For Midterm

1
For Midterm 2

• Ch. 4 (2D Motion) General Problems (Problems
  50-76 p. 101 - 102)
• Ch. 5 (Newtons Laws) General Problems
  (Problems 45 -67 p. 135 - 136)

2
Physics 1020Applications of Newtons Laws(Ch. 6
of Walker)

• Frictional Forces
• Strings and Springs
• Translational Equillibrium
• Connected Objects
• Circular Motion

3
Applications of Newtons Law

• Assumptions
• Objects can be modeled as particles
• Masses of strings or ropes are negligible
• When an object is being pulled by a rope, the
  rope exerts a force on the object. Its
  direction is along the rope, away from the
  object. The magnitude of this force is called
  the tension.
• Interested only in the external forces acting ON
  the object do not include reaction forces

4
Problem-Solving Hints Newtons Laws

• Conceptualize the problem
• Draw a diagram (include convenient coordinate
  axes)
• Categorize the problem
• Equilibrium (SF 0) or Newtons Second Law (SF
  m a)
• Analyze
• Draw free-body diagrams for EACH object
• Include only forces acting ON the object

5
Equilibrium Example

• A traffic light weighing 122 N hangs from a
  cable tied to two other cables fastened to a
  support, as shown. The upper cables make angles
  of 37 and 53 with the horizontal. These upper
  cables are not as strong as the vertical cable
  and will break if the tension in them exceeds 100
  N. Does the traffic light remain in this
  situation, or will one of the cables break?
• Conceptualize the traffic light
• Categorize as an equilibrium problem
• No movement, so acceleration is zero

6
Equilibrium, Example, cont

• Analyze
• Need two free-body diagrams
• Apply equilibrium equation to the light and find

7
Equilibrium, Example, cont

• Analyze, cont.
• Apply equilibrium equations to the knot and find
  and

8
Equilibrium, Example, cont

• Analyze, cont
• Solve the 1st of these for T2
• Substitute into the 2nd equation to give

Both less than 100 N, so cables do not break!
9
Objects Experiencing a Net Force

• If an object that can be modeled as a particle
  experiences an acceleration, there must be a
  nonzero net force acting on it.
• Draw a free-body diagram
• Apply Newtons Second Law in component form

10
Nonzero Net Force - Example

• Forces acting on the crate
• A tension, the magnitude of force
• The gravitational force,
• The normal force, , exerted by the floor

• Apply Newtons Second Law in component form
• Solve for the unknown(s)

11
Inclined Plane Recipes
Memorize!!!!

• Forces acting on the object
• The normal force acts perpendicular to the plane
• The gravitational force acts straight down
• Choose the coordinate system with x along the
  incline and y perpendicular to the incline
• Replace the force of gravity with its components

12
Inclined Plane Recipes, cont.

• Newtons 2nd law
• From 1st equation
• Check special cases to see that answer makes
  sense

What is acceleration in each of these cases?
13
Forces of Friction (Ch. 6 Sect. 6-1)

• When an object is in motion on a surface or
  through a viscous medium, there will be a
  resistance to the motion
• This is due to the interactions between the
  object and its environment
• This resistance is called the force of friction

14
Static Friction
Static friction acts to keep the object from
moving If increases, so does If
decreases, so does
15
Static Friction

• 0 ? s ? fs, max
• fs, max µs N
• µs is called the coefficient of static friction
• µs - dimensionless

• s ? µs N where the equality holds when the
  surfaces are on the verge of slipping
• Called impending motion
• The force of static friction between 2 surfaces
  is parallel to the surface of contact and has the
  direction opposite to the motion (Does not act in
  the direction of the normal force!!!!)

16
Static Friction (more magic!)

• How does the static friction know how big it has
  to be?
• Same thing as for the normal force!
• As you apply a force to move an object, you start
  to distort the stuff at the contact points of
  the object
• Floor, table, air, water whatever
• The more force you apply, the more distortion
• If you push hard enough you break the contact
  points
• At this point, the kinetic friction model comes
  into play

17
Kinetic Friction

• The force of kinetic friction acts when the
  object is in motion
• Although µk can vary with speed, we shall neglect
  any such variations
• k µk N
• µk - the coefficient of kinetic friction
• µk dimensionless
• Typical values are in the range between 0 and 1
• Simpler to work with than the static case because
  of the equality

18
Static vs. Kinetic
19
Forces of Friction, Summary

• The direction of the frictional force is (in
  general) opposite the direction of motion (or
  impending motion) and parallel to the surfaces in
  contact
• The coefficients of friction are nearly
  independent of the area of contact
• The coefficient of kinetic friction (mk) it
  typically smaller than the coefficient of static
  friction (ms)
• The maximum magnitude of static friction, s, is
  generally greater than the magnitude of kinetic
  friction, k
• The coefficient of friction (µ) depends on the
  surfaces in contact
• fs, max µs N fk µk N
• These equations relate the magnitudes of the
  forces, they are not vector equations

!
!
20
Friction in Newtons Laws Problems

• Friction is a force, so it simply is included in
  the SF in Newtons Laws
• The rules of friction allow you to determine the
  direction and magnitude of the force of friction

21
Example (simple)Problem 4 p. 165

• When you push a 1.80-kg book resting on a
  tabletop, it takes 2.25 N to start the book
  sliding. Once it is sliding, however, it takes
  only 1.50 N to keep the book moving with constant
  speed. What are the coefficients of static and
  kinetic friction between the book and the
  tabletop?
• P. 5 p. 165. What is the frictional force exerted
  on the book when you push on it with a force of
  0.75 N?

22
Example (more difficult) Problem 7 p. 166

• To move a large crate across a rough floor, you
  push down on it at an angle of 21, as shown.
  Find the force necessary to start the crate
  moving, given that the mass of the crate is 32 kg
  and the coefficient of static friction between
  the crate and the floor is 0.57.

23
HOMEWORK !!!!

• Example 6-2 page 141 (Making a big splash)

24
Check you understanding

• A physics student is pulling upon a rope which is
  attached to a wall. In the bottom picture, the
  physics student is pulling upon a rope which is
  held by the Strongman. In each case, the force
  scale reads 500 Newtons. The physics student is
  pulling
• with more force when the rope is attached to the
  wall.
• with more force when the rope is attached to the
  Strongman.
• the same force in each case.

25
Check you understanding

• Consider the addition of two forces, both having
  a magnitude of 10 Newtons. What is their sum?

26
(No Transcript)
27
Multiple Objects (Sect. from 6-2 to 6-4)

• When two or more objects are connected or in
  contact, Newtons laws may be applied to the
  system as a whole and/or to each individual
  object.
• Whichever you use to solve the problem, the other
  approach can be used as a check.

28
Strings and Springs (Sect. 6-2)

• Assumptions
• Masses of strings or ropes are negligible
• Pulleys have no mass, no friction
• A pulley changes the direction of the tension but
  does not change its magnitude

When an object is being pulled by a rope, the
rope exerts a force on the object. Its
direction is along the rope, away from the
object. The magnitude of this force is called
the tension.
29
Conceptual Checkpoint 6-2a p.148How do the scale
readings compare?

• The scale at left reads 9.81 N. Is the reading of
  the scale at right
• greater that 9.81N
• equal to 9.81N
• less that 9.81 N

30
Multiple Objects

• v const a 0

31
Multiple Objects

• Forces acting on the objects
• Tension (same for both objects since one string)
• Gravitational force
• Each object has the same acceleration since they
  are connected
• Draw the free-body diagrams
• Apply Newtons Laws
• Solve for the unknown(s) - Here a and T)

32
Atwoods Machine (Ex. 6-7 p. 156)
Masses m1 and m2 are attached to an ideal
massless string and hung as shown around an ideal
massless pulley.
Fixed Pulley

• Find the accelerations, a1 and a2, of the masses.
• What is the tension in the string T ?

y
T1
T2
m1
a1
m2
a2
33
Atwoods Machine...

• Draw free body diagrams for each object
• Applying Newtons Second Law ( y -components)
• T1 - m1g m1a1
• T2 - m2g m2a2
• But T1 T2 T since pulley is ideal
• and a2 -a1 -a.since the masses are
  connected by the string

Free Body Diagrams
T1
T2
y
a1
a2
m2g
m1g
34
Atwoods Machine...

• T - m1g m1 a (a)
• T - m2g -m2 a (b)
• Two equations two unknowns
• we can solve for both unknowns (T and a).
• subtract (a) - (b)
• g(m2 m1 ) a(m1 m2 )
• a

35
Atwoods Machine

• T - m1g m1 a (a)
• T - m2g -m2 a (b)
• divide (a) by (b)

36
Atwoods Machine...

• So we find

37
Is the result reasonable? Check limiting
cases!

• Special cases
• i.) m1 m2 m a 0 and T mg. OK!
• ii.) m2 or m1 0 a g and T 0.
  OK!
• Atwoods machine can be used to determine g (by
  measuring the acceleration a for given masses).

-
38
Example (Pr. 31 p. 167)

• After a skiing accident, your leg is in a cast
  and supported in a traction device, as shown.
  Find the magnitude of the force exerted by the
  leg on the small pulley. (By Newtons third law,
  the small pulley exerts an equal and opposite
  force on the leg.) Let the mass m be 2.50 kg.

39
Multiple Objects, Example 2

• Three blocks of mass 3m, 2m, and m are connected
  by strings and pulled with constant acceleration
  a. What is the relationship between the tension
  in each of the strings?

(a) T1 gt T2 gt T3 (b) T3 gt T2 gt T1
(c) T1 T2 T3
40
Solution

• Draw free body diagrams!!

T3 3ma
T1 gt T2 gt T3
41
Multiple Objects - Example 3

• First treat the system as a whole
• Apply Newtons Laws to the individual blocks
• Solve for unknown(s) Here a and P12
• Check P21 P12

42
Multiple Objects - Example 2, cont
P21 P12 P12 -P21 a1 a2 a

• First treat the system as a whole
• Apply Newtons 2nd law to m2
• Check. Apply Newtons 2nd law to m1

as before.
43
Example (Pr. 36 and 39 p. 168)

• Find the acceleration of the masses shown in
  Figure, given that m11.0 kg, m22.0 kg and
  m33.0 kg
• Find the tension in each of the strings in Figure.

44
Next lecture

• Springs
• Circular Motion