1. Forces in 2-D
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2. Using Newton's second law in more than one dimension
So far we have considered processes where we need to consider motion along only one axis.
Most everyday processes involve forces that are not all along the axis of motion.
Now we learn to apply Newton's laws to these more complex processes.
This requires breaking forces into their vector components to find the forces along each axis.
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3. Example: Determining forces and acceleration
Alice pushes on a 2.0-kg book, exerting a 21.0-N force 45o above the horizontal as the book slides down a slippery vertical wall.
Sketch and translate
Simplify and diagram
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4. Newton's second law in component form
Apply Newton's second law to situations where forces are not along only one axis.
What is the acceleration if the object is accelerating in the x direction?
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5. Processes involving inclines
To solve problems on an incline, we choose the x-axis to be parallel to the incline and the y-axis to be perpendicular to the incline.
This choice makes the acceleration along the y-axis zero, simplifying the mathematical solution because the acceleration is nonzero only along the x-axis.
With this choice, the angle of gravitational force exerted by Earth on the system relative to the x-axis plus the angle of the incline relative to the horizontal is 90°.
AKA Think of your axis as rotatable!
The x axis is always WITH the direction of major movement.
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6. Book work number 18 example
You exert a force of known magnitude F on a grocery cart of total mass m. The force you exert on the cart points at an angle  below the horizontal. If the cart starts at rest, determine an expression for the speed of the cart after it travels a distance d. Ignore friction.

7. Objects linked together move with the same-magnitude acceleration
For objects linked together by a rope to move together, we need a rope that does not stretch.
All ropes stretch at least a little, but we make an assumption that the stretch is small enough to ignore.
These boxes have the EXACT SAME acceleration!
This includes DIRECTION!!
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8. Objects linked together move with the same-magnitude acceleration (Cont'd)
We assume the string is very light. If a pulley is present, we assume its mass is very small and it rotates with ignorable friction.
These assumptions allow us to state that the pulley shifts the motion but not the magnitude of the force exerted by the string.
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9. Book work number 33… sort of
The 60 kg block of a modified Atwood machine sits on a table, while a 15 kg block hangs off a pulley. The 60 kg block accelerates along the table, while the 15kg block accelerates down. What is the magnitude of the acceleration and the force that the cable exerts on the 15kg block. Ignore Friction.

10. Book work number 33… sort of
The 60 kg block of a modified Atwood machine sits on an incline, while a 15 kg block hangs off a pulley. The 60 kg block accelerates down and to the left along the ramp, while the 15kg block accelerates up. What is the magnitude of the acceleration and the force that the cable exerts on the block if the angle of the incline is 23. Ignore Friction.

11. Book Work number 34
A person holds a 200 g block that is connected to a 250 gram block by string going over a light pulley with no friction. After the person releases the 200 g block. It starts moving upwards and the heavier block descends. What is the acceleration of each block? What is the tension force exerted on each block?

12. Stop here and finish on Thursday 11/9
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13. Static friction
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14. Static friction
Static friction force is parallel to the surfaces of two objects that are not moving in relation to each other and opposes the tendency of one object to move across the other.
Static friction force changes magnitude to prevent motion, up to a maximum value called the maximum static friction force.
When the external force exceeds this static friction force, the object starts moving.
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15. Static friction can help us move
Sometimes friction is a necessary phenomenon for movement—for example, when walking on a flat horizontal sidewalk.
Static friction prevents your shoe from sliding at the start and end of a step.
This is also how car wheels propel a car forward.
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16. Tip about static friction and the normal force
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17. Relationship between normal force and friction force
The ratio of the maximum friction force to the normal force is constant in all trials.
The proportionality constant is different for different surfaces; the proportionality depends on the types of contacting surfaces.
The proportionality constant is greater for two rough surfaces contacting each other and smaller for smoother surfaces.
This ratio is the coefficient of static friction.
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18. Coefficient of static friction
The coefficient of static friction is a measure of the relative difficulty of sliding two surfaces across each other.
The easier it is to slide one surface on the other, the smaller the coefficient is.
This coefficient has no unit and typically has a value between 0 and 1.
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19. Static friction force
The static friction force changes magnitude to prevent motion – up to a max value.
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20. Assumptions for our static friction model
Our equation is reasonable only in situations in which the following conditions hold:
Relatively light objects are resting on relatively firm surfaces.
The objects never cause the surfaces to deform significantly (for example, they do not involve a car tire sinking into mud).
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21. Kinetic friction
Kinetic indicates that the surfaces in contact are moving relative to each other.
A similar relationship exists as between the friction force and the normal force, but with two important differences:
Under the same conditions, the magnitude of the kinetic friction force is always lower than the maximum static friction force.
The resistive force exerted by the surface on the moving object has a constant value.
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22. Assumptions for our kinetic friction model
Our equation is reasonable only in situations in which the following conditions hold:
It cannot be used for rolling objects.
It makes the same assumption about the rigidity of the surfaces as the model for static friction.
The objects cannot be moving at high speed.
This equation does not have general applicability, but it is useful for rigid surfaces and objects moving at everyday speeds.
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