# 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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#### Solution Summary

This posting contains the solution to the given problems.

Sample questions are demonstrated.

1. During the later portion of the swing phase of a walking stride the knee is extended from 35 degrees (initial) to 10 degrees (final) over a time period of 0.1 seconds. What was the angular velocity?

2. If an object achieves an angular acceleration of 12 rad/s2 from a moment (torque) of 300 Nm, what is the object's moment of inertia?

3. What would be the angular speed of a diver who completed 5 ½ revolutions (i.e., angular distance) in 2s? Convert this speed to degrees/s, and rad/s.

4. Number of degrees and revolutions in 1.4 radians.

5. Number of radians in 80 degrees.

6. Determine the linear distance (along the arc) that an object 1.3m from the axis of rotation spans in 56 degrees.

7. Calculate the moment of inertia of an object consisting of a 30kg mass, located 0.76m from an axis.

8. The moment of inertia about the center of mass of an 60 kg object is 1.4 kgm^2. If the moment of inertia about the object's proximal end is 40 kgm^2, what is the distance between the proximal end and the center of mass?

9. A gymnast takes off with an angular momentum of 60 kgm^2/s. What will be her angular velocity during a somersault, if her moment of inertia in the tucked position is 5 kgm^2. Be careful with the units.

10. If a person pushed on a door with a force of 500 N and a moment arm of 0.75 meter, what would be the moment (torque)?

11. A force is applied to a lever 15cm from the axis of rotation. What happens to the moment (torque) created if the same force is moved closer to the axis and applied a distance of say 5 cm from the axis?

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