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)
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)
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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