(f ) work done by a force, (vi) rad/s
(g) angular velocity, (vii) m
(h) angular acceleration, (viii) rad
(i) force, F (ix) N
1.5 The machine of Figure P1.5 has a vertical displacement x(t). The machine has a component which rotates with a constant angular speed . The center of mass of the rotating component is a distance efrom the axis of rotation. The center of mass of the rotating component is as shown at t⫽0. Determine the vertical component of the acceleration of the rotating component.
ω
x(t)
e
FIGURE P1.5
(a)
r
(b) FIGURE P1.6
1.6 The rotor of Figure P1.6 consists of a disk mounted on a shaft. Unfortunately, the disk is unbalanced, and the center of mass is a distance efrom the center of the shaft. As the disk rotates, this causes a phenomena called “whirl”, where the disk bows. Let rbe the instantaneous distance from the center of the shaft to the original axis of the shaft and be the angle made by a given radius with the horizontal. Determine the acceleration of the mass center of the disk.
1.7 A 2 tonne truck is traveling down an icy, 10° hill at 80 km/h when the driver sees a car stalled at the bottom of the hill 76 m away. As soon as he sees the stalled car, the driver applies his brakes, but due to the icy conditions, a braking force of only 2000 N is generated. Does the truck stop before hitting the car?
1.8 The contour of a bumpy road is approximated by
What is the amplitude of the vertical acceleration of the wheels of an automobile as it travels over this road at a constant horizontal speed of 40 m/s?
1.9 The helicopter of Figure P1.9 has a horizontal speed of 33 m/s and a horizontal acceleration of 1 m/s2. The main blades rotate at a constant speed of 135 rpm. At the instant shown, determine the velocity and acceleration of particle A.
y(x) = 0.03sin(0.125x) m
1.10 For the system shown in Figure P1.10, the angular displacement of the thin disk is given by rad. The disk rolls without slipping on the surface. Determine the following as functions of time.
(a) The acceleration of the center of the disk.
(b) The acceleration of the point of contact between the disk and the surface.
(c) The angular acceleration of the bar.
(d) The vertical displacement of the block.
(Hint:Assume small angular oscillations of the bar. Then sin 艐.) u(t) = 0.03sin(30t + p
4)
45°
64 cm
A 135 rpm
33 m/s
1 m/s2
FIGURE P1.9
Thin disk of radius 10 cm
Rigid link θ(t) = 0.03 sin(30t + –)
Rigid link 20 cm
4 π
φ
30 cm
FIGURE P1.10
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
1.11 The velocity of the block of the system of Figure P1.11 is sin 20t m/s downward.
(a) What is the clockwise angular displacement of the pulley?
(b) What is the displacement of the cart?
1.12 A 30-kg block is connected by an inextensible cable through the pulley to the fixed surface, as shown in Figure P1.12. A 20 kg weight is attached to the pulley, which is free to move vertically. A force of magnitude N tows the block. The system is released from rest at
(a) What is the acceleration of the 30-kg block as a function of time?
(b) How far does the block travel up the incline before it reaches a velocity of 60 cm/s?
t = 0.
P = 500(1 + e-t) y#
= 0.02
r2 r1 r1 = 10 cm r2 = 30 cm
y = 0.02sin20t m/s
FIGURE P1.11 FIGURE P1.12
20 kg
30 kg
45° µ = 0.3
P
1.13 Repeat Problem 1.12 for a force of
1.14 Figure P1.14 shows a schematic diagram of a one-cylinder reciprocating one- cylinder engine. If at the instant of time shown the piston has a velocity vand an acceleration a, determine (a) the angular velocity of the crank and (b) the angular acceleration of the crank in terms of v, a, the crank radius r, the connecting rod length and the crank angle
1.15 Determine the reactions at Afor the two-link mechanism of Figure P1.15. The roller at Crolls on a frictionless surface.
u.
/,
P = 100t N.
1.16 Determine the angular acceleration of each of the disks in Figure P1.16.
l
, a
r θ
B
C A
3.6 kg
2.4 kg 30°
2 m
2.6 m/s 1.4 m/s2 3 m
FIGURE P1.14
FIGURE P1.16
FIGURE P1.15
20 kg 60 cm
(a) (b)
30 kg 4 kg · m2
180 N 60 cm
270 N 4 kg · m2
1.17 Determine the reactions at the pin support and the applied moment if the bar of Figure P1.17 has a mass of 50 g.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
1.18 The disk of Figure P1.18 rolls without slipping. Assume if (a) Determine the acceleration of the mass center of the disk.
(b) Determine the angular acceleration of the disk.
P = 18 N.
3 m 1 m
θ = 10°
α = 14 rad/s2
ω = 5 rad/s M
FIGURE P1.17 FIGURE P1.18
P 1.8 kg 20 cm
1.19 The coefficient of friction between the disk of Figure P1.18 and the surface is 0.12. What is the largest force that can be applied such that the disk rolls without slipping?
1.20 The coefficient of friction between the disk of Figure P1.18 and the surface is 0.12. If what are the following?
(a) Acceleration of the mass center.
(b) Angular acceleration of the disk.
1.21 The 3 kg block of Figure P1.21 is displaced 10 mm downward and then released from rest.
(a) What is the maximum velocity attained by the 3-kg block?
(b) What is the maximum angular velocity attained by the disk?
P = 22 N,
5 kg 3 kg
20 cm
4000 N/m 0.25 kg · m2
FIGURE P1.21
1.22 The center of the thin disk of Figure P1.22 is displaced 15 mm and released.
What is the maximum velocity attained by the disk, assuming no slipping between the disk and the surface?
r r = 25 cm 20,000 N/m
m = 2 kg
FIGURE P1.22 FIGURE P1.23
k
m
µ
1.23 The block of Figure P1.23 is given a displacement and then released.
(a) What is the minimum value of such that motion ensues?
(b) What is the minimum value of such that the block returns to its equilibrium position without stopping?
1.24 The five-blade ceiling fan of Figure P1.24 operates at 60 rpm. The distance between the mass center of a blade and the axis of rotation is 0.35 m. What is its total kinetic energy?
d d
d
60 rpm G
G 13 mm
m = 1.21 kg
m = 4.7 kg I = 0.96 kg · m2
I = 5.14 kg · m2 Blade
Motor FIGURE P1.24
1.25 The U-tube manometer shown in Figure P1.25 rotates about axis A-Aat a speed of 40 rad/s. At the instant shown, the column of liquid moves with a
100 cm
40 rad/s v = 20 m/s
Specific gravity = 1.4 Area = 3 × 10–4 m2
60 cm 20 cm
FIGURE P1.25
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
speed of 20 m/s relative to the manometer. Calculate the total kinetic energy of the column of liquid in the manometer.
1.26 The displacement function for the simply supported beam of Figure P1.26 is
wherec⫽0.003 m and tis in seconds. Determine the kinetic energy of the beam.
y(x, t) = c sinapx
L bcosap2A EI rAL4tb
x
y(x, t) E = 200 × 104 N/m2 I = 1.73 × 10–7 m4 ρ = 7400 kg/m3 A = 1.6 × 10–4 m2
3.1 m
FIGURE P1.27
FIGURE P1.28 FIGURE P1.26
1.27 The block of Figure P1.27 is displaced 1.5 cm from equilibrium and released.
(a) What is the maximum velocity attained by the block?
(b) What is the acceleration of the block immediately after it is released?
12,000 N/m
65 kg
1.28 The slender rod of Figure P1.28 is released from the horizontal position when the spring attached at Ais stretched 10 mm and the spring attached at Bis unstretched.
(a) What is the acceleration of the bar immediately after it is released?
(b) What is the maximum angular velocity attained by the bar?
B A
y
1000 N/m
m = 1.2 kg 1200 N/m
1 m
1.29 Let xbe the displacement of the left end of the bar of the system in Figure P1.29.
Let represent the clockwise angular rotation of the bar.
(a) Express the kinetic energy of the system at an arbitrary instant in terms of and .
(b) Express the potential energy of an arbitrary instant in terms of and u.x u
# x#
u
k k
θ x
4
3L F(t)
FIGURE P1.29
1.30 Repeat Problem 1.29 using as coordinates x1, which is the displacement of the mass center, andx2, which is the displacement of the point of attachment of the spring that is a distance 3L/4 from the left end.
1.31 Let represent the clockwise angular displacement of the pulley of the system in Figure P1.31 from the system’s equilibrium position.
(a) Express the potential energy of the system at an arbitrary instant in terms of . (b) Express the kinetic energy of the system at an arbitrary instant in terms of .u
#
2r
2 m Ip m
k
2 k r
θ
FIGURE P1.31
1.32 A 20 tonne railroad car is coupled to a 15 tonne car by moving the 20 tonne car at 8 km/h toward the stationary 15 tonne car.
(a) What is the resulting speed of the two-car coupling?
(b) What would the resulting speed be if the 15 tonne car is moving at 8 km/h toward a stationary 20 tonne car?
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
1.33 The 15 kg block of Figure P1.33 is moving with a velocity of 3 m/s at t ⫽0 when the force F(t) is applied to the block.
(a) Determine the velocity of the block at t⫽2 s.
(b) Determine the velocity of the block at t⫽4 s.
(c) Determine the block’s kinetic energy at t ⫽4 s.
v
F(t) 15 kg 30 N
3 t
µ = 0.08 FIGURE P1.33
1.34 A 400 kg forging hammer is mounted on four identical springs, each of stiffness k⫽4200 N/m. During the forging process, a 110 kg hammer, which is part of the machine, is dropped from a height of 1.4 m onto an anvil, as shown in Figure P1.34.
(a) What is the resulting velocity of the entire machine after the hammer is dropped?
(b) What is the maximum displacement of the machine?
1.4 m
Drop hammer
Workpiece
Anvil
FIGURE P1.34
1.35 The motion of a baseball bat in a ballplayer’s hands is approximated as a rigid- body motion about an axis through the player’s hands, as shown in Figure P1.35.
The bat has a centroidal moment of inertia I. The player’s “bat speed” is , and the velocity of the pitched ball is v. Determine the distance from the player’s hand along the bat where the batter should strike the ball to minimize the
v
1.36 A playground ride has a centroidal moment of inertia of 23 km . m2. Three children of weights 222 N, 222 N, and 222 N are on the ride, which is rotating at 60 rpm. The children are 76 cm from the center of the ride. A father stops the ride by grabbing it with his hands. What is the impulse felt by the father?
Problems 1.37 through 1.39 present different problems that are to be formulated in non- dimensional form. For each problem answer the following.
(a) What are the dimensions involved in each of the parameters?
(b) How many dimensionless parameters does the Buckingham Pi theorem predict are in the non-dimensional formulation of the relation between the natural frequencies and the other parameters?
(c) Develop a set of dimensionless parameters.
1.37 The natural frequencies of a thermally loaded fixed-fixed beam (Figure P1.37) are a function of the material properties of the beam, including:
E, the elastic modulus of the beam , the mass density of the beam , the coefficient of thermal expansion The geometric properties of the beam are A, its cross-sectional area
I, its cross section moment of inertia L, its length
Also,
, the temperature difference between the installation and loading
¢T
FIGURE P1.35 G
a b
ω
L
E, I, A, P, α, ∆T
FIGURE P1.37
impulse felt by the his/her hands. Does the distance change if the player
“chokes up” on the bat, reducing the distance from Gto his/her hands.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
1.38 The drag force Fon a circular cylinder due to vortex shedding is a function of
the velocity of the flow
the dynamic viscosity of the fluid the mass density of the fluid the length of the cylinder
the diameter of the cylinder
1.39 The principal normal stress due to forcing of a beam with a concentrated harmonic excitation is a function of
, the amplitude of loading the frequency of the loading the elastic modulus of the beam the mass density of the beam the beam’s cross-sectional area
the beam’s cross-sectional moment of inertia the beam’s length
the location of the load along the axis of the beam
1.40 A MEMS system is undergoing simple harmonic motion according to
(a) What is the period of motion?
(b) What is the frequency of motion in Hz?
(c) What is the amplitude of motion?
(d) What is the phase and does it lead or lag?
(e) Plot the displacement.
1.41 The force that causes simple harmonic motion in the mass-spring system of Figure P1.41 is F(t)⫽35 sin 30tN. The resulting displacement of the mass is x(t)⫽0.002
(a) What is the period of the motion?
(b) The amplitude of displacement is where is the amplitude of the force and is a dimensionless factor called the magnification factor.
Calculate M.
(c) Mhas the form
where is called the natural frequency. If then otherwise Calculate vn.
f = 0.
f = p;
vn 6 v, vn
M =
1
`1 - av vnb2` M
F0 X =
F0 kM sin(30t - p)m.
x(t) = 33.1sin(2 *105t + 0.48) + 4.8cos(2 * 105t + 1.74)4 mm a,
L, I, A, r, E, v, F0 D, L, r, m, U,
1.42 The displacement vector of a particle is
(a) Describe the trajectory of the particle.
(b) How long does it take the particle to make one circuit around the path?
r(t) = 32sin20t i + 3cos20t j4 mm
m
35 sin 30t 3.5 ⫻ 104 N/m
FIGURE P1.41
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).