slide25

Normal Force

But what if the object and scale are accelerating??

  • If they are accelerating downward, the apparent weight on the scale will be less than the actual weight (see Ex. 8-7).
slide26

Normal Force

But what if the object and scale are accelerating??

  • If they are in free fall, the apparent weight on the scale will be zero (see Ex. 8-8).
slide28

What is Friction?

  • Definition: the contact force between two surfaces sliding against each other that opposes their relative motion
slide29

What is Friction?

  • explained by Newton’s 3rd Law
  • necessary for forward motion
  • necessary for rolling and spinning objects
slide30

Traction

  • friction that makes walking, rolling, and similar motions possible
  • notation: ft
  • also describes friction that prevents unwanted motion
slide31

Friction

  • opposes motion
  • rougher surfaces tend to have more friction
  • very smooth surfaces have increased friction
slide32

Friction

  • What affects its magnitude?
    • mass
    • area of surface contact does not affect it
    • greater on level surfaces than slopes
slide33

Friction

  • Friction is proportional to the mass and to the normal force on the object
  • f = μN
  • μ is called the coefficient of friction
slide34

Friction

  • μ is unique for each particular pair of surfaces in contact
  • μ is also dependent on the object’s state of motion
slide35

Kinetic Friction

  • More force is needed to start an object moving, than to keep it moving
  • μk is the coefficient of kinetic friction—the object is already moving
slide36

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):
    • is oriented parallel to the contact surface
    • opposes the motion of the system of interest
slide37

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):
    • depends in some ways on the kinds of materials in contact and the condition of the surfaces
slide38

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):
    • is generally independent of the relative speed of the sliding surfaces
slide39

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):
    • is generally independent of the surface area of contact between the surfaces
slide40

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):
    • is directly proportional to the normal force acting on the sliding object
slide41

Static Friction

  • friction between stationary objects
  • friction will prevent objects from sliding until the force parallel to the surface exceeds the static friction
slide42

Static Friction

  • 0 ≤ fs ≤ fs max
  • If the applied force parallel to the surface is less than fs max, static friction will cancel out applied force. No movement occurs.
slide43

Static Friction

  • If F > fs max, the surfaces will begin to slide.
  • magnitude for maximum static friction between two materials in contact:
  • fs max = μsN
slide44

Static Friction

  • Properties of static friction:
    • can be any value between zero and a maximum value characteristic for the materials in contact
slide45

Static Friction

  • Properties of static friction:
    • is oriented parallel to the contact surface
    • opposes the motion of the system of interest
slide46

Static Friction

  • Properties of static friction:
    • depends on the kinds of materials and condition of the contact surfaces
    • is normally independent of contact surface area
slide48

Rolling Friction

  • Defined: the sum total of all points of friction that retard the freedom of motion of the wheel, including the friction forces between the wheel and the surface over which it rolls
slide49

Rolling Friction

  • notation: fr
  • magnitudes: Fprop = Fapp – fr
  • forces: Fprop = Fapp + fr
slide50

Inclined-Plane Dynamics

  • Assign coordinate systems to each system element so that the x-axis is aligned to the sliding surface and pointing up the slope. If there are multiple objects, axes should point in the same general direction relative to their motion.
slide51

Inclined-Plane Dynamics

  • Resolve all forces acting on each element of the system into their components relative to the coordinate system for that system element.
slide52

Inclined-Plane Dynamics

  • Determine the maximum static friction possible for the two materials at the angle of incline.
slide53

Inclined-Plane Dynamics

  • Sum the nonfriction forces parallel to the sliding surface for the entire system and compare to the maximum static friction for the system to determine the dynamic state of the system.
slide54

Inclined-Plane Dynamics

  • If the system is accelerating, calculate the kinetic friction.
  • Sum the x-component forces, including kinetic friction, to find acceleration according to Newton’s 2nd Law.
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