Concept explainers
Ends A and D of the two solid steel shafts AB and CD are fixed, while ends B and C are connected to gears as shown. Knowing that the allowable shearing stress is 50 MPa in each shaft, determine the largest torque T that can be applied to gear B.
Fig. P3.157
Find the largest torque (T) that can be applied to gear B.
Answer to Problem 157RP
The largest torque (T) that can be applied to gear B is
Explanation of Solution
Given information:
The allowable shearing stress
The diameter of the shaft AB
The diameter of the shaft CD
The radius of the gear B
The radius of the gear C
The length of the shaft AB
The length of the shaft CD
Assume that clockwise direction is negative and anticlockwise direction is positive.
Calculation:
Gear B and C:
Calculate the rotation in gear B
Substitute
Show the free body diagram of the gear C as in Figure 1.
Calculate the force in the shaft CD (F) using the formula:
Show the free body diagram of the gear B as in Figure 2.
Calculate the torque produced in the shaft AB
Substitute
Substitute
Shaft AB:
Calculate the radius of the shaft AB
Substitute
Calculate the polar moment of inertia in the shaft AB
Substitute
Calculate the radius of the shaft CD
Substitute
Calculate the polar moment of inertia in the shaft CD
Substitute
Calculate the rotation about gear B
Substitute
Calculate the rotation about gear C
Substitute
Calculate the torque in CD
Substitute
Calculate the total torque (T) using the formula:
Consider the equation (2),
Substitute
Substitute
Calculate the torque in the shaft AB
Substitute
Substitute
Calculate the torque in the shaft CD
Substitute
Calculate the total torque (T):
Substitute
From the above calculated torque in the shaft AB and CD, take the lesser value.
Thus, the largest torque (T) that can be applied to gear B is
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