VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
11th Edition
ISBN: 9781259633133
Author: BEER
Publisher: MCG
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Chapter 17.1, Problem 17.11P

(a)

To determine

The number of revolutions of gear C required for its angular velocity to increase from 100 to 450 rpm.

(a)

Expert Solution
Check Mark

Answer to Problem 17.11P

The number of revolutions of gear C required for its angular velocity to increase from 100 to 450 rpm is 6.35rev_.

Explanation of Solution

Given information:

The mass (mA) of gear A is 2.4 kg.

The mass (mB) of gear B is 2.4 kg.

The mass (mC) of gear C is 12 kg.

The radius of gyration (kA) of gear A is 60 mm.

The radius of gyration (kB) of gear B is 60 mm.

The radius of gyration (kC) of gear C is 150 mm.

The radius (rA) of gear A is 80 mm.

The radius (rB) of gear B is 80 mm.

The radius (rC) of gear C is 200 mm.

The initial angular velocity (ωC)1 of gear C is 100 rpm.

The final angular velocity (ωC)2 of gear C is 450 rpm.

The couple acting on gear C is 10Nm.

Calculation:

Find the mass moment of inertia (IA) of gear A using the equation:

IA=mAkA2

Substitute 2.4 kg for mA and 60 mm for kA.

IA=(2.4)(60mm×1m1,000mm)2=8.64×103kgm2

Find the mass moment of inertia (IB) of gear B using the equation:

IB=mBkB2

Substitute 2.4 kg for mB and 60 mm for kB.

IB=(2.4)(60mm×1m1,000mm)2=8.64×103kgm2

Find the mass moment of inertia (IC) of gear C using the equation:

IC=mCkC2

Substitute 12 kg for mC and 150 mm for kA.

IC=(12)(150mm×1m1,000mm)2=270×103kgm2

Find the initial angular velocity (ωA)1 of gear A using the kinematics:

rA(ωA)1=rC(ωC)1(ωA)1=rCrA(ωC)1

Substitute 200 mm for rC, 80 mm for rA, and 100 rpm for (ωC)1.

(ωA)1=20080(100revmin×2πrad1rev×1min60s)=26.18rad/s

Find the initial angular velocity (ωB)1 of gear B using the kinematics:

rB(ωB)1=rC(ωC)1(ωB)1=rCrB(ωC)1

Substitute 200 mm for rC, 80 mm for rB, and 100 rpm for (ωC)1.

(ωB)1=20080(100revmin×2πrad1rev×1min60s)=26.18rad/s

Find the initial kinetic energy (T1) of the system using the equation:

T1=(T1)A+(T1)B+(T1)C=12IA(ωA)12+12IB(ωB)12+12IC(ωC)12

Here, (T1)A is the initial angular velocity of gear A, (T1)B is the initial angular velocity of gear B, and (T1)C is the initial angular velocity of gear C.

Substitute 8.64×103kgm2 for IA, 26.18rad/s for (ωA)1, 8.64×103kgm2 for IB, 26.18rad/s for (ωB)1, 270×103kgm2 for IC, and 100 rpm for ωC.

T1={12[(8.64×103)(26.18)2+12(8.64×103)(26.18)2+12(270×103)](100revmin×2πrad1rev×1min60s)2}=2.9609+2.9609+14.8044=20.726J

Find the final angular velocity (ωA)2 of gear A using the kinematics:

rA(ωA)2=rC(ωC)2(ωA)2=rCrA(ωC)2

Substitute 200 mm for rC, 80 mm for rA, and 450 rpm for (ωC)2.

(ωA)2=20080(450revmin×2πrad1rev×1min60s)=117.81rad/s

Find the final angular velocity (ωB)2 of gear B using the kinematics:

rB(ωB)2=rC(ωC)2(ωB)2=rCrB(ωC)2

Substitute 200 mm for rC, 80 mm for rB, and 450 rpm for (ωC)2.

(ωB)2=20080(450revmin×2πrad1rev×1min60s)=117.81rad/s

Find the initial kinetic energy (T2) of the system using the equation:

T2=(T2)A+(T2)B+(T2)C=12IA(ωA)22+12IB(ωB)22+12IC(ωC)22

Here, (T2)A is the final angular velocity of gear A, (T2)B is the final angular velocity of gear B, and (T2)C is the final angular velocity of gear C.

Substitute 8.64×103kgm2 for IA, 117.81rad/s for (ωA)1, 8.64×103kgm2 for IB, 117.81rad/s for (ωB)1, 270×103kgm2 for IC, and 450 rpm for ωC.

T2={[12(8.64×103)(117.81)2+12(8.64×103)(117.81)2+12(270×103)](450revmin×2πrad1rev×1min60s)2}=59.958+59.958+299.8=419.7J

Find the work done (U12) due to couple acting on gear C using the equation:

U12=MCθC

Substitute 10Nm for MC.

U12=10θC

Find the number of revolutions of gear C (θC) required for its angular velocity to increase from 100 to 450 rpm.

Apply Principle of work and energy for system.

T1+U12=T2

Substitute 20.726J for T1, 419.7 J for T2, and 10θC for U12.

20.726+10θC=419.7θC=39.897rad×1rev2πradθC=6.35rev

Thus, the number of revolutions of gear C required for its angular velocity to increase from 100 to 450 rpm is 6.35rev_.

(b)

To determine

The tangential force acting on gear A.

(b)

Expert Solution
Check Mark

Answer to Problem 17.11P

The tangential force acting on gear A is 7.14N_.

Explanation of Solution

Calculation:

Find the rotation of gear A using the equation:

rAθA=rCθCθA=rCrAθC

Substitute 200 mm for rC, 80mm for rA, and 39.897rad for θC.

θA=20080(39.897)=99.743rad

Find the tangential force (Ft) acting on gear A.

Apply Principle of work and energy for gear A.

(T1)A+MAθA=(T2)A12IA(ωA)12+MAθA=12IA(ωA)22

Substitute 8.64×103kgm2 for IA, 26.18rad/s for (ωA)1, 99.743 rad/s for θA, and 117.81rad/s for (ωA)2.

12(8.64×103)(26.18)2+MA(99.743)=12(8.64×103)(117.81)22.9609+99.743MA=59.958MA=0.57144Nm

Find the tangential force (Ft) acting on gear A using the equation:

Ft=MArA

Substitute 0.57144Nm for MA and 80 mm for rA.

Ft=0.57144(80mm×1m1,000mm)=7.14N

Thus, the tangential force acting on gear A is 7.14N_.

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Chapter 17 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 17.1 - PROBLEM 17.6 The flywheel of a punching machine...Ch. 17.1 - Prob. 17.7PCh. 17.1 - Prob. 17.8PCh. 17.1 - The 10-in.-radius brake drum is attached to a...Ch. 17.1 - Prob. 17.10PCh. 17.1 - Prob. 17.11PCh. 17.1 - Prob. 17.12PCh. 17.1 - Prob. 17.13PCh. 17.1 - The double pulley shown has a mass of 15 kg and a...Ch. 17.1 - Gear A has a mass of 1 kg and a radius of gyration...Ch. 17.1 - Prob. 17.16PCh. 17.1 - Prob. 17.17PCh. 17.1 - A slender 9-lb rod can rotate in a vertical plane...Ch. 17.1 - An adapted golf device attaches to a wheelchair to...Ch. 17.1 - Prob. 17.20PCh. 17.1 - A collar with a mass of 1 kg is rigidly attached...Ch. 17.1 - A collar with a mass of 1 kg is rigidly attached...Ch. 17.1 - Two identical slender rods AB and BC are welded...Ch. 17.1 - Prob. 17.24PCh. 17.1 - Prob. 17.25PCh. 17.1 - Prob. 17.26PCh. 17.1 - Greek engineers had the unenviable task of moving...Ch. 17.1 - A small sphere of mass m and radius r is released...Ch. 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...Ch. 17.1 - Prob. 17.33PCh. 17.1 - A bar of mass m = 5 kg is held as shown between...Ch. 17.1 - The 1.5-kg uniform slender bar AB is connected to...Ch. 17.1 - The motion of the uniform rod AB is guided by...Ch. 17.1 - Prob. 17.37PCh. 17.1 - Prob. 17.38PCh. 17.1 - The ends of a 9-lb rod AB are constrained to move...Ch. 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...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 - 17.45 The uniform rods AB and BC weigh 2.4 kg and...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 - Prob. 17.51PCh. 17.2 - The 350-kg flywheel of a small hoisting engine has...Ch. 17.2 - Prob. 17.2IMDCh. 17.2 - Prob. 17.3IMDCh. 17.2 - Prob. 17.52PCh. 17.2 - Prob. 17.53PCh. 17.2 - Prob. 17.54PCh. 17.2 - A uniform 144-lb cube is attached to a uniform...Ch. 17.2 - Prob. 17.56PCh. 17.2 - Prob. 17.57PCh. 17.2 - Prob. 17.58PCh. 17.2 - Prob. 17.59PCh. 17.2 - Each of the double pulleys shown has a centroidal...Ch. 17.2 - Each of the gears A and B has a mass of 675 g and...Ch. 17.2 - Prob. 17.62PCh. 17.2 - Prob. 17.63PCh. 17.2 - Prob. 17.64PCh. 17.2 - Prob. 17.65PCh. 17.2 - Show that, when a rigid body rotates about a fixed...Ch. 17.2 - Prob. 17.68PCh. 17.2 - A flywheel is rigidly attached to a 1.5-in.-radius...Ch. 17.2 - A wheel of radius r and centroidal radius of...Ch. 17.2 - Prob. 17.71PCh. 17.2 - 17.72 and 17.73 A 9-in.·radius cylinder of weight...Ch. 17.2 - 17.72 and 17.73 A 9-in.·radius cylinder of weight...Ch. 17.2 - Two uniform cylinders, each of mass m = 6 kg and...Ch. 17.2 - Prob. 17.75PCh. 17.2 - Prob. 17.76PCh. 17.2 - A sphere of radius r and mass m is projected along...Ch. 17.2 - A bowler projects an 8.5-in.-diameter ball...Ch. 17.2 - Prob. 17.79PCh. 17.2 - A satellite has a total weight (on Earth) of 250...Ch. 17.2 - Two 10-lb disks and a small motor are mounted on a...Ch. 17.2 - Prob. 17.82PCh. 17.2 - Prob. 17.83PCh. 17.2 - Prob. 17.84PCh. 17.2 - Prob. 17.85PCh. 17.2 - Prob. 17.86PCh. 17.2 - Prob. 17.87PCh. 17.2 - Prob. 17.88PCh. 17.2 - A 1.8-kg collar A and a 0.7-kg collar B can slide...Ch. 17.2 - Prob. 17.90PCh. 17.2 - A small 4-lb collar C can slide freely on a thin...Ch. 17.2 - Rod AB has a weight of 6 lb and is attached to a...Ch. 17.2 - Prob. 17.93PCh. 17.2 - Prob. 17.94PCh. 17.2 - The 6-lb steel cylinder A of radius r and the...Ch. 17.3 - A uniform slender rod AB of mass m is at rest on a...Ch. 17.3 - Prob. 17.5IMDCh. 17.3 - Prob. 17.6IMDCh. 17.3 - At what height h above its center G should a...Ch. 17.3 - A bullet weighing 0.08 lb is fired with a...Ch. 17.3 - In Prob. 17.97, determine (a) the required...Ch. 17.3 - A 16-lb wooden panel is suspended from a pin...Ch. 17.3 - Prob. 17.100PCh. 17.3 - A 45-g bullet is fired with a velocity of 400 m/s...Ch. 17.3 - A 45-g bullet is fired with a velocity of 400 m/s...Ch. 17.3 - Prob. 17.103PCh. 17.3 - Prob. 17.104PCh. 17.3 - Prob. 17.105PCh. 17.3 - A uniform slender rod AB is at rest on a...Ch. 17.3 - A bullet of mass m is fired with a horizontal...Ch. 17.3 - Determine the height h at which the bullet of...Ch. 17.3 - A uniform slender bar of length L = 200 mm and...Ch. 17.3 - A uniform slender rod of length L is dropped onto...Ch. 17.3 - A uniform slender rod AB has a mass m, a length L,...Ch. 17.3 - 17.113 The slender rod AB of length L = 1 m forms...Ch. 17.3 - The trapeze/lanyard air drop (t/LAD) launch is a...Ch. 17.3 - The uniform rectangular block shown is moving...Ch. 17.3 - The 40-kg gymnast drops from her maximum height of...Ch. 17.3 - Prob. 17.117PCh. 17.3 - Prob. 17.118PCh. 17.3 - A 1-oz bullet is fired with a horizontal velocity...Ch. 17.3 - For the beam of Prob. 17.119, determine the...Ch. 17.3 - Prob. 17.121PCh. 17.3 - Prob. 17.122PCh. 17.3 - A slender rod AB is released from rest in the...Ch. 17.3 - Prob. 17.124PCh. 17.3 - Block A has a mass m and is attached to a cord...Ch. 17.3 - Prob. 17.126PCh. 17.3 - 17.127 and 17.128Member ABC has a mass of 2.4 kg...Ch. 17.3 - 17.127 and 17.128Member ABC has a mass of 2.4 kg...Ch. 17.3 - Prob. 17.129PCh. 17.3 - Prob. 17.130PCh. 17.3 - A small rubber ball of radius r is thrown against...Ch. 17.3 - Sphere A of mass m and radius r rolls without...Ch. 17.3 - In a game of pool, ball A is rolling without...Ch. 17 - A uniform disk, initially at rest and of constant...Ch. 17 - The 8-in.-radius brake drum is attached to a...Ch. 17 - A uniform slender rod is placed at corner B and is...Ch. 17 - The motion of the slender 250-mm rod AB is guided...Ch. 17 - Prob. 17.141RPCh. 17 - Disks A and B are made of the same material, are...Ch. 17 - Disks A and B are made of the same material, are...
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