Vector Mechanics For Engineers
Vector Mechanics For Engineers
12th Edition
ISBN: 9781259977237
Author: BEER
Publisher: MCG
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Chapter 17.2, Problem 17.91P
To determine

(a)

Calculate the angular velocity of the ring when collar passes through 900 ? Collar is free to slide on a ring and ring is attached to a vertical shaft. Which rotates in a fixed bearing. Initially, collar is at top of the ring θ=0 at 35 rad/sec of angular velocity. Refer figure.

Vector Mechanics For Engineers, Chapter 17.2, Problem 17.91P , additional homework tip  1

Expert Solution
Check Mark

Answer to Problem 17.91P

The angular velocity of the ring when coller is at 900 from the top is 15 rad/sec.

Explanation of Solution

Given:

The weight of collar = ωc=4lb=1.814kg

Mass of ring = ωr=6lb=2.721kg

The radius of the ring,

Rr=10in=0.254mθ=0vc=0

The angular velocity (ωc) of the collar = 35 rad/sec.

Concept used:

Conservation of angular momentum.

Conservation of energy.

Calculation:

Vector Mechanics For Engineers, Chapter 17.2, Problem 17.91P , additional homework tip  2

Position (1): θ=0vc=0

Position (2): θ=900(vc)4=vy=Rω2

By conservation of angular momentum,

IRω1=IRω2+mcvyRmRR2ω1=(mR+2mc)R2ω2ω2=mRω1(mR+2mc)

Potential energy at position 1 and position 2

v1=mcgRv2=0

The kinetic energy of position 1 and position 2

T1=IRω122=12mRR2ω12T2=IRω122+12mc(vx2+vY2)=mRR2ω224+mcR2ω222+mcvy22

Applying conservation of energy principle,

T1+v1=T2+v2mRR2ω124+mcgR=mRR2ω224+mcR2ω222+mcvy22

But,

Mass of collar, mc=wcg=0.1849kg

Mass of ring, mR=wRg=0.277kg

ω2=mRω1(mR+2mc)=0.277×35(0.277+2×0.1849)=14.98ω2=15rad/sec

Conclusion:

Using conservation of angular momentum we are able to find, the angular velocity of the ring for θ=900 is 15 a rad/sec.

To determine

(b)

What will be the velocity of collar with respect to ring? Collar is free to slide on a ring and ring is attached to a vertical shaft. Which rotates in a fixed bearing. Initially, collar is at top of the ring θ=0 at 35 rad/sec of angular velocity. Refer figure.

Expert Solution
Check Mark

Answer to Problem 17.91P

The corresponding velocity of collar relative to ring is 6.242 m/s.

Explanation of Solution

Given:

The weight of color = ωc=4lb=1.814kg

Mass of ring = ωr=6lb=2.721kg

The radius of the ring,

Rr=10in=0.254mθ=0vc=0

The angular velocity (ωc) of the collar = 35 rad/sec.

Concept used:

Conservation of angular momentum.

Conservation of energy.

Calculation:

Potential energy at position 1 and position 2

v1=mcgRv2=0

The kinetic energy of position 1 and position 2

T1=IRω122=12mRR2ω12T2=IRω122+12mc(vx2+vY2)=mRR2ω224+mcR2ω222+mcvy22

Applying conservation of energy principle,

T1+v1=T2+v2mRR2ω124+mcgR=mRR2ω224+mcR2ω222+mcvy22

mRR2ω124+mcgR=(mR4+mc2)R2ω22+mcvy225.472+0.4607=2.347+0.092vy2vy=6.242m/s.

Conclusion:

Using conservation of energy principle we are able to find the velocity of collar relative to ring at θ=900 is 6.242m/s

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Chapter 17 Solutions

Vector Mechanics For Engineers

Ch. 17.1 - Prob. 17.6PCh. 17.1 - Prob. 17.7PCh. 17.1 - Prob. 17.8PCh. 17.1 - Prob. 17.9PCh. 17.1 - Prob. 17.10PCh. 17.1 - Each of the gears A and B has a mass of 10 kg and...Ch. 17.1 - Solve Prob. 17.11, assuming that the 6 N m couple...Ch. 17.1 - The gear train shown consists of four gears of the...Ch. 17.1 - Prob. 17.14PCh. 17.1 - Prob. 17.15PCh. 17.1 - Prob. 17.16PCh. 17.1 - The 15-kg rear hatch of a vehicle opens as shown...Ch. 17.1 - A slender 9-lb rod can rotate in a vertical plane...Ch. 17.1 - Prob. 17.19PCh. 17.1 - Prob. 17.20PCh. 17.1 - A collar with a mass of 1 kg is rigidly attached...Ch. 17.1 - Prob. 17.22PCh. 17.1 - Prob. 17.23PCh. 17.1 - The 30-kg turbine disk has a centroidal radius of...Ch. 17.1 - A 100-kg solid cylindrical disk, 800 mm in...Ch. 17.1 - Prob. 17.26PCh. 17.1 - Prob. 17.27PCh. 17.1 - Prob. 17.28PCh. 17.1 - Prob. 17.29PCh. 17.1 - A half-cylinder with mass m and radius r is...Ch. 17.1 - Prob. 17.31PCh. 17.1 - Two uniform cylinders, each of weight W=14 lb and...Ch. 17.1 - Two uniform cylinders, each of weight W=14 lb and...Ch. 17.1 - A bar of mass m=5 kg is held as shown between four...Ch. 17.1 - The 1.5-kg uniform slender bar AB is connected to...Ch. 17.1 - Prob. 17.36PCh. 17.1 - A 5-m-long ladder has a mass of 15 kg and is...Ch. 17.1 - Prob. 17.38PCh. 17.1 - Prob. 17.39PCh. 17.1 - The mechanism shown is one of two identical...Ch. 17.1 - The mechanism shown is one of two identical...Ch. 17.1 - Each of the two rods shown is of length L=1 m and...Ch. 17.1 - The 4-kg rod AB is attached to a collar of...Ch. 17.1 - If in Prob. 17.43 the angular velocity of the...Ch. 17.1 - The uniform rods AB and BC are of mass 3 kg and 8...Ch. 17.1 - The uniform rods AB and BC weigh 2.4 kg and 4 kg,...Ch. 17.1 - The 80-mm-radius gear shown has a mass of 5 kg and...Ch. 17.1 - Prob. 17.48PCh. 17.1 - Three shafts and four gears are used to form a...Ch. 17.1 - Prob. 17.50PCh. 17.1 - The drive belt on a vintage sander transmits 12 hp...Ch. 17.2 - Slender bar A is rigidly connected to a massless...Ch. 17.2 - A 1-m-long uniform slender bar AB has an angular...Ch. 17.2 - The 350-kg flywheel of a small hoisting engine has...Ch. 17.2 - A sphere of radius r and mass m is placed on a...Ch. 17.2 - Prob. 17.F3PCh. 17.2 - Prob. 17.52PCh. 17.2 - Prob. 17.53PCh. 17.2 - Prob. 17.54PCh. 17.2 - Prob. 17.55PCh. 17.2 - Prob. 17.56PCh. 17.2 - A disk of constant thickness, initially at rest,...Ch. 17.2 - Prob. 17.58PCh. 17.2 - A cylinder of radius r and weight W with an...Ch. 17.2 - Each of the double pulleys shown has a centroidal...Ch. 17.2 - Prob. 17.61PCh. 17.2 - Prob. 17.62PCh. 17.2 - Prob. 17.63PCh. 17.2 - A tape moves over the two drums shown. 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