Questions on Center of mass, Torque and Levers

Please refer to the attachment for the mentioned figures.

1. Three masses, 2.0 kg, 3.0 kg, and 6.0 kg, are located at points (3.0, 0), (6.0, 0), and (-4.0, 0), respectively, in meters from the origin. Where is the center of mass of this system?

2. A dumbbell, Figure below, has a connecting bar of negligible mass. Find the location of the center of mass
a. if m1 and m2 are each 5.0 kg, and
b. if m1 is 5.0 kg and m2 is 10.0 kg.

3. A water molecule consists of an oxygen atom and two hydrogen atoms. Oxygen has a mass of 16 atomic units (u) and each hydrogen has a mass of 1 u. The hydrogen atoms are each at an average distance of 96 pm ( 96 x 10^-12 m) from the oxygen atom, and are separated from one another by an angle of 104.5 degree. Find the center of mass of a water molecule.

4. Two children are balanced on a see-saw of negligible mass. (This assumption is made to keep the example simple.) The first child, sitting on left side of the pivot, has a mass of 26.0 kg and sits 1.60 m from the pivot. The second child, sitting on right side of pivot, has a mass of 32.0 kg and 1.3 m from the pivot. Calculate the position of center of mass of the two children:
a. from the first child
b. from the pivot point.

5. Three balls A, B, and C, with masses of 3 kg, 1 kg, and 1 kg, respectively, are connected by massless rods. The balls are located as in Figure below. What are the coordinates of the center of mass?

6. Alley club-ax consists of a symmetrical 8 kg stone attached to the end of a uniform 2.5 kg stick that is 98 cm long. The dimensions of the club-ax are shown in Figure blow. How far is the center of mass from the handle end of the club-ax?

7. A 46.4 N force is applied to the outer edge of a door of width 1.26 m in such a way that it acts
a. Perpendicular to the door.
b. At the angle of 43.0 degree with respect to the door surface.
c. So that the line of action of the force passes through the axis of the door hinges.
Find the torque for these three cases.

8. When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.85 m from the hinges. What torque are you exerting relative to the hinge?

9. If the load placed in the wheel barrow weighs 20 N and is 40 cm from the wheel, what force must be applied to the handles at a distance of 120 cm in order to balance the load?

10. A child of mass 40.0 kg is sitting on a horizontal see-saw at a distance of 2.0 m from the supporting axis. What is the magnitude of the torque about the axis due to the weight of the child?

11. In figure below, a force with a magnitude of F = 55 N is applied to a door; however the lever arms are different in the three cases a, b, and c. Take the distance between the hinge (black dot) and the door knob as 0.8 m, and in case b take lever arm as 0.6 m. Find the torque in each case.

12. Figure below shows the ankle joint and the Achilles tendon attached to the heel at point P. The tendon exerts a force F with a magnitude of F = 720 N . Determine the torque (magnitude and direction) of this force about the joint, which is located 3.6 x 10^−2 m away from point P.

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