1. Laws of Motion Friction and Drag

2. Friction
The roughness of one object’s surface collides with the roughness of the other object’s surface. Friction is the sum all the microscopic collisions due the the imperfections in each surface.
The variable for friction: f or Ff
There are two types of friction
Static friction: Friction acting on stationary objects
Kinetic friction: Friction acting on moving objects

3. Static Friction f F
When an object is stationary on a surface, it DOES NOT mean that static friction is present.
In order for static friction to be present all the other forces, and components of forces, parallel to the surface must create a net force (excluding friction) that is attempting to move the object. If the net force due to all other forces fails to move the object, then static friction is acting to balance the attempt to move the object, and the object remains stationary.
Static friction will then be equal in magnitude to the sum of all other forces parallel to the surface, but it will act in the opposite direction.

4. Static Friction f F Static friction is the strongest form of friction.
When the object is at rest on the surface, it is as though the imperfections in the surfaces sink in toward each other locking the surfaces together.
In order to move the object, you must increase the magnitude of the forward force until it is sufficient to push the object up and out of this locked condition.

5. Kinetic Friction F f
As the object begins to move kinetic friction replaces static friction.
Kinetic friction is weaker, and the magnitude of the friction force decreases.

6. Kinetic Friction F f
As the object begins to move kinetic friction replaces static friction.
Kinetic friction is weaker, and the magnitude of the friction force decreases.
In the diagram above the object the applied forward force is greater than the force of kinetic friction. The object will accelerate.
To move the object at constant velocity, you would need to reduce the forward force until it is in equilibrium with friction. The applied force needed to keep the object moving at constant velocity would be smaller than the force needed to start the object moving in the first place.

7. Friction Equation f Friction force
μ Coefficient of friction: Reflects how rough the surfaces are. The higher the coefficient, the stickier the surfaces are to one another, and the stronger the friction force will be. Coefficients have no units.
μs Coefficient of static friction
μk Coefficient of kinetic friction
N Normal force: Force that the surfaces press against each other with. The greater the normal force the stronger the friction force will be.
Less than or equal to
The inequality only affects the calculations of static friction, fs ≤ μsN. Unfortunately, this makes static friction more challenging to work with. The magnitude of static friction calculated using the equation may not be the actual value of static friction.
Kinetic friction always solves as an equality, fk = μkN

8. The Problem With Static Friction
Example: A 2.0 kg block is initially at rest on a rough horizontal surface with coefficients of friction μs = and μk =
When the equation for static friction is set as an equality, it can only solve for the value of maximum static friction.
However, when an object is at rest the actual magnitude of static friction may be less than this maximum value.
We cannot solve an inequality, so how is static friction determined when its magnitude is less than the value of maximum static friction?
When an applied forward force, +F , attempts to move an object forward, along a rough surface, friction, −f , opposes forward motion.
If the object is initially at rest and the magnitude of the applied force is less than the magnitude of maximum static friction, then the object cannot be moved. The object remains at rest (ΣF = 0) due to static friction, fs .
The object is in equilibrium and forces must be in balance.

9. The Problem With Static Friction
To summarize: There are three possible static friction problems.
When an object is initially at rest and the applied forward force is less than the value of maximum static friction.
The inequality applies However, it cannot be solved.
Instead, summing forces results in a balanced force equation that solves for static friction.
Determining the magnitude of maximum static friction.
Now the equality applies.
A special case: The object is initially at rest and the forward applied force is increased until its equal to the value of maximum static friction. Under these circumstances both of the preceding equations apply

10. Graphing Friction fs max f F1 2 3 4 5 6 f
fs max
In a hypothetical experiment an applied force will be varied (independent variable) and be plotted on the x-axis, while the response of the friction force (dependent variable) will be plotted on the y-axis.
We will use the same values as the preceding example for static friction.
A 2.0 kg block is initially at rest on a rough surface with coefficients of friction μs = and μk = 0.10
Maximum static friction is the largest magnitude of friction possible This is the strongest form of friction at its maximum value.
Marking the magnitude of maximum static friction as a dashed horizontal line visually shows the limits the friction force y-values.

11. Graphing Friction fs max f F 41 2 3 4
When the applied force is less than maximum static friction use the balanced force equation.
If the applied force is zero, then static friction is also zero.
If the applied force is increased to 1 N , 2 N , and then 3 N static friction will respond by increasing to maintain equilibrium and balanced forces.
All of these values are less than the magnitude of static friction. The friction equation is NOT useful as the applied forces are LESS THAN the max static friction, and the equality would apply. F 1 2 3 4 5 6

12. Graphing Friction fs max f F 41 2 3 4
When the magnitude of the applied force is increased to the same value as maximum static friction, then both equations will solve for static friction.
This is an important and special case.
How can the value of maximum static friction be determined experimentally?
When friction keeps an object stationary, we know that static friction is equal to the applied force, fs = F . However, static friction can be less than maximum. How do we know if this particular applied force is equal to maximum static friction or not?
Increase the applied force very slowly, until you see the object just begin to move. In this example static friction cannot be greater than 4 N. When the applied force increases to a value that is slightly larger (Example: N) than max static friction, then the forces will be unbalanced and the object will just barely begin to move. The value N is nearly equal to 4 N. Therefore, the applied force needed to barely move an object approximately equals the magnitude of max static friction. F 1 2 3 4 5 6

13. Graphing Friction fs max f F 41 2 3 4
What happens when the magnitude of the applied force is increased to values that are greater than max static friction?
The magnitude of friction cannot ever exceed the value of max static friction. An applied force greater than 4 N will result in unbalanced forces, and acceleration.
Acceleration involves motion, which means that kinetic friction replaces static friction.
Good news! Kinetic friction does not have the same problem as static friction. The equality always solves for kinetic friction. However, the value of the coefficient of friction is different. Kinetic friction is weaker than static friction.
As long as the applied forward force is greater than 4 N the magnitude of kinetic friction will remain constant. In this example it will remain constant at 2 N. F 1 2 3 4 5 6

14. Graphing Friction fs max f F 4 What about constant velocity?1 2 3 4
What about constant velocity?
Constant velocity is another important special case.
At constant velocity the object is in dynamic equilibrium, ΣF = 0 , and forces are in balance.
The object is also moving, which involves kinetic friction. Kinetic friction always solves as an equality, and it will have the same value as it does when accelerating.
Where does this plot on the graph?
It plots where both the applied force and kinetic friction have equal magnitudes.
Constant velocity F 1 2 3 4 5 6

15. Solving Friction F f Stationary, F < fs CANNOT SOLVE1 2 3 4 F 5 6 f
Max static friction
Friction is summarized in the graph at the right.
The equations, and when to use them, will be summarized in the table below.
Acceleration
Stationary, but less than max static friction
Constant velocity
Balance Forces
Friction Equation
Stationary, F < fs
CANNOT SOLVE
Stationary, F = fs
Constant Velocity
Accelerating
CANNOT SOLVE

16. Example 1 Assess: x-direction, constant velocity, Fx = 0
A 5.0 kg box is pulled at constant velocity by a 10 N force along a rough surface. Determine the coefficient of friction. m F
Assess: x-direction, constant velocity, Fx = 0
y-direction, stationary, Fy = 0
Diagram: Free body diagram Alternate diagram
Sum forces:
Solve: F N Fg f
The normal force is now needed, and this means solving in the y-direction
Substitute the normal force and continue working in the x-direction

17. Example 1
Determine the acceleration of a mass placed on a rough 30o incline, μs = and μk = 0.10 m
Assess: Parallel to incline, acceleration, Fx = ma
Perpendicular to incline, stationary, Fy = 0
Diagram: Free body diagram Alternate diagram
Sum forces:
Solve: N Fg f
+Fg sin θ −f

18. A 5. 0 kg box is pulled on a rough surface, μk = 0
A 5.0 kg box is pulled on a rough surface, μk = 0.10, by a 10 N force at 37o. Determine the acceleration of the mass. m F
Example 3
Assess: x-direction, acceleration, Fx = ma
y-direction, stationary, Fy = 0
Diagram: Free body diagram Alternate diagram
Sum forces:
Solve: F N Fg f Fy Fx N Fg f

19. Drag
Friction slows objects due to microscopic collisions that occur when surfaces rub past one another. When objects move in a fluid (gas or liquid) microscopic collisions occur between the object and the molecules of the fluid. Like friction these tiny forces, from countless collisions, add up to an overall force, known as drag, that opposes the forward motion of an object.
Drag, D , can be thought of as the friction of a fluid. The most common form of drag is air resistance (atmospheric drag). The magnitude of air resistance depends on several factors.
the density, 𝜌 , of air
the surface area, A , of the object that is colliding with air molecules.
the speed, v , of the object
Air resistance, Dair or Fair , is approximately:
You are not responsible for this equation. It is included only to show that air resistance is proportional to the square of speed. Doubling speed quadruples air resistance.

20. Graphing Drag
Picture yourself in a car initially at rest. Your hand is sticking out of the open window. At this instant you feel no air resistance.
You depress the car’s accelerator. As speed increases the pressure on your hand (air resistance) increases.
The speedometer needle moves very quickly at first, but as air resistance increases it becomes more and more difficult for the car to accelerate. The speedometer needle moves, so the car is still accelerating. However, the needle seems to move slower with each passing second. Acceleration is decreasing. The magnitude of speed continues to increase, but the increase in speed each second is less and less.
Eventually the force of air resistance equals the force of the cars engine, and forces are now balanced. Since the car is moving when equilibrium is reached the car continues forward, but at constant velocity (dynamics equilibrium). The current speed is the car’s maximum final velocity, and is known as terminal velocity.

21. Graphing Drag
Picture yourself in a car initially at rest. Your hand is sticking out of the window. At this instant you feel no air resistance. You step on the gas. As speed increases the pressure on your and (air resistance) increases according to:
Simplify this equation by setting the all constants equal to b : b = ¼ ρA
Sum Force:
Solve:
And we have a problem. This is a differential expression requiring calculus to solve. AP Physics C is responsible for this, but AP Physics 1 is not. However, both class need to understand this conceptually. Time to imagine how it works.

22. Graphing Drag v Δv Δv Δv Δv t Terminal Velocity
After depressing the accelerator, the speed of the car begins to increase. The pressure on your hand (air resistance) also increases, but the increase is proportional to the square of velocity (when speed doubles air resistance quadruples). To imagine how the graph of velocity versus time would look, imagine what the needle of the car’s speedometer (magnitude of velocity) is doing.
At first the needle moves very quickly. However:
as speed increases air resistance increases
increasing the negative drag force decreases the net force
decreasing net force decreases acceleration
decreasing acceleration means that the change in velocity each second is decreasing
If the change in velocity is decreasing the needle of the speedometer is slowing each second. The speedometer indicates an ever increasing higher value Therefore, the magnitude of speed and velocity are increasing (even though the change in speed is decreasing).
Speed increases until drag equals the applied force and terminal velocity is reached. Terminal velocity is the final (maximum) velocity when air resistance is present.
Terminal Velocity v t Δv Δv Δv Δv

23. Comparing Friction and Drag
Depends on
1. Roughness of surface. (rougher = greater friction)
2. How hard surfaces push against one another (does not depend on surface area).
3. Depends on if the object is moving or standing still, but not its speed.
Motion ?
Value ?
1. Roughness of surface. (rougher = greater friction)
2. How hard surfaces push against one another (does not depend on surface area).
3. Depends on if the object is moving or standing still, but not its speed.
1. Density of the fluid. (denser = greater drag)
2. Does depend on surface area. A larger surface runs into more fluid molecules causing greater slowing.
3. The objects speed.

24. Comparing Friction and Drag
Depends on
1. Roughness of surface. (rougher = greater friction)
2. How hard surfaces push against one another (does not depend on surface area).
3. Depends on if the object is moving or standing still, but not its speed.
1. Density of the fluid. (denser = greater drag)
2. Does depend on surface area. A larger surface runs into more fluid molecules causing greater slowing.
3. The objects speed
Motion ?
Standing still: static friction or
Moving: kinetic friction
Must be moving. There is no drag when an object is stationary.
Value ?

25. Comparing Friction and Drag
Depends on
1. Roughness of surface. (rougher = greater friction)
2. How hard surfaces push against one another (does not depend on surface area).
3. Depends on if the object is moving or standing still, but not its speed.
1. Density of the fluid. (denser = greater drag)
2. Does depend on surface area. A larger surface runs into more fluid molecules causing greater slowing.
3. The objects speed
Motion ?
Standing still: static friction or
Moving: kinetic friction
Must be moving. There is no drag when an object is stationary.
Value ?
The type of motion determines the formula and coefficients needed to determine the force of friction.
Drag is dependent on the speed of the object. Drag increases with speed following a set function for the fluid.