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# Simplifying Problems - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Simplifying Problems' - mateo

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Presentation Transcript

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

• common forces in free-body diagrams include:

• tension forces

• gravity (weight)

• normal force

• friction

• ideal strings...

• have no mass; therefore, do not affect acceleration

• do not stretch

• ideal strings...

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

• exert forces only in line with the string

• hold objects at fixed distances

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

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.

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).

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

Select a coordinate system for each object.

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

• 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.

• used to change the direction of tension in a string

• has the following characteristics:

• 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.

• Its axle is frictionless.

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

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

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.

The free-body diagrams are drawn first.

Be especially careful with the components this time!

Are the pulleys moving in the direction you calculated?

• 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.

• 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.

• 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

• 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.

• 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.

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).

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).

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).

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

• explained by Newton’s 3rd Law

• necessary for forward motion

• necessary for rolling and spinning objects

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

• notation: ft

• also describes friction that prevents unwanted motion

• opposes motion

• rougher surfaces tend to have more friction

• very smooth surfaces have increased friction

• What affects its magnitude?

• mass

• area of surface contact does not affect it

• greater on level surfaces than slopes

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

• f = μN

• μ is called the coefficient of friction

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

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

• 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

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

• is oriented parallel to the contact surface

• opposes the motion of the system of interest

• 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

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

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

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

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

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

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

• friction between stationary objects

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

• If F > fs max, the surfaces will begin to slide.

• magnitude for maximum static friction between two materials in contact:

• fs max = μsN

• Properties of static friction:

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

• Properties of static friction:

• is oriented parallel to the contact surface

• opposes the motion of the system of interest

• Properties of static friction:

• depends on the kinds of materials and condition of the contact surfaces

• is normally independent of contact surface area

• 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

• notation: fr

• magnitudes: Fprop = Fapp – fr

• forces: Fprop = Fapp + fr

• 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.

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

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

• 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.

• 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.