Question

Gear C has a mass of 5 kg and a centroidal radius of gyration of 75 mm. The uniform bar AB has a mass of 3 kg and gear D is stationary. If the system is released from rest in the position shown, determine (a) the angular acceleration of gear C, (b) the acceleration of point B.

Fig. P16.108

Answer 1

7949-16-108P AID: 1825 | 22/11/2013

Assume that the contact point between gear C and D is E.

Calculate the tangential acceleration of point E .

…… (1)

Here, tangential acceleration of point B is , and relative tangential acceleration between point E and B is .

Calculate the tangential acceleration of point B.

Here, radius of rotation of bar AB is , and angular acceleration of bar AB is .

Substitute 200 mm for .

Calculate the relative tangential acceleration between point E and B.

Here, radius of gear C is , and angular acceleration of gear C is .

Substitute 100 mm for .

Since the gear D is fixed, its tangential acceleration is zero.

Take clockwise acceleration as positive.

Substitute 0 for , for , and for in Equation (1).

Show the free body diagram of gear C as in Figure (1).

$Z:\01_Solutions_Authoring\Mechanical_Engineering\7949\04_Artwork\7949-16-108P_fig1.jpg$

Show the kinetic diagram of gear C as in Figure (2).

$Z:\01_Solutions_Authoring\Mechanical_Engineering\7949\04_Artwork\7949-16-108P_fig2.jpg$

Calculate the moment of inertia of gear C .

Here, mass of gear C is , and radius of gyration of gear C is .

Substitute 5 kg for , and 75 mm for .

Take moment about point B.

Since the system of external forces is equivalent to system of effective forces, express the system of external moments as:

…… (2)

Take counterclockwise moment as positive.

From Figure (1), calculate .

Here, force exerted on gear C is Q.

From Figure (2), calculate .

In Equation (2), substitute 0.1Q for , for , and for .

Show the free body diagram of gear C and bar AB as in Figure (3).

$Z:\01_Solutions_Authoring\Mechanical_Engineering\7949\04_Artwork\7949-16-108P_fig3.jpg$

Show the kinetic diagram of gear C and bar AB as in Figure (4).

$Z:\01_Solutions_Authoring\Mechanical_Engineering\7949\04_Artwork\7949-16-108P_fig4.jpg$

a)

Calculate the moment of inertia of bar AB .

Here, mass of bar AB is , and length of bar AB is .

Substitute 3 kg for , and 0.2 m for .

Calculate the acceleration of bar AB .

Substitute 0.1 m for .

Take moment about point A.

Since the system of external forces is equivalent to system of effective forces, express the system of external moment as:

…… (3)

Take clockwise moment as positive.

From Figure (3), calculate .

Here, weight of bar AB is , and weight of gear C is .

From Figure (4), calculate .

Here, acceleration of point B is .

In Equation (3), substitute for , and for .

Substitute for , for , 3 kg for , 5 kg for , for a, for , for , and for .

Substitute for Q, for , and 9.81 for g.

Hence, the angular acceleration of gear C is .

b)

Calculate the acceleration of point B.

Substitute for , and for .

Hence, the acceleration of point B is .