Chapter 9 Statics. Two 50 lb. children sit on a see-saw. Will they balance in all three cases?

Chapter 9 Statics

Two 50 lb. children sit on a see-saw. Will they balance in all three cases?

Review of Forces: Ex 1

A dentist places braces on a person’s teeth that exert 2.00 N in each direction as shown. Calculate the net force on the teeth.

Look at each force separately and resolve it into x and y components.

Fx = (2.00 N)(cos 20o) = 1.88 N

Fy = (2.00 N)(sin 20o) = 0.684 N

For the other force:

Fx = -1.88 N

Fy = 0.684 N

2.0 N70 o

20 o

The x-components cancel and the y-components add

Fx = 1.88 N – 1.88 N = 0

Fy = 0.684 N + 0.684 N = 1.37 N2.0 N

Fnet = 1.37 N

2.0 N

Review of Forces: Ex 2Calculate the force exerted by the traction device

shown below.

Calculate the x and y-components:

Top force:

Fx = (200 N)(cos 37o) = 160 N

Fy = (200 N)(sin 37o) = 120 N

Bottom force:

Fx = (200 N)(cos 37o) = 160 N

Fy = (200 N)(sin 37o) = 120 N (downward)

Fy = 120 N –120 N = 0

Fx = 160 N + 160 N = 320 N

Why doesn’t the patient slide out of the bed?

Review of TorqueMr. Saba can’t quite budge this rock. How can

he increase his torque so he can move it?

= FR

Conditions for Static Equilibrium

= 0

F = 0

F = 0: Ex 1

A 90.0 kg man cannot do a pull-up. His best efforts produce a scale reading of 23 kg. What force is he exerting?

Fx = 0

Fy = FB + Fs – mg

Since he is not moving F = 0

0 = FB + FS – mg

FB = mg – FS

FB = (90 kg)(9.8 m/s2) – (23 kg)(9.8 m/s2)

FB = 660 N

F = 0: Ex 2

Calculate the F1 and F2 assuming the chandelier is not moving.

Fx = 0

Fy = 0

Fx = 0 = F2 – (F1)(cos 60o)

Fy = 0 = (F1)(sin 60o) - mg

0 = F2 – (F1)(cos 60o) two eqns, two unknowns

0 = (F1)(sin 60o) – mg

0 = (F1)(sin 60o) – (200 kg)(9.8 m/s2)

0 = (F1)(0.866) – 1960 N

1960 N = (F1)(0.866)

F1 = 2260 N

0 = F2 – (F1)(cos 60o)

F2 = (F1)(cos 60o)

F2 = (2263)(cos 60o) = 1130 N

A 200 g mass is hung by strings as shown. Calculate the tension in string A and string B.

(A = 1.13 N, B = 1.51 N)

t = 0: Example 1

Two children sit on a see-saw as shown. The board of the see-saw has a mass of 2.00 kg centered at the pivot. Where should the 25.0 kg child sit so that they are in perfect balance?

2.50 m

?

30.0 kg 25.0 kg

Let’s pick the pivot as the origin

= 0

0=(245 N)(x) – (294 N)(2.5 m) – (FN)(0) + (mg)(0)

0 = (245 N)(x) – (294 N)(2.5 m)

(245 N)(x) = 735 m-N

x = 3.0 m

mg

294 N 245 N

FN2.5 m x

A crane lifts a 1000.0 kg car as shown in the picture below, 8.00 m from the pivot. Calculate the force that the motor of the crane must provide 2.00 m from the pivot. (39,200 N)

A teacher (56.0 kg) sits at the end of a 4 m long see saw. Where should a 203 kg owlbear sit to perfectly balance the seesaw? (55 cm)

F = 0 and t = 0: Example 1

A heavy printing press is placed on a large beam as shown. The beam masses 1500 kg and the press 15,000 kg. Calculate the forces on each end of the beam.

Fy = 0 and = 0 (we’ll ignore Fx)

Fy = 0 = F1 + F2 - (1500 kg)(g) – (15,000 kg)(g)

0 = F1 + F2 – 1.617 X 105 N

F1 + F2 = 1.617 X 105 N

We’ll choose F1 as the pivot

= 0 0 = (F1)(0 m) – (1500kg)(g)(10m) - (15,000 kg)(g)(15 m) + (F2)

(20 m)

0 = -2.352 X 106 m-N + (F2)(20 m)

(F2)(20 m) = 2.352 X 106 m-N

F1 + F2 = 1.617 X 105 N

(F2)(20 m) = 2.352 X 106 m-N

(F2) = 2.352 X 106 m-N/(20m) = 1.176 X 105 N

F1 + 1.176 X 105 N = 1.617 X 105 N

F1 = 4.41 X 104 N

F = 0 and t = 0: Example 3

The beam below has a mass of 1200 kg. Calculate F1 and F2 for the cantilever as shown. Assume the center of gravity is at 25 m.

F1 F2

20.0 m 30.0 m

(1200 kg)(g)

Fy = 0 and = 0

Fy = 0 = F1 + F2 - (1200 kg)(g)

0 = F1 + F2 – 11,760 N

F1 + F2 = 11,760 N

We’ll choose F1 as the pivot

= 0 0 = (F1)(0 m) + (F2)(20m) - (11,760 N)(25 m)

0 = (F2)(20m) – 294,000 N

(F2)(20m) = 294,000 N

F1 + F2 = 11,760 N

(F2)(20m) = 294,000 N

F2 = 294,000 N/20 m = 14,700 N

F2 = 14,700 N

F1 = 11,760 N - F2

F1 = 11,760 N - 14,700 N

F1 = -2940 N

(we picked the wrong direction for Force 1)

The following system is completely in balance. Calculate the mass of Mass C and the value of the force at the support point (S).

(217 g, 3.34 N)

F = 0 and t = 0: Example 4A sign of mass M = 280

kg is suspended from a 25.0 kg beam that is 2.20 m long. The angle between the sign and the wire is 30o. Calculate FH and FT, the forces at the hinge and the tension in the wire.

Fx = 0

0 = FHcos - FTcos30o

0 = FHcos – 0.866FT

Fy = 0

0 = FHsin + FTsin30o – (25 kg)(g) – (280 kg)(g)

0 = FHsin + 0.5FT – 2989 N

= 0 (choose hinge as origin)

0 = (25kg)(g)(1.1m) + (280kg)(g)(2.2m) – (FT)(sin30o)(2.2m)

T hree equations, three unknowns

0 = FHcos – 0.866FT

0 = FHsin + 0.5FT – 2989 N

0 = 6306 – 1.1FT

FT = 5730 N, FH = 4960 N, = 1.43o

F = 0 and t = 0: Example 5

Calculate the force on the hinge and the tension in the cord for the following sign. The beam is 2.00 m long, though the sign is hung at 1.80 m.

Fx = 0

0 = FHcos – FTcos20o

FHcos – 0.940FT

Fy = 0

0 = FHsin + FTsin20o – (20 kg)(g) – (150 kg)(g)

0 = FHsin + 0.342FT– 1666N

= 0 (choose hinge as origin)

0 = (20kg)(g)(1m) + (150kg)(g)(1.8m) – (FT)(sin30o)(2m)

T hree equations, three unknowns FHcos – 0.940FT

0 = FHsin + 0.342FT– 1666N

0 = 2842 – 0.684FT

FT = 4155 N, FH = 3914 N, = 3.59o

F = 0 and t = 0: Example 6A crane is designed to

hold a maximum of 10,000 kg (M). The top crossbeam has a mass of 450 kg. Calculate the forces exerted by the post (Fp) and the angled beam (FA).

15.0 m 5.0 m

M

FA

Fp

FA

FpFpy

Fpx

FAy

FAx

Fx =0

0 = FAcos35o - FPsin

0 = 0.819FA - FPsin

Fy =0

0 = FA sin35o – Fp cos – mg – Mg

0 = FA sin35o – Fp cos – (450 kg)(g) –(10,000kg)(g)

0 = 0.574FA – Fp cos – 1.02 X 105 N

= 0 (choose top of post as origin)0 = (450 kg)(g)(10m) – (10,000 kg)(g)(20m) + (FA)(sin35o)(15m)

0 = 2.00 X 106 – 8.6 FA

Three Equations, three unknowns

0 = 0.819FA - FPsin

0 = 0.574FA – Fp cos – 1.02 X 105 N

0 = 2.00 X 106 – 8.6 FA

FA = 2.34 X 105 N, Fp = 1.95 X 105 N, = 80.7o

The Ladder: Example 1A 5-m long ladder leans

against a wall at a point 4 m above the ground. The ladder has a mass of 12.0 kg and is uniform. Assume the wall is frictionless, but the ground is not. Calculate the force from the wall (Fw) and the force from the ground (FG).

Working with the Triangle

52 = 42 + x2

x2 = 52 – 42

x = 3

sin = 4/5

= 53o

5 m4 m

x

5 m4 m

3 m

53o

37o

Fx = 0

0 = FGx – Fw

0 = Fgcos - Fw

Fy = 0

0 = FGy – mg

0 = Fgsin - 118

Back to the ladder

Tilting the Ladder

53o

37o

37o

mg

Fw

FG

53o

53o

37o

FGFw

mg

Calculating the Torque forces

Choose the point at ground as the pivot = 0

0 = (5m)(Fw)(sin53o) – (2.5m)(mg)(sin 37o)

0 = 3.99Fw - 177 m-N

53o

37o

FGFw

mg

Fwsin53o

mgsin37o

Three equations, three unknowns

0 = Fgcos - Fw

0 = Fgsin - 118

0 = 3.99Fw - 177

= 69.3o

Fg = 126 N

Fw = 44.3 N

The Ladder: Example 2

Mr. Fredericks (56.0 kg) leans a 3.00 m, 20.0 kg ladder against his house at an angle of 65.0o with the ground. He can safely climb 2.50 m up the ladder before it slips. Calculate the coefficient of friction between the ground and the ladder.

Fx = 0

0 = Fgcos - Fw

Fy = 0

0 = Fgsin – (20.0 kg)(g) – (56.0 kg)(g)

0 = Fgsin – 744.8

Choose the point at ground as the pivot = 00 = (20)(g)(1.5)(sin25o) + (56)(g)(2.5m)(sin 25o)

- (FW)(3)(sin 65o)

0 = 704 – 2.72Fw

Three equations, three unknowns

0 = Fgcos - Fw

0 = Fgsin – 744.8

0 = 704 – 2.72Fw

FW = 259 N, Fg = 788 N, = 70.8o

Dealing with Friction

Fgx = Fgcos 70.8o

Fgx = 259 N = Ffr

Fgy = Fg sin 70.8o

Fgy = 744 N = FN

Ffr = FN

259 = 744

= 0.35

The Ladder: Example 3

A man who weighs 800 N climbs to the top of a 6 meter ladder that is leaning against a smooth wall at an angle of 60°. The non-uniform ladder weighs 400 N and its center of gravity is 2 meters from the foot of the ladder. What must be the minimum coefficient of static friction between the ground and the foot of the ladder if it is not to slip?

Fx = 0

0 = Fw - Fgcos

Fy = 0

0 = Fgsin – 800 – 400

0 = Fgsin – 1200

 Working with the Triangle

Tilt the Ladder FG400 N

800 N

Fw

60o

60o

30o

30o

60o

30o 30o

FG

Fw

800 N 400 N

Calculating the Torque forces

Choose the ground as the pivot

= 0

0 =(2)(400sin30o) + (6)(800sin30o) - (6)(Fwsin60o)

0 = 2800 – 5.2Fw

60o

30o 30o

FG

Fw

800 N 400 N800sin30o 400sin30o

Fwsin60o

Three equations, three unknowns

0 = Fw - Fgcos

0 = Fgsin – 1200

0 = 2800 – 5.2Fw

Fw = 538 N, Fg = 1315 N, = 65.9o

Dealing with Friction

Fgx = Fgcos65.9o

Fgx = 537 N = Ffr

Fgy = Fg sin65.9o

Fgy = 1200 N = FN

Ffr = FN

537 = 1200

= 0.45

Stable vs. Unstable Equilibrium

A body will fall if its center of gravity is no longer above the base

Humans adjust their posture to keep CG above their base

Try leaning against a wall and lifting one leg.

Elasticity: Hooke’s Law

• Hooke’s Law – usually used with a spring

• Can consider anything to be like a spring

• F = kL (F=kxspring)

• k = proportionality (spring) constant

• Can’t stretch things forever

Elastic region – material will still bounce back

Plastic region – material will not return to original length (but has not broken)

F = kL

This is only linear

in the proportional

region

Elastic Region: Young’s Modulus(E)

Stress = Force = F

Area A

Strain = Change in length = L

Original length Lo

E = stress

strain

E = F/A or F = E L

L/Lo A Lo

Young’s Modulus: Example 1

A 1.60 m long steel piano wire has a diameter of 0.20 cm. How great is the tension in the wire if it stretches 0.30 cm when tightened?

A = r2 = (0.0010 m)2 = 3.1 X 10-6 m2

F = E L

A Lo

F = AE L

Lo

F = AE L

Lo

F = (3.1 X 10-6m2)(200 X 1011N/m2)(0.0030 m)

1.60 m

F = 1200 N

Young’s Modulus: Example 2

A steel support rod of radius 9.5 mm and length 81 cm is stretched by a force of 6.2 X 104 N (about 7 tons). What is the stress? What is the elongation?

Area = r2 = (0.0095 m2

Stress = Force = 6.2 X 10-4m = 2.2 X 108 N/m2

Area 2.84 X 10-4 m2

F = EL

A Lo

L = FL = (6.2 X 104 N)(0.81m)EA (200 X 109 N/m2)(2.84 X 10-4

m2)

L = 8.84 X 10-4 m = 0.89 mm

A 100 kg block hangs at the end of a 20 cm copper wire. The wire has a diameter of 5.05 X 10-4 m.

a)Calculate the area of the wire

b)Calculate the stress on the wire.

c)If the wire stretches 8.9 mm, calculate the Young’s Modulus for copper.

(1.1 X 1011 N/m2)

The Three Types of Stress

Stretching Squeezing Horizontal

Other Modulus’

Shear Modulus – Used for shear stress

Bulk Modulus – Used for even compression on all sides (an object when submerged)

Fracture

• Breaking Point

• Uses– Tensile Strength – Stretching– Compressive Strength – under a load– Shear Stress – Shearing

• Safety Factor – reciprocal that is multiplied by the tensile strength

• Ex: A safety factor of 3 means you will only use 1/3 of the maximum stress

Fracture: Example 1

A concrete column 5 m tall will have to support 1.2 X 105 N (compression). What area must it have to have a safety factor of 6?

Max stress = (1/6)(compressive strength)

Max stress = (1/6)(20 X 106 N/m2)= 3.3X106 N/m2

Stress = F

A

Stress = F

A

Area = F = (1.2 X 105 N) = 3.64 X 10-2 m

Stress 3.3 X 106N/m2

How much will the column compress under the load?

F = EL

A Lo

L = FL = (1.2 X 105 N)(5 m)

EA (20 X 109 N/m2)(3.64 X 10-2 m2)

L = 8.3 X 10-4 m = 0.83 mm

Fracture: Example 2

Spider-man’s webbing has a tensile strength of 600 X 106 N/m2 and he wishes to use a safety factor of 3. What is the diameter of the webbing if the maximum force at the bottom of a swing is 1500 N?

Maximum Stress

(1/3)(600 X 106 N/m2) = 200 X 106 N/m2

Stress = Force

Area

Area = Force = 1500 N = 7.5 X 10-6m2

Stress 200X106N/m2

Area = r2

r = (A/)1/2 = 1.55 X 10-3 m or 1.55 mm

Diameter = 3.10 mm

Concrete

• Concrete is much stronger under compression than tension– Tensile Strength – 2 X 106N/m2

– Compressive Strength – 20 X 106N/m2

• Prestressed concrete – rods or mesh are stretched when the concrete is poured. Released after concrete dries.

• Now under compression

2. 1.7 N

4. 1.7 m

6. 6.52 kg

8. 6130 N, 8570 N

10. 3.0 m from the adult

12. FT2 = 3900 N, FT1 = 3400 N

14. 560 N

16. 48 N (only vertical components balance wt.)

18. 1.1 m (pivot on the side of lighter boy)

20. F1 = -2100 N(down), F2 = 3000 N (up)

24. a) 210 Nb) 2000 N

26. FT = 542 N, FhingeH = 409 N, FhingeV=-5.7 (down)

28. min = tan-1(1/2)

44. 1.91 cm

46. 0.029 mm

50. 9.6 X 106 N/m2

56. FTmax = 390 N

58. a) 4.4 X 10-5 m2 b) 2.7 mm