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The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when θ = 0°, determine its angular velocity just before it strikes the specimen S , θ = 90°.
Given:
M =50 kg kA =1.75 m
d =1.25 m
g 9. m s^2
Solution:
0 + M g d
= M kA^2 ω 2^2
ω 2 2 g d kA^2
= ω 2 2. rad s
Problem 18- The pulley of mass M (^) p has a radius of gyration about O of kO. If a motor M supplies a force to the cable of P = a ( b − ce − dx ), where x is the amount of cable wound up, determine the speed of the crate of mass M (^) c when it has been hoisted a distance h starting from rest. Neglect the mass of the cable and assume the cable does not slip on the pulley. Given: M (^) p = 10 kg a =800 N M (^) c = 50 kg b = 3 kO = 0.21 m c = 2 r = 0.3 m d^1 m
h =2 m
Solution:
Guesses vc 1 m s
599
Problem 18-
Given 0
h
a b (^ − c e −^ dx ) x
d 1 2
M (^) p kO^2
vc r
2 1 2
= + M (^) c vc^2 + M (^) c g h
vc = Find ( v c ) vc 9.419 m
s
Problem 18-
The uniform pipe has a mass M and radius of gyration about the z axis of kG. If the worker pushes on it with a horizontal force F , applied perpendicular to the pipe, determine the pipe’s angular velocity when it has rotated through angle θ about the z axis, starting from rest. Assume the pipe does not swing.
Units Used:
Mg = 10 3 kg
Given:
M =16 Mg θ =90 deg kG = 2.7 m r =0.75 m F = 50 N l =3 m
Solution:
0 + F l θ^1 2
= M kG^2 ω^2
ω 1 M
M F l π kG
= ω 0.0636 rad s
*Problem 18-
The slender rod of mass mrod is subjected to the force and couple moment. When it is in the position shown it has angular velocity^ ω 1. Determine its angular velocity at the instant it has rotated downward 90°. The force is always applied perpendicular to the axis of the rod. Motion occurs in the vertical plane.
600
mrod a^2 3
ω 1^2 + F a 2 π+ M 2 π^1 2
mrod a^2 3
= ω 2^2
ω 2 =Find( ω (^) 2 ) ω 2 11.2 rad s
Problem 18-
The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving sheave A with constant torque M , determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves.
Units Used: Mg =1000 kg
Given:
mE = 1.80 Mg d =10 m mC = 2.30 Mg r =0.35 m mS = 150 kg k =0.2 m
M = 100 N m⋅ g 9.81 m s^2
Solution:
M d r
− (^) ( m (^) E − mC ) g d^1 2
mE + mC 2 mSk
2
r^2
= v^2
v
2 M
d r
⎡⎢ −( m (^) E − mC ) g d
mE + mC 2 mSk
2
r^2
= v 4.973 m s
Problem 18-
The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving
sheave A with torque a^ θ^2 + b , determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves.
602
Given
Units Used:
Mg =1000 kg
Given:
mE =1.80 Mg mC =2.30 Mg mS =150 kg
a =0.06 N m⋅ b =7.5 N m⋅ d =12 m r =0.35 m k =0.2 m
g 9.81 m s^2
Solution:
Guess v 1 m s
Given 0
d r a θ^2 + b θ
⎮⌡ d^ −^ (^ mE − mC ) g d
mE + mC 2 mSk
2
r^2
= v^2
v = Find ( ) v v 5.343 m s
*Problem 18-
The wheel has a mass M 1 and a radius of gyration kO. A motor supplies a torque M^ = ( a θ + b ), about the drive shaft at O. Determine the speed of the loading car, which has a mass M 2 , after it travels a distance s = d.^ Initially the car is at rest when s = 0 and^ θ^ = 0°. Neglect the mass of the attached cable and the mass of the car’s wheels.
603
kG =0.375 ft
Solution:
Guess ω 1 rad s
Given M d ro
W
g
( ω ro )^2 2
W
g
k
ri + ro ro
⎛⎜ d
2 = +
ω =Find( ω) ω 7.08 rad s
Problem 18-
The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine its angular velocity at the instant its mass center G has moved distance d down along the inclined plane. The disk rolls without slipping.
Given:
md =20 kg θ =30 deg
M = 30 N m⋅ r =0.2 m
d = 0.8 m g 9.81 m s^2
k 150 N m
Solution: Guess ω 1 rad s
Initial stretch in the spring k d 0 = md g sin( θ )
d (^0)
md g sin(^ θ) k
= d 0 =0.654 m
Given
M d r
− ⎡⎣( d + d 0 )^2 − d 0^2 ⎤⎦^1 2
md ( ω r )^2 2
⎛⎜ md r^2
= + ω^2
605
ω =Find(^ ω) ω 11. rad s
Problem 18-
The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine how far the center of mass of the disk travels down along the incline, measured from the equilibrium position, before it stops. The disk rolls without slipping.
Given:
md =20 kg M =30 N m⋅
k 150
N
m
θ =30 deg r =0.2 m
g 9. m s^2
Solution: Guess d =3 m
Initial stretch in the spring k d 0 = md g sin( θ^ )
d (^0)
md g sin(^ θ) k = d 0 =0.654 m
Given M d r
- md g d sin(^ θ) k 2 − ⎡⎣( d + d 0 )^2 − d 0^2 ⎤⎦= 0
d = Find ( ) d d =2 m
*Problem 18-
The linkage consists of two rods AB and CD each of weight W 1 and bar AD of weight W 2. When θ = 0, rod AB is rotating with angular velocity ω O. If rod CD is subjected to a couple moment M
606
Given:
W 1 = 8 lb a =2 ft
W 2 = 10 lb b =3 ft
ω 0 2 rad s
= θ 1 =45 deg
M = 15 lb ft⋅ P =20 lb
g 32.2 ft s^2
Solution:
U P a sin( θ (^) 1 ) + M θ 1 2 W 1 a 2
= − (^) ( 1 −cos( θ 1 ))− W 2 a (^) ( 1 −cos( θ (^) 1 ))
Guess ω 1 rad s
Given 1 2
W 1
g
a^2 3
ω 0^2 2
W 2
g
W 1
g
a^2 3
ω^2 2
W 2
g
= + ( a ω)^2
ω =Find( ω) ω 5.916 rad s
Problem 18-
The spool has weight W and radius of gyration kG. A horizontal force P is applied to a cable wrapped around its inner core. If the spool is originally at rest, determine its angular velocity after the mass center G has moved distance d to the left. The spool rolls without slipping. Neglect the mass of the cable.
Given:
W = 500 lb d =6 ft
608
kG = 1.75 ft ri =0.8 ft
P = 15 lb ro =2.4 ft
g 32. ft s^2
Solution: (^) Guess ω 1 rad s
Given P
ro + ri ro
d
W
g
kG^2 ω^2
W
g
= + ( r (^) o ω)^2 ω =Find(^ ω) ω 1. rad s
Problem 18-
The double pulley consists of two parts that are attached to one another. It has a weight W (^) p and a centroidal radius of gyration kO and is turning with an angular velocity ω clockwise. Determine the kinetic energy of the system. Assume that neither cable slips on the pulley.
Given:
W (^) P = 50 lb r 1 =0.5 ft
W (^) A = 20 lb r 2 =1 ft
W (^) B = 30 lb kO =0.6 ft
ω 20 rad s
Solution:
KE
I ω^2
= + W (^) B vB^2
KE
W P
g
kO^2 ω^2
W A
g
W B
g
= + ( r 1 ω)^2
KE =283 ft lb⋅
609
Given:
W 1 =1500 lb W 2 =115 lb M =2000 lb ft⋅ kO =0.95 ft h =10 ft r =1.25 ft
Solution:
Guess v 1 ft s
Given M h r
W 1
g
v^2 2
W 2
g
kO^2 v r
2 = + + W 1 h
v = Find ( ) v v 6.41 ft s
Problem 18-
The assembly consists of two slender rods each of weight W (^) r and a disk of weight W (^) d. If the spring is unstretched when^ θ^ =^ θ 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant θ = 0. The disk rolls without slipping.
Given:
W (^) r =15 lb W (^) d =20 lb θ 1 =45 deg
k 4 lb ft
L =3 ft r =1 ft
Solution:
Guess ω 1 rad s
611
Given (^2) W r
L
sin( θ (^) 1 ) 1 2
− k (^) ( 2 L − 2 L cos( θ 1 ))^22 2
W (^) r g
⎛⎜ L 2
= ω^2
ω =Find( ω) ω 4.284 rad s
Problem 18-
The uniform door has mass M and can be treated as a thin plate having the dimensions shown. If it is connected to a torsional spring at A , which has stiffness k , determine the required initial twist of the spring in radians so that the door has an angular velocity^ ω^ when it closes at^ θ^ = 0° after being opened at θ = 90° and released from rest. Hint: For a torsional spring M = k θ, where k is the stiffness and θ is the angle of twist.
Given:
M = 20 kg a =0.8 m
k 80 N m⋅ rad
= b =0.1 m
ω 12 rad s
= c =2 m
P =0 N
Solution:
Guess θ 0 =1 rad Given
θ 0 +90 deg
θ 0 − k θ θ
d 1 2
= M a^2 ω^2
θ 0 =Find( θ 0 ) θ 0 =1.659 rad
*Problem 18-
The uniform slender bar has a mass m and a length L. It is subjected to a uniform distributed load w 0 which is always directed perpendicular to the axis of the bar. If it is released from the position shown, determine its angular velocity at the instant it has rotated 90°. Solve the problem for rotation in (a) the horizontal plane, and (b) the vertical plane.
612
Problem 18-
The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when θ = θ 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod.
Given:
W =1500 lb θ 1 =60 deg
L =13 ft h =12 ft
a =7 ft
g 32.2 ft s^2
Solution:
W L 2 sin( θ (^) 1 ) h^ − a 2
W
g
= vA^2
vA 2 g L 2 sin( θ (^) 1 ) h^ − a 2
= vA 14.2 ft s
Problem 18-
The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when θ = 0°, determine its angular velocity just before it strikes the specimen S ,^ θ^ = 90°,^ using the conservation of energy equation.
Given:
M =50 kg kA =1.75 m
d =1.25 m
g 9.81 m s^2
614
Solution:
0 + M g d 0 1 2 = + M kA^2 ω 2^2 ω 2^2 g d kA^2
ω 2 2.83 rad s
*Problem 18-
The soap-box car has weight W (^) c including the passenger but excluding its four wheels. Each wheel has weight W (^) w radius r , and radius of gyration k , computed about an axis passing through the wheel’s axle. Determine the car’s speed after it has traveled distance d starting from rest. The wheels roll without slipping. Neglect air resistance. Solve using conservation of energy.
Given:
W (^) c = 110 lb d =100 ft
W (^) w =5 lb θ =30 deg
r = 0.5 ft g 32.2 ft s^2
k =0.3 ft
Solution:
0 + (^) ( W (^) c + 4 W (^) w ) d sin( θ^ ) 0 1 2
W (^) c + 4 W (^) w g
W (^) w g
⎛⎜ k 2
v r
2 = +
v
2 ( W (^) c +^4 W^ w ) d^ sin(^ θ^ ) g
W (^) c + 4 W (^) w 4 W (^) w k
2
r^2
= v 55.2 ft s
Problem 18-
The assembly consists of two slender rods each of weight W (^) r and a disk of weight W (^) d. If the spring is unstretched when θ = θ 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant θ = 0. The disk rolls without slipping. Solve using the conservation of energy.
615
Problem 18-
The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when θ = θ 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod. Solve using the conservation of energy.
Given:
W =1500 lb θ 1 =60 deg
L =13 ft h =12 ft
a =7 ft
g 32.2 ft s^2
Solution:
0 W L
W
g
vA^2 W h^ − a 2
vA 2 g L 2 sin( θ (^) 1 ) h^ − a 2
= vA 14.2 ft s
*Problem 18-
The system consists of disk A of weight W (^) A , slender rod BC of weight W (^) BC , and smooth collar C of weight W (^) C. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e. θ = 0°.^ The system is released from rest when θ = θ 0. Solve using the conservation of energy.
Given:
W (^) A = 20 lb L =3 ft W (^) BC = 4 lb r =0.8 ft
617
W (^) C = 1 lb g 32. ft s^2
θ 0 =45 deg
Solution:
Guess vC 1 ft s
Given
0 W BC
L
- cos( θ (^) 0 )+ W (^) C L cos( θ 0 )
W C
g
vC^2
W BC
g
L^2
vC L
2 = + + 0
vC = Find (^) ( v (^) C ) vC 13. ft s
Problem 18-
The spool has mass mS and radius of gyration kO. If block A of mass mA is released from rest, determine the distance the block must fall in order for the spool to have angular velocity ω. Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord.
Given:
ms = 50 kg ri = 0.2 m g 9. m s^2
mA = 20 kg ro =0.3 m
ω 5 rad s = kO =0.280 m
Solution:
Guesses d = 1 m T =1 N Given
0 + 0
ms kO^2 ω^2
= + mA ( r (^) i ω)^2 − mA g d
0 + 0 − T d
= mA ( r (^) i ω)^2 − mA g d
d T
= Find ( d T , ) d = 0.301 m T =163 N
618
