Simplifying Problems

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 Free-Body Diagrams

ideal strings... have no mass; therefore, do not affect acceleration do not stretch Ideal Strings

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

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

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. 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). Connected Objects

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

Select a coordinate system for each object. It is not necessary for all objects to use the same coordinate system. Connected Objects

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.

Transmitting Mechanical Forces

Ideal Pulleys used to change the direction of tension in a string has the following characteristics:

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.

Ideal Pulleys Its axle is frictionless. The motion of the string around the pulley is frictionless.

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

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.

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?

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.

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.

Normal Force On a flat surface, the normal force has a magnitude equal to the object’s weight, but with the opposite direction. N = - F w norm = - F wy

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.

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.

Normal Force 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??

Normal Force 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??

Normal Force If they are in free fall, the apparent weight on the scale will be zero (see Ex. 8-8). But what if the object and scale are accelerating??

Friction

Definition: the contact force between two surfaces sliding against each other that opposes their relative motion What is Friction?

explained by Newton’s 3 rd Law necessary for forward motion necessary for rolling and spinning objects What is Friction?

friction that makes walking, rolling, and similar motions possible notation: f t also describes friction that prevents unwanted motion Traction

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

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

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

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

More force is needed to start an object moving, than to keep it moving μ k is the coefficient of kinetic friction—the object is already moving Kinetic Friction

Properties of the kinetic frictional force (f k = μ k N): is oriented parallel to the contact surface opposes the motion of the system of interest Kinetic Friction

Properties of the kinetic frictional force (f k = μ k N): depends in some ways on the kinds of materials in contact and the condition of the surfaces Kinetic Friction

Properties of the kinetic frictional force (f k = μ k N): is generally independent of the relative speed of the sliding surfaces Kinetic Friction

Properties of the kinetic frictional force (f k = μ k N): is generally independent of the surface area of contact between the surfaces Kinetic Friction

Properties of the kinetic frictional force (f k = μ k N): is directly proportional to the normal force acting on the sliding object Kinetic Friction

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

0 ≤ f s ≤ f s max If the applied force parallel to the surface is less than f s max, static friction will cancel out applied force. No movement occurs. Static Friction

If F > f s max, the surfaces will begin to slide. magnitude for maximum static friction between two materials in contact: f s max = μ s N Static Friction

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

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

More Applications of Friction

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

Rolling Friction notation: f r magnitudes: F prop = F app – f r forces: F prop = F app + f r

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. 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. Inclined-Plane Dynamics

Determine the maximum static friction possible for the two materials at the angle of incline. 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. 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 2 nd Law. Inclined-Plane Dynamics