
Concept explainers
In Prob. 15.199, determine (a) the common angular acceleration of gears A and B, (b) the acceleration of the tooth of gear A that is in contact with gear C at point 1.

(a)
The common angular acceleration of the gears
Answer to Problem 15.200P
The common angular acceleration of the gears
Explanation of Solution
Given Information:
The angular velocity of the gear
Draw the schematic diagram of the given system.
Figure-(1)
Write the expression for the velocity of point 1.
Here, the angular velocity of the gear
Write the expression for the velocity of point 1 when gear
Here, the angular velocity of the shaft unit carries gear
Write the expression for the velocity of point 2.
Here, the angular velocity of the gear
Write the expression for the velocity of point 2 when gear
Write the expression for the angular velocity of the inclined shaft unit which carries gear
Here, the angular velocity of the shaft in x- direction is
Draw the diagram to show the motion of the shaft is
Figure-(2)
Write the expression for the velocity of point
Here, the position vector at point
Write the expression for the velocity at point
Here, the angular velocity of shaft
Write the expression for the common angular acceleration of the gear
Calculation:
Consider the unit vector along
From Figure-(1) the coordinate of the point 1 at the intersection of the gears
From Figure-(1) the coordinate of the point 2 at the intersection of the gears
Since, the shaft
Substitute
Substitute
Substitute
Compare the terms along the x- direction in Equation (X).
Compare the terms along the z- direction in Equation (X).
Since, the shaft
Substitute
Substitute
Substitute
Compare the terms along the z- direction in Equation (X).
Substitute
Substitute
Substitute
From Figure-(2) the coordinate of the point
Substitute
Substitute
Substitute
Substitute
Conclusion:
The common angular acceleration of the gear

(b)
The acceleration of the tooth of the gear
Answer to Problem 15.200P
The acceleration of the tooth of the gear
Explanation of Solution
Write the expression for the acceleration of the gear
Calculation:
Substitute
Conclusion:
The acceleration of the tooth of the gear
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Chapter 15 Solutions
Vector Mechanics for Engineers: Dynamics
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