By definition, an object is in equilibrium when it is either at rest or moving with constant velocity, i.e., with no CHAPTER 12

acceleration. The following are examples of objects in static equilibrium ...

STATIC EQUILIBRIUM AND ELASTICITY • Conditions for static equilibrium ! Center of gravity ! Examples of equilibrium • Couples Balanced bottle

Garden of the Gods Colorado Springs

• Elasticity ! Stress and strain ! Young’s modulus ! Hooke’s Law and the elastic limit ! Shear modulus

Alexei

Therefore, two conditions are necessary for a body to be in static equilibrium ... • The net external force acting on the body is zero, ! i.e., ∑ i Fi = 0,

Question 12.1: A see-saw consists of a board of length 4.0

so there is no translational motion

m pivoted at its center. A 28 kg child sits at one end of the board. Where should a 40 kg child sit to balance the

• The net external torque about any point is zero, ! i.e., ∑ i τi = 0, so there is no rotational motion Let’s do a problem as an illustration ...

seesaw, i.e., have the see-saw in static equilibrium?

First, identify all the forces ... FA

40kg

y

FB

28kg x F 4m

d θ

θ

The conditions for static and rotational equilibrium: ! ∑ i Fyi = 0 and ∑ i τi = 0.

Question 12.2: At airports you see people running

Therefore, the pivot must supply an upward force so that

around pulling their luggage on small trolleys.

the net force on the board is zero, i.e.,

Sometimes they arrange their cases as shown on the left;

F − (28 kg)g − (40 kg)g = 0

sometimes as shown on the right.

∴F = (68 kg)g = 666.4 N. Define ccw torques as positive and taking torques about

(a) If FA and FB are the forces necessary to hold the

the pivot point we have:

carts in the two cases, which force is the smaller or are

(28 kg)g × (2 m) − (40 kg)g × d = 0 56 kg ⋅ m ∴d = = 1.40 m. 40 kg So, for equilibrium Cathy must stand 1.40 m from the pivot point.

the equal? (b) What difference, if anything, does increasing or decreasing the angle θ make?

FA

x3

FB

We assumed the weight force acts through the center of mass ... strictly speaking it should be the center of weight. What’s the connection between the two?

mg

θ

Mg x1

center of gravity (center of weight)

x2

xCG

(a) Take torques about the contact point with the ground. ! At equilibrium ∑ i τi = 0, i.e.,

O

" W=

∑ i mi g i

O

xi " w i = mi g i

F.x 3 − Mg.x1 − mg.x 2 = 0.

Take some origin, O, and “break” the object into small

∴F =

τ! = ∑i mi gi x i = Wx CG , where x CG is the center of gravity (weight) defined as ∑ wx ∑ wx x CG = i i i = i i i . W ∑ i wi

(Mg.x1 + mg.x 2 ) x3

.

M, m, x1 and x 3 are the same on the left and on the right, but x 2 (left) > x 2 (right) . ∴FA > FB, so FB is the smaller force. (b) Note, x i = d i cosθ. Substituting the equation for F, cosθ cancels out. So, changing θ mkes no difference.

elements of mass mi. The total torque about O is:

(This is similar to the definition of the center of mass.) The center of mass is at the same point as the center of weight, i.e., x CM ≡ x CG , only if g is constant, then τ! = (∑ i mi x i ) g = Mgx CM = WxCM .

O

! W=

∑ i mi g i (a)

If the origin is taken at the center of gravity, then the object produces zero torque about that point. The center of gravity is then ... the point about which the gravitational force on the object produces zero torque no matter what the orientation of the object. It could also be called ... the center of weight! If g is constant over the object then the center of gravity and the center of mass occur at the same point.

(b)

Discussion problem 12.1: Why is it you cannot touch your toes without falling over if you have your heels against a wall as in (a) ... yet, ordinarily, you have no problem, as in (b)? If you don’t believe me ... try it!

1m

! F

Ty

! T

Tx

2m

y

1m

x 2m

4 kg

Fy

I physics! 20 kg

Question 12.3: A sign hangs in front of a store. The sign has a mass of 20 kg and it hangs at the end of a horizontal rod of length 2 m and mass 4 kg, which is hinged at the wall. The rod is supported by a wire attached to a point on the wall 1 m above the rod. (a) What is the tension in the wire? (b) What is the magnitude and direction of the force the rod exerts on the wall?

4 kg

I physics!

Fx Force of the wall on the rod

20 kg

(a) The wall must supply a force on the rod ... how do we know that? However, we have no idea in what direction so choose it arbitrarily to begin with. Note, by Newton’s 3rd Law the force the wall exerts on the rod is equal and opposite to the force the rod exerts on the wall. For static and rotational equilibrium:

! ∑ i Fxi = 0, ∑ i Fyi = 0 and ∑ i τi = 0.

Take torques about the hinge ... a great idea since we ! don’t know anything about F. Then Ty × (2 m) − (4 kg)g × (1 m) − (20 kg)g × (2 m) = 0 ∴Ty =

(44 kg ⋅ m)g = 215.8 N. (2 m)

! F

1m

Ty

! T 2m

Tx θ

! F

1m

y

! T

Ty Tx

2m

y

x Fy

4 kg Fx

x Fy

I physics!

4 kg Fx

I physics!

20 kg

Ty

(1 m) 1 But = tan θ = = . Tx (2 m) 2 ∴ Tx = 2 Ty , i.e., Tx = −431.6 N. T = Tx2 + Ty2 = (−431.6 N)2 + (215.8 N)2 = 482.5 N ! Note ... we found T even though we ! know nothing about F! Of course, the wire must be strong enough to withstand a force of 482.5 N.

20 kg

(b) For static and rotational equilibrium:

! ∑ i Fxi = 0, ∑ i Fyi = 0 and ∑ i τi = 0.

So ∑ i Fxi = Fx + Tx = Fx − 431.6 N = 0 ∴Fx = 431.6 N, and ∑ i Fyi = Fy + Ty − (20 kg)g − (4 kg)g = 0 ∴Fy = (24 kg)g − (215.8 N) = 19.6 N. Fy

! F = (431.6ˆi + 19.6ˆj) N φ Fx

⎛ Fy ⎞ ∴ φ = tan −1⎜ ⎟ = 2.6" ⎝ Fx ⎠

Therefore, the force of the rod on the wall is ! F = −(431.6ˆi + 19.6ˆj) N.

! F

1m

! T

Ty Tx

2m

Question 12.4: A square plate is made by welding

y

together four smaller square plates, each of side ℓ. Each x

Fy

4 kg Fx

I physics! 20 kg

of the four squares is made from a different material so they have different weights, as shown in the figure. (a) Find the position of the center of gravity, relative

! Note: we could find F another way ... (a) by taking

to the point (0, 0).

torques about the right hand end of the rod, since the ! system is in static equilibrium, ∑ i τi = 0 about any point,

the angle between the vertical and the left-hand side of

i.e., (4 kg)g × (1 m) − Fy × (2 m) = 0. (4 kg ⋅ m)g ∴Fy = = 19.6 N. (2 m) (b) By taking torques about the top end of the wire, i.e., Fx × (1 m) − (4 kg)g × (1 m) − (20 kg)g × (2 m) = 0. (44 kg ⋅ m)g ∴Fx = = 431.6 N. (1 m)

(b) If the plate is suspended from the point P, what is the plate? P ℓ

60 N

20 N

ℓ

50 N

30 N

ℓ

ℓ

(0,0)

P ℓ

60 N

20 N

(a) By definition, the center of gravity (weight), with respect

ℓ

(0, 0)

50 N

30 N

ℓ

ℓ

to (0, 0) is given by ∑ wx x CG = i i i W

⎛ ℓ ℓ 3ℓ 3ℓ⎞ ⎜ (60 N) × + (50 N) × + (20 N) × + (30 N) × ⎟ ⎝ 2 2 2 2⎠ = (60 N) + (50 N) + (20 N) + (30 N) = 0.8125ℓ, and y CG =

∑ i wi y i W

(b) When hung from P, a vertical line through P passes through the CG. ℓ φ

⎛ 3ℓ ℓ 3ℓ ℓ⎞ ⎜ (60 N) × + (50 N) × + (20 N) × + (30 N) × ⎟ ⎝ 2 2 2 2⎠ = (60 N) + (50 N) + (20 N) + (30 N) = 1.000ℓ. Therefore, the coordinates of the CG relative to (0, 0) are (0.8125ℓ, ℓ).

∴tan φ =

0.8125ℓ (0, 0)

P

CG

0.8125ℓ , ℓ

i.e., φ = 39.1" .

Let the ladder, length ℓ, rest at an angle θ against the wall. Identify all of the force acting on the ladder. " F1

Since the wall is frictionless, the force of

ℓ

Question 12.5: A uniform ladder rests against a frictionless, vertical wall. If the coefficient of static friction between the bottom of the ladder and the floor is 0.3, what is the smallest angle at which the ladder can rest against the wall without slipping?

" Fn

ℓ sin θ

" w

the wall acting on the " ladder is F1, which is ⊥ to the wall, i.e., a normal

θ " fs

force. The ground exerts

two forces on the ladder; a " 2 2 normal force Fn and a " " static frictional force f s. The weight of the ladder is w, ℓ cosθ

ℓ cosθ

which acts through the CG, i.e., the middle of the ladder. For static and rotational equilibrium:

# ∑ i Fxi = 0, ∑ i Fyi = 0 and ∑ i τi = 0.

and

∴ ∑ i Fxi = f s − F1 = 0 ... ... ... (1) ∑ i Fyi = Fn − w = 0 ... ... ... (2)

Taking torques about the foot of the ladder: ! ℓ ∑ i τi = −w⎛⎝ cosθ⎞⎠ + F1 (ℓsin θ) = 0 ... (3) 2 ! F1

! w

ℓ sin θ

θ

! fs

ℓ cosθ

F1 = f s = µ sFn = µ s w. ∴θ = tan −1 12µ s

(

= 59.0# . 2

w . 2F1

But from (1) and (2):

ℓ ! Fn

∴tan θ =

ℓ cosθ

2

)

Question 12.6: In the previous problem there was no friction at the wall but there was static friction between the ground and the foot of the ladder. If, instead, there was friction at the wall but no friction at the ground, what difference would it make?

Again, identify all the forces acting on the ladder. Summing the vertical and horizontal forces we get ! F1

! Fn

! fs

! w

∑ Fy = Fn − w + f s ... (1) and ∑ Fx = −F1 ... ... ... (2). Question 12.7: A box X, of mass 10 kg, is placed on a

Now eq (1) can be zero (if f s = w − Fn ). However, eq

box Y, of mass 5 kg, so that the center of box X lies

(2) can never be zero if the ladder is leaning against the

directly over the left hand edge of box Y. If the two

wall as it consists of only one term ( F1). Consequently,

boxes are then placed over the edge of a table, what is the

the ladder can never be in equilibrium, i.e., if there’s no

maximum possible overhang, h, without the boxes falling,

friction with the ground it will always slip!

if the sides of the boxes is 0.50 m?

We ignore the arrangement when the ladder is vertical!

OR ... take torques about O, above the edge of the table. h L

For ‘balance’ the CG of the two

2

h

boxes combined must lie directly over the edge of the table, i.e., the

xCG 10 kg

‘pivot’ point. 5 kg

L

2

O

Let the length of the side of the boxes be L, and find the CG with

10 kg

respect to the center line of the upper box (X), which is the same as the left hand edge of lower box (Y): ∑ i mi gx i (10 kg)g × 0 + (5 kg)g × L 2 L x CG = = = . (15 kg)g 6 ∑i mi g Therefore, total overhang is, L L 2L h= + = = 0.33 m. 6 2 3

5 kg L

2

At equilibrium, ∑ i τi = 0. L ∴(10 kg)g × ⎛ h − ⎞ − (5 kg)g × (L − h ) = 0, ⎝ 2⎠ i.e., 10h − 5L − 5L + 5h = 0. 10L 2(0.5 m) ∴h = = = 0.33m. 15 3

1.0 m

2.0 m 50 kg 100 kg 150 kg 1.0 m FL 2.0 m

Question 12.8: The figure shows a cart of mass 100 kg loaded with six identical packing cases, each of mass

100 kg

FR

At equilibrium: ∑ Fy = 0 and ∑ τ = 0. ∴ (FL + FR ) − (50 kg)g − (100 kg)g −(150 kg)g − (100 kg)g = 0,

cases distributed between the left hand and the right hand

i.e., ∴ (FL + FR ) = 3924 N. Taking torques about the axle of the right hand wheel

wheels?

(ccw is positive):

50 kg. How is the total weight of the cart and packing

(50 kg)g × (2 m) + (100 kg)g × (1 m) + (100 kg)g × (1 m) − FL × (2 m) = 0. (300 kg ⋅ m)g ∴FL = = 1471.5 N (2 m) and

FR = 3924 N − 1471.5 N = 2452.5 N.

Stress and strain (Young’s modulus) Area A

A couple

L " x1 O•

" F1

F

F

ℓ " x2

ΔL + L " F2

Two forces form a couple if they are equal but opposite and their line of action is separated (by a distance ℓ): " " i.e., F1 = F2 = F. Then the torque about any point O is: " " " " " τ = ( x1 × F1 ) + ( x 2 × F2 ) " i.e., τ = x1F − x 2F = (x1 − x 2 )F = ℓF So the torque produced by a couple is the same about

Objects deform when subjected to a force. In the case of the length of a metal bar or wire ... The stress ⇒ F A , and the strain ⇒ ΔL L . F stress A Young’s modulus ⇒ Y = = ΔL strain L

( ) ( )

Units: Force/area ⇒ N ⋅ m−2 (scalar) • Thomas Young (1773-1829)

all points in space, i.e., no matter where O is located ...

The example above is tensile stress. There is also

it depends only on their perpendicular separation.

compressive stress. Young’s modulus is often the same in both cases, but there are exceptions (e.g., bone and concrete).

Discussion problem 12.2: However, the strain is reversible over only a limited

An aluminum wire and a steel wire

range of stress ... tensile stress

L Elastic limit

with the same lengths (L) and diameters (d), are joined to form a

Fracture

wire of length 2L. One end of the

Plastic behavior

wire is fixed to the ceiling and the L

Hooke’s Law (linear)

Plastic deformation

strain

... up to the elastic limit. Beyond that the material shows plastic behavior until it reaches the point of fracture (tensile strength). The initial linear behavior is known as Hooke’s Law ... discovered by Robert Hooke (1635-1703).

w

other is attached to a weight (w). Neglecting the weight of the wires, which one of the following statements is true?

A: The strains in the two wires are the same. B: The stresses in the two wires are the same. C: The stress in the aluminum wire is greater than the stress in the steel wire. D: The stress in the aluminum wire is less than the stress in the steel wire. Ysteel = 200 × 109 N ⋅ m−2 and YAl = 70 × 109 N ⋅ m−2 .

( ) ( )

F stress A . (a) The Young’s modulus is Y = = ΔL strain L Question 12.9: A steel wire of length 1.50 m and diameter 1.00 mm is joined to an aluminum wire of

d = 1.00mm 1.50m

aluminum

1.50m

steel

identical dimensions to make a composite wire of length 3.00 m.

= 6.25 × 107 N ⋅ m−2 .

( )( )

(a) What is the length of the composite wire if it is used to support a mass of 5 kg? (b) What is the maximum load the composite wire could withstand? Assume the weights of the wires are negligible. Aluminum:

But ΔL = L Y F A . ∴ΔL = (6.25 × 107 N ⋅ m−2 ) L Y .

5kg

• Aluminum: ΔL1 = (6.25 × 107 N ⋅ m−2 )

YAl = 70 × 109 N ⋅ m−2 . Tensile strength = 90 × 106 N ⋅ m−2 .

Steel:

The stress in each wire is: (5 kg)g F = A π × (0.5 × 10 −3 m)2

Ysteel = 200 × 109 N ⋅ m−2 . Tensile strength = 520 × 106 N ⋅ m−2 .

(1.50 m)

(70 × 109 N ⋅ m−2 )

= 1.34 × 103 m ⇒ 1.34 mm. • Steel: ΔL2 = (6.25 × 107 N ⋅ m−2 )

(1.50 m) (200 × 109 N ⋅ m−2 )

= 0.47 × 103 m ⇒ 0.47 mm. Therefore, the increase in length is 0.00181 m, so the new length is 3.00181 m.

Shear modulus

Area

F L θ

F

x Δx

(b) The tensile strength of aluminum is less than that of steel. Consequently, as the load increases, it will be the

F

θ

aluminum wire that fails. Mg

6 −2 A = 90 × 10 N ⋅ m . ∴M = (90 ×106 N ⋅ m−2 ) A g

The maximum stress is F A =

π(0.5 ×10 −3 m)2 = (90 × 106 N ⋅ m−2 ) 9.81 m/s2 = 7.21 kg.

The shear strain is The stress is

F A

Δx = tanθ L

(note which area!)

Shear modulus ⇒ Ms = =

shear stress shear strain

(F A ) = (F A ) . (Δx L) tanθ

Units: Force/area ⇒ N ⋅ m−2 (scalar).

=A