Concept explainers
Solve Prob. 17.11, assuming that the 6 N·m couple is applied to gear B.
17.11 Each of the gears A and B has a mass of 10 kg and a radius of gyration of 190 mm, while gear C has a mass of 2.5 kg and a radius of gyration of 80 mm. If a couple M of constant magnitude 6 N·m is applied to gear C, determine (a) the number of revolutions of gear C required for its angular velocity to increase from 450 rpm to 1800 rpm, (b) the corresponding tangential force acting on gear A.
Fig. P17.11
(a)
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Find the number of revolutions required for gear C.
Answer to Problem 17.12P
The number of revolutions required for the gear C for the work to be done is
Explanation of Solution
Given information:
The mass of the gear A is
The mass of the gear B is
The radius of gyration of the gear A is
The radius of gyration of the gear B is
The mass of the gear C is
The radius of gyration of the gear C is
The radius of the gear A is
The radius of the gear B is
The radius of the gear C is
The magnitude of the couple moment applied at point B is
Calculation:
Find the mass moment of inertia of gear A
Substitute 10 kg for
Find the mass moment of inertia of gear B
Substitute 10 kg for
Find the mass moment of inertia of gear C
Substitute 2.5 kg for
The gears A and C are in contact.
Use the kinematics concept;
Substitute 250 mm for
The gears B and C are in contact.
Use the kinematics Equation:
Substitute 250 mm for
Find the total kinetic energy
Substitute
When the angular velocity is at 450 rpm:
Substitute 450 rpm for
When the angular velocity is at 1800 rpm:
Substitute 1800 rpm for
Find the work done
Here, the number of revolution at gear B is
Substitute
Write the equation of work and energy for the system using the equation.
Substitute 146.03 J for
Find the number of revolution at gear C
Substitute 58.1 rev for
Therefore, the number of revolutions required for the gear C for the work to be done is
(b)
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Find the tangential force acting on gear A.
Answer to Problem 17.12P
The tangential force acting on gear A is
Explanation of Solution
Given information:
The mass of the gear A is
The mass of the gear B is
The radius of gyration of the gear A is
The radius of gyration of the gear B is
The mass of the gear C is
The radius of gyration of the gear C is
The radius of the gear A is
The radius of the gear B is
The radius of the gear C is
The magnitude of the couple moment is
Calculation:
Refer to part (a) calculation;
The number of revolution in the gear A is equal to the number of revolution in the gear B.
Substitute 365.08 rad for
Find the total kinetic energy
When the angular velocity is at 450 rpm;
Substitute
When the angular velocity is at 1800 rpm;
Substitute
Find the work done
Here, the magnitude of couple moment at gear A is
Substitute 365.08 rad for
Write the equation of work and energy for the system using the equation.
Substitute 64.133 J for
Find the tangential force
Substitute
Therefore, the tangential force acting on gear A is
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Chapter 17 Solutions
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