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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