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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Simplifying Problems

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Simplifying problems

Simplifying Problems


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


Simplifying problems

Free-Body Diagrams

  • common forces in free-body diagrams include:

    • tension forces

    • gravity (weight)

    • normal force

    • friction


Simplifying problems

Ideal Strings

  • ideal strings...

    • have no mass; therefore, do not affect acceleration

    • do not stretch


Simplifying problems

Ideal Strings

  • ideal strings...

    • exert only pulling forces—you can’t push on a string!

    • exert forces only in line with the string


Simplifying problems

Ideal Strings

  • hold objects at fixed distances

  • all objects connected by the string are pulled with the same speed and acceleration


Simplifying problems

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.


Simplifying problems

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).


Simplifying problems

Connected Objects

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


Simplifying problems

Connected Objects

Select a coordinate system for each object.

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


Simplifying problems

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.


Simplifying problems

Transmitting Mechanical Forces


Simplifying problems

Ideal Pulleys

  • used to change the direction of tension in a string

  • has the following characteristics:


Simplifying problems

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.


Simplifying problems

Ideal Pulleys

  • Its axle is frictionless.

  • The motion of the string around the pulley is frictionless.


Simplifying problems

Ideal Pulleys

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


Simplifying problems

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.


Simplifying problems

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?


Simplifying problems

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.


Simplifying problems

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.


Simplifying problems

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


Simplifying problems

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.


Simplifying problems

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.


Simplifying problems

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).


Simplifying problems

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).


Simplifying problems

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).


Simplifying problems

Friction


Simplifying problems

What is Friction?

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


Simplifying problems

What is Friction?

  • explained by Newton’s 3rd Law

  • necessary for forward motion

  • necessary for rolling and spinning objects


Simplifying problems

Traction

  • friction that makes walking, rolling, and similar motions possible

  • notation: ft

  • also describes friction that prevents unwanted motion


Simplifying problems

Friction

  • opposes motion

  • rougher surfaces tend to have more friction

  • very smooth surfaces have increased friction


Simplifying problems

Friction

  • What affects its magnitude?

    • mass

    • area of surface contact does not affect it

    • greater on level surfaces than slopes


Simplifying problems

Friction

  • Friction is proportional to the mass and to the normal force on the object

  • f = μN

  • μ is called the coefficient of friction


Simplifying problems

Friction

  • μ is unique for each particular pair of surfaces in contact

  • μ is also dependent on the object’s state of motion


Simplifying problems

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


Simplifying problems

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


Simplifying problems

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


Simplifying problems

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):

    • is generally independent of the relative speed of the sliding surfaces


Simplifying problems

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):

    • is generally independent of the surface area of contact between the surfaces


Simplifying problems

Kinetic Friction

  • Properties of the kinetic frictional force (fk = μkN):

    • is directly proportional to the normal force acting on the sliding object


Simplifying problems

Static Friction

  • friction between stationary objects

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


Simplifying problems

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.


Simplifying problems

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


Simplifying problems

Static Friction

  • Properties of static friction:

    • can be any value between zero and a maximum value characteristic for the materials in contact


Simplifying problems

Static Friction

  • Properties of static friction:

    • is oriented parallel to the contact surface

    • opposes the motion of the system of interest


Simplifying problems

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


Simplifying problems

More Applications of Friction


Simplifying problems

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


Simplifying problems

Rolling Friction

  • notation: fr

  • magnitudes: Fprop = Fapp – fr

  • forces: Fprop = Fapp + fr


Simplifying problems

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.


Simplifying problems

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.


Simplifying problems

Inclined-Plane Dynamics

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


Simplifying problems

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.


Simplifying problems

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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