problems for practice

PROBLEMS FOR PRACTICE

1. The four masses m1, m2, m3 and m4 having their radii of rotation as 200 mm, 150 mm, 250 mm and 300 mm are 200 kg, 300 kg, 240 kg and 260 kg in magnitude respectively. The angles between the successive masses are 45°, 75° and 135° respectively. Find the position and magnitude of the balancing mass required, if the radius of rotation is 200 mm. 

[A.U., Nov/Dec 2012] 

[Ans. 116 kg; 201.48° from m1 anticlockwise]

[Hint: Refer Example 12.1]

2. Determine the mass and its position at 150 mm radius to balance the following coplanar force system: 6 kg at 100 mm radius; 12 kg at 75 mm radius 90° counter clockwise from the first radius; 15 kg at 100 mm radius 240° counter clockwise from the first radius. Find also the mass of each of the two balance masses which can be substituted for the single one already found if these are at 75 mm radius and positioned respectively at 30° and 150° counter clockwise from 6 kg mass. 

[Ans. 6.4 kg; 4,5 kg; 2,87 kg]

3. A rotor has the following properties:

If the shaft is balanced by two counter masses located at 100 mm radii and revolving in planes midway of planes 1 and 2, and midway of 3 and 4, determine the magnitudes of the masses and their respective angular positions. 

[Ans. 6.65 kg; 25o; 5.2 kg; 275°]

4. A shaft carries four masses A, B, C and D of magnitude 200 kg, 300 kg, 400 kg and 200 kg respectively and revolving at radii 80 mm, 70 mm, 60 mm and 80 mm in planes measured from A at 300 mm, 400 mm and 700 mm. The angles between the cranks measured anticlockwise are A to B 45°, B to C 70° and C to D 120°. The balancing masses are to be placed in planes X and Y. The distance between the planes A and X is 100 mm, between X and Y is 400 mm and between Y and D is 200 mm. If the balancing masses revolve at a radius of 100 mm, find their magnitudes and angular positions. 

[Hint: Refer Example 12.2] 

[A.U., Nov/Dec 2011] 

[Ans. my = 355 kg, 145° from mA clockwise; mY 182.5 kg, 12° from mA clockwise]

5. Four masses A, B, C and D are completely balanced, masses C and D makes angles of 90° and 195° respectively with B in the same sense. The rotating masses have the following properties:

Planes B and C are 250 mm apart. Determine: (1) the mass A and its angular position, and (ii) the positions of planes A and D.

 [Ans. (i) 17.37 kg, 294.6° measured CCW from OX; (ii) lA = 380 mm; lD = -310] 

6. A, B, C and D are four masses carried by a rotating shaft at radii 100, 125, 200 and 150 mm respectively. The planes in which the masses revolve are spaced 600 mm apart and the mass of B, C and D are 10 kg, 5 kg and 4 kg respectively. Find the required mass A and relative angular settings of the four masses so that the shaft be in complete balance. 

[Hint: Refer Example 12.6] 

[A.U., Apr/May 2011] 

7. A shaft with 3 m span between two bearings carries two masses of 10 kg and 20 kg acting at the extremities of the arms 0.45 m and 0.6 m long respectively. The planes in which these masses rotate are 1.2 m and 2.4 m respectively from the left end bearing supporting the shaft. The angle between the arms is 60°. The speed of rotation of the shaft is 200 rpm. If the masses are balanced by two countermasses rotating with the shaft at radii of 0.3 m and placed at 0.3 m from each bearing centres, estimate the magnitude of the two balance masses and their orientation with respect to x-axis, i.e., mass of 10 kg.

[Ans. 10 kg and 41 kg at 190° and 235° from x-axis in the anticlockwise direction]

8. A shaft is supported in bearings 1.8 m apart and projects 0.45 m beyond bearings at each end. The shaft carries three pulleys one at each end and one at the middle of its length. The mass of end pulleys is 48 kg and 20 kg and their centre of gravity are 15 mm and 12.5 mm respectively from the shaft axis. If the pulleys are arranged so as to give static balance, determine:

1. relative angular positions of the pulleys, and

2. dynamic forces produced on the bearings when the shaft rotates at 300 rpm.

[Ans. Angle between B and A = 161°; A and C = 76°; C and B = 123°; 2.533 N, 533 N]