ENGR.MECH.:STAT.+DYNAMICS
ENGR.MECH.:STAT.+DYNAMICS
15th Edition
ISBN: 9780134780955
Author: HIBBELER
Publisher: RENT PEARS
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Chapter 18, Problem 1FP

The 80-kg wheel has a radius of gyration about its mass center O of ko = 400 mm. Determine its angular velocity after it has rotated 20 revolutions starting from rest.

Chapter 18, Problem 1FP, The 80-kg wheel has a radius of gyration about its mass center O of ko = 400 mm. Determine its

Expert Solution & Answer
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To determine

The angular velocity of the wheel after it has rotated 20 revolutions starting from rest.

Answer to Problem 1FP

The angular velocity of the wheel after it has rotated 20 revolutions starting from rest is 24.3rad/s .

Explanation of Solution

Given:

The mass of wheel is 80kg .

The radius of gyration of wheel about its mass center O is kO=400mm .

The radius of the wheel is 0.6m .

The force acting on the wheel is P=50N .

Write the formula for mass moment of inertia (IO) .

IO=mkO (I)

Here, m is the mass and kO is the radius of gyration about mass center O.

Write the formula for kinetic energy (T) (Rotation about fixed axis).

T=12IOω2 (II)

Here, IO is the mass moment of inertia and ω is the angular velocity.

Write the formula for work done by the couple moment. (M) .

UM=Prθ (III)

Here, P is the force or load, r is the radius and θ is the angular displacement in radians .

Write the formula for Principle of Work and Energy.

T1+U12=T2 (IV)

Here, T1 is initial kinetic energy, T2 is final kinetic energy and U12 is total work done by all the forces.

Conclusion:

Calculate the mass moment of inertia (IO) .

Substitute 80kg for m and 400mm for kO in Equation (I).

IO=80kg×(400mm×1m1000mm)2=12.8kgm2

Calculate the initial and final kinetic energy of the wheel.

At initial:

When the wheel starts from rest.

The initial kinetic energy becomes zero.

T1=0

At final:

Substitute 12.8kgm2 for IO in Equation (II).

T2=12(12.8kgm2)ω2=6.4ω2

Calculate the angular displacement (θ) .

θ=No.of revolutions×2π= 20×2π= 40π

Calculate the work done by the couple moment (M) .

Substitute 50N for P , 0.6m for r and 40π for θ in Equation (III).

UM=50N×0.6m×40π=3769.9111J

Here, the total work done is U12=UM .

U12=3769.9111J

Calculate the angular velocity of the wheel after it has rotated 20 revolutions starting from rest.

Substitute 0 for T1 , 3769.9111J for U12 and 6.4ω2 for T2 in Equation (IV).

0+3769.9111J=6.4ω2ω2=3769.91116.4ω=589.0486ω=24.2703rad/s

Thus, the angular velocity of the wheel after it has rotated 20 revolutions starting from rest is 24.3rad/s .

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Chapter 18 Solutions

ENGR.MECH.:STAT.+DYNAMICS

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