Annotations on Chapter 6: Applications of Newton's Laws

Chapter 6: Applications of Newton's Laws

Units of Chapter 6

Frictional Forces
Strings and Springs
Translational Equilibrium
Connected Objects
Circular Motion

6-1: Normal Forces

The normal force is defined as the force exerted by a surface on an object.
Its relationship with weight is variable: it can be equal to, greater than, or less than the object's weight.
The normal force is always perpendicular to the surface in contact.

6-1: Frictional Forces

Friction originates from surfaces that are not perfectly smooth.
- Kinetic Friction: The type of friction experienced by surfaces sliding against one another.
- Kinetic Frictional Force:
- It is directly proportional to the normal force.
- The constant $M_k$, called the coefficient of kinetic friction, represents the relationship.
The static frictional force prevents an object from starting to move when a force is applied.
- The maximum value of static friction exists, but the force can vary from zero up to this maximum value; this is adjusted based on what is required to ensure equilibrium.
- Static friction does not depend on the area of contact or the relative speed of the surfaces.

Table 6-1: Typical Coefficients of Friction

| Materials | $Mk$ (Kinetic Friction) | $M{ ext{e}}$ (Static Friction) |
|---------------------------|--------------------|---------------------|
| Rubber on concrete (dry) | 0.80 | 1.4 |
| Steel on steel | 0.57 | 0.74 |
| Glass on glass | 0.40 | 0.94 |
| Wood on leather | 0.40 | 0.50 |
| Copper on steel | 0.36 | 0.53 |
| Rubber on concrete (wet) | 0.25 | 0.30 |
| Steel on ice | 0.06 | 0.10 |
| Waxed ski on snow | 0.05 | 0.10 |
| Teflon on Teflon | 0.04 | 0.04 |
| Synovial joints in humans | 0.003 | 0.01 |

6-2: Strings and Springs

The tension in a real rope varies along its length due to the rope's own weight.
For this analysis, all ropes, strings, and wires are assumed to be massless unless stated otherwise.
An ideal pulley changes only the direction of tension without impacting its magnitude.
Hooke’s Law states that the force exerted by a spring varies proportionally to the displacement of the spring from its rest position:
- The proportionality constant $k$ is termed the spring constant.
When an object is in translational equilibrium, the net force acting upon it is zero, which allows for the calculation of unknown forces.

6-4: Connected Objects

If forces are exerted on connected objects, their accelerations will be identical.
When two objects are connected by a string with known force and masses, one can determine the acceleration and tension.
- Each object can be treated as a separate system for analysis.
With the presence of a pulley, adopting the coordinate system in line with the string simplifies calculations.

6-5: Circular Motion

An object moving in a circle requires a central force; without this force, it would move in a straight line.
The direction of this force is consistently directed towards the center of the circle.
Algebraic manipulation can yield the magnitude of acceleration, and thus the necessary force to maintain an object of mass $m$ in circular motion with radius $r$:
- Resulting in the equation: F = rac{mv^2}{r}
- Possible sources for this centripetal force include tension in a string, normal force, friction, etc.

Conceptual Questions Practice

Question 1: A person pulls/pushes a sled with force $F$ at angle $ heta$. Which scenario results in a greater normal force?
- Options:
  a) Case 1
  b) Case 2
  c) It's the same for both.
  d) It depends on the magnitude of force $F$.
  e) It depends on the ice surface.
Question 2: Two identical blocks, one flat and one inclined, which has the greater normal force?
- Options:
  a) Case A
  b) Case B
  c) Both the same ($N = mg$)
  d) Both the same ($0 < N < mg$)
  e) Both the same ($N = 0$)
Question 3: For moving a sled on level ground, which is the easiest method?
- Options:
  a) Pushing from behind
  b) Pulling from the front
  c) Both are equivalent
  d) It's impossible to move the sled
  e) Suggesting walking instead
Question 4: If a box rests on a board raised at an angle, why does it eventually slide down?
- Options:
  a) Gravity's parallel component increased
  b) Coefficient of static friction decreased
  c) Normal force exerted decreased
  d) Both a) and c)
  e) All of a), b), and c)
Question 5: A car skids on a curve; what describes the situation?
- Options:
  a) Engine insufficient to maintain direction
  b) Insufficient friction between tires and road
  c) Car too heavy for turn
  d) An external cause (deer)
  e) None of the above
Question 6: On a Ferris wheel, how does the normal force change at the top when in motion?
- Options:
  a) Normal force equals weight
  b) Normal force less than weight
  c) Normal force greater than weight
  d) None of the above

Problem Solving Practice

Example 6-1: Pass the Salt
- The 50g salt shaker slides with an initial speed of 1.15 m/s and decelerates to rest over 0.840 m. Calculate:
  a) Coefficient of kinetic friction
  b) Time to stop if initial speed is 1.32 m/s
Example 6-3: Slightly Tilted
- A 95.0kg crate slides off a truck bed from incline exceeding $0 = 23.3°$.
- Calculate coefficient of static friction.
Example 6-6: Setting a Broken Leg with Traction
- A traction device using three pulleys is examined. If the force on the foot is 165N, find mass $m$ providing this tension.
Example 6-15: Rounding a Corner
- A 1200kg car on a 45m radius curve with $ = 0.82$, find maximum speed without skidding.
Quick Example 6-17: Normal Force in a Dip
- Driving at 17.0 m/s through a circular dip of 65.1m, determine:
  a) Whether normal force is greater, less, or equal to weight
  b) Calculate normal force on an 80.0kg passenger in the dip.