1 - Oakland University



Statics of particles and rigid bodies

∑F = R = 0 and ∑MO = 0 MO = rO × F

g = 9.81 m/s2 = 32.2 ft/s2

Normal stress, deformation

σ = P/A ε = δ/L σ = Eε δ = PL / AE

Friction

static equilibrium: f ≤ μsN ; if moving, f = μkN

belt friction: T2 = T1 eμβ

Particle kinematics

v = dx/dt a = dv/dt = d2x/dt2 = v(dv/dx)

uniform rectilinear motion: x = x0 + v0t

uniformly accelerated rectilinear motion:

v = v0 + a0t x = x0 + v0t + ½ a0t2 v2 = v02 + 2a0(x-x0)

relative motion: xB = xA + xB/A vB = vA + vB/A aB = aA + aB/A

intrinsic coordinates: v = v et a = dv/dt et + v2/ρ en

Particle kinetics

Newton’s Second Law: ∑F = ma

intrinsic coordinates: ∑Ft = m dv/dt ∑Fn = m v2/ρ

work and energy: TA + UAB = TB

kinetic energy: T = ½mv2

work done by spring : U = ½k δ2 ; by weight: -Wy

conservation of energy: VA + TA = VB + TB

if not conserved, then: VA + TA = VB + TB + energy lost

impact and momentum: mv1 + ∫Fdt = mv2

coefficient of restitution: e = (v´B-v´A)/(vA-vB) = v´B/A/vA/B

Rigid body kinematics – plane motion

translation: vA = vB aA = aB

rotation: v = ωk × r a = αk × r – ω2r

uniform rotation: θ = θ0 + ω0t

uniformly accelerated rotation:

ω = ω0 + α0t θ = θ0 + ω0t + ½ α0t2 ω2 = ω02 + 2α0(θ-θ0)

general plane motion: vB = vA + ωk × rB/A a= aA + αk × rB/A – ω2rB/A

rolling: v = ωr a = αr

pulleys: ω1r1= ω2r2 gears: ω1r1= -ω2r2

mass moments of inertia:

parallel axis theorem: IO = IG + md2 ; radius of gyration: k2 = I/m

rod: IG = mL2 / 12 disk: IG = mr2 / 2

ring: IG = mr2 sphere: IG = 2mr2 / 5

Rigid body kinetics – plane motion

Newton’s Second Law: ∑F = ma ∑MG = IGαk

non-centroidal rotation: ∑Mo = Ioαk

kinetic energy: T = ½mvG2 + ½ IG ω2 ; non-centroidal rotation: T = ½ IO ω2

total momentum: mvG + IGω ; non-centroidal rotation: Ioω

1. Given force F = (3i + 5j + 4k)lb passing through point (4,12,6)ft, find the y-component of the moment of F about the origin.

a. -2 ft-lb

b. 0

c. 2 ft-lb

d. 14 ft-lb

2. Given F = (3i + 5j + 4k)N and r = (6i – 3j + 12k)m, find F∙r.

a. -21 N-m

b. (-72i + 12j + 39k)N-m

c. 42 N-m

d. 51 N-m

3. The block shown is acted upon by a force P = 7 lb. If μs=0.8 and μk=0.5, determine the acceleration of the block.

a. 0

b. 3.22 m/s2

c. 6.44 m/s2

d. 25.8 m/s2

4. Determine the force in member AB.

a. 900 N (T)

b. 900 N (C)

c. 1200 N (T)

d. 1200 N (C)

5. The reaction at B is:

a. 3.5 lb ↑

b. 3.5 lb ↓

c. 16.5 lb ↑

d. 16.5 lb ↓

6. Find the force in member BD of the simple truss.

a. 750 N (C)

b. 1000 N (C)

c. 1000 N (T)

d. 3000 N (C)

7. A block will slide down a 30° incline if the coefficient of static friction is less than

a. 0.500

b. 0.577

c. 0.866

d. 0.625

8. What force F is required for equilibrium?

a. W / 4

b. W / 6

c. W / 8

d. W / 9

9. One complete revolution of a rope about a post is used to hold a boat. If μs=0.5, what maximum force by the boat can be resisted if the holder provides a force of 100 N?

a. 962 N

b. 1920 N

c. 2310 N

d. 3850 N

10. A 30-mm diameter rod supports a tensile force of 35 kN. What is the normal stress in the rod?

a. 12.4 MPa

b. 36.4 MPa

c. 42.8 MPa

d. 49.5 MPa

11. A weight of 40 kN is suspended from a 20-m long, 20-mm diameter steel cable. If E = 207 GPa, determine the elongation of the cable.

a. 3.1 mm

b. 8.7 mm

c. 10.8 mm

d. 12.3 mm

12. The Work and Energy principle is a relationship between

a. Force, acceleration and time

b. Force, speed and distance

c. Force, velocity and time

d. All of the above

13. Impulse and Momentum is a relationship between

a. Force, acceleration and time

b. Force, speed and distance

c. Force, velocity and time

d. All of the above

14. A ball is given an initial velocity of 40 m/s straight up. Ignoring friction, how high will the ball go?

a. 40 m

b. 60.8 m

c. 81.6 m

d. 122 m

15. A wheel rotates at 20 rad/s. How many revolutions will it rotate in 4 s, after it begins to decelerate at 10 rad/s2?

a. 3.18 rev

b. 4.25 rev

c. 6.37 rev

d. 7.40 rev

16. A car is traveling at a constant speed around a circular curve having a radius of curvature of ρ = 250 m. If the magnitude of the car’s acceleration is 1.5 m/s2, determine the speed at which the car is traveling.

a. 19.4 m/s

b. 23.7 m/s

c. 32.8 m/s

d. 34.1 m/s

17. The center of the wheel is moving to the right with a speed of 2 m/s. If no slipping occurs at the ground, determine the velocity of point B at the instant shown.

a. 1.33 rad/s cw

b. 4 m/s →

c. 3 m/s →

d. 2 m/s →

18. If the system shown is released from rest, find the tension in the cord which is wrapped around the 50-kg cylinder.

a. 80 N

b. 99 N

c. 100 N

d. 109 N

19. The 20-kg mass in Problem 18 drops from rest. If it falls through 2 m, what is the angular speed of the cylinder?

a. 8.4 rad/s

b. 9.3 rad/s

c. 10.4 rad/s

d. 12.2 rad/s

20. The 35-kg block strikes the unstretched spring. The maximum deflection of the spring is:

a. 1.67 m

b. 2.80 m

c. 4.78 m

d. 6.33 m

21. The 6-lb ball is fired from rest using a spring as shown. Determine how far the spring must be compressed so that when the ball reaches a height of 8 ft it has a speed of 6 ft/s.

a. 0.80 ft

b. 1.27 ft

c. 1.60 ft

d. 2.56 ft

22. A ball is dropped vertically downward onto a horizontal surface from a height of 52 m. If the coefficient of restitution e = 0.8, determine the speed of the ball after it bounces.

a. 20.5 m/s

b. 25.6 m/s

c. 30.6 m/s

d. 32.0 m/s

23. The clock pendulum consists of a 1-kg slender rod A and a 4-kg moveable disk B. Determine the mass moment of inertia of the pendulum about the fixed point O.

a. 0.10 kg-m2

b. 1.97 kg-m2

c. 2.30 kg-m2

d. 3.14 kg-m2

24. In the belt and pulley system shown, if pulley A is rotating at 1740 rpm, what is the angular speed of pulley D? Pulley C is attached to and rotates with pulley B.

a. 196 rpm

b. 870 rpm

c. 1390 rpm

d. 2180 rpm

25. In the gear system shown, if gear A is rotating at 1740 rpm clockwise, determine the angular velocity of gear E. Gear B is attached to and rotates with gear C.

a. 61.9 rpm ccw

b. 61.9 rpm cw

c. 96.7 rpm ccw

d. 96.7 rpm cw

-----------------------

10 lb

7 lb

3

4

1200 N

B

A

C

A

B

C

D

E

3 m

3 kN

4 m

4 m

4 m

10 lb

3 lb

6 ft

6 ft

6 ft

A

B

F

W

8 ft

k = 40 lb/ft

1.5 m

A

B

2 m/s

20 kg

50 kg

400 mm

40 m/s

k = 20 kN/m m/s

1 m

0.7 m

50 mm

O

A

B

C

D

rA = 15 mm

rB = 50 mm

rC = 15 mm

rD = 40 mm

A

B

C

D

E

rA = 4 mm

rB = 18 mm

rC = 4 mm

rD = 20 mm

rE = 25 mm

6 ft/s

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