Concept explainers
Gear A weighs 1 lb and has a radius of gyration of 1.3 in.; gear B weighs 6 lb and has a radius of gyration of 3 in.; gear C weighs 9 lb and has a radius of gyration of 4.3 in. Knowing a couple M of constant magnitude of 40 lb·in. is applied to gear A, determine (a) the angular acceleration of gear C, (b) the tangential force that gear B exerts on gear C.
Fig. P16.37
(a)
Find the angular acceleration of the gear C
Answer to Problem 16.37P
The angular acceleration of the gear C
Explanation of Solution
The weight of the gear A
The weight of the gear B
The weight of the gear C
The radius of gyration of the gear A
The radius of gyration of the gear B
The radius of gyration of the gear C
The couple of the constant magnitude applied to gear A (M) is
The radius of the gear A
The radius of the outer gear B
The radius of the inner gear B
The radius of the gear C
The angular acceleration of the gear A is
The angular acceleration of the gear B is
The angular acceleration of the gear C is
Calculation:
Consider the acceleration due to gravity (g) is
Convert the unit of the couple (M):
Convert the unit of the radius of the gear A
Convert the unit of the radius of the outer gear B
Convert the unit of the radius of the inner gear B
Convert the unit of the radius of the gear C
Calculate the mass of the gear A
Substitute
Calculate the mass of the gear B
Substitute
Calculate the mass of the gear C
Substitute
Calculate the mass moment of inertia of the gear A
Substitute
Calculate the mass moment of inertia of the gear B
Substitute
Calculate the mass moment of inertia of the gear C
Substitute
The point of contact between A and B:
Substitute
The point of contact between B and C:
Substitute
Therefore, the angular acceleration of the gear A is
Show the free body diagram of the gear A as in Figure 1.
Here,
Refer to Figure 1.
Calculate the moment about point A by applying the equation of equilibrium:
Substitute
Show the free body diagram of the gear B as in Figure 2.
Here,
Refer to Figure 2.
Calculate the moment about point B by applying the equation of equilibrium:
Substitute
Show the free body diagram of the gear C as in Figure 3.
Here,
Refer to Figure 3.
Calculate the moment about point C by applying the equation of equilibrium:
Calculate the angular acceleration of the gear C
Substitute
Hence, the angular acceleration of the gear C
(b)
Find the tangential force which gear B exerts on gear C.
Answer to Problem 16.37P
The tangential force which gear B exerts on gear C
Explanation of Solution
The weight of the gear A
The weight of the gear B
The weight of the gear C
The radius of gyration of the gear A
The radius of gyration of the gear B
The radius of gyration of the gear C
The couple of the constant magnitude applied to gear A (M) is
The radius of the gear A
The radius of the outer gear B
The radius of the inner gear B
The radius of the gear C
The angular acceleration of the gear A is
The angular acceleration of the gear B is
The angular acceleration of the gear C is
Calculation:
Refer the part (a).
Calculate the tangential force which gear B exerts on gear C
Substitute
Hence, the tangential force which gear B exerts on gear C
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