The center of gravity By Abdurasul Yarbekov

The center of gravity

By Abdurasul Yarbekov

The center of gravity

A body such as a brick or a uniform rod has its center of gravity at its middle or center. The center of gravity of something like an automobile, however, is not at its center or middle because its weight is not evenly distributed throughout. Its center of gravity is located nearer the heavy engine.

The center of gravity of any body is that point at which all of its weight can be considered to be concentrated.

The center of gravity

Each particle of the object has a certain weight, and therefore, exerts a torque about whatever point the object is suspended from. There is only a sin­gle point in an object about which all these torques cancel out no matter how it is oriented, this is its center of gravity.

For a non-uniform body or one with a nonsymmetrical shape, the sums become integrals

Example

A meter stick of negligible weight has a 10 N weight hung from each end. Where and with what force, should the meter stick be picked up such that it remains horizontal while it moves upward at a constant velocity? If weight w2 in the preceding discussion is changed to 20 N, where will the center of gravity of the system be located?

Example

Three masses are located in a rectangular coordinate system as follow, a 3 kg mass is at (4, - 2), a 2 kg mass at (- 3, 4), and a 4 kg mass is at (3, 3). Find the co­ordinates of the center of gravity, where all distances are in meters.

Example

In this example, we show how to find the location of a person’s center of gravity. Suppose your lab partner has a height L of 173 cm (5 ft, 8 in) and a weight w of 715 N (160 lb). You can determine the position of his center of gravity by having him stretch out on a uniform board supported at one end by a scale, as shown in Figure 4. If the board’s weight wb is 49 N and the scale reading F is 3.50 · 102 N, find the distance of your lab partner’s center of gravity from the left end of the board.

Example

A carpenter's square has the shape of L as shown in Figure, Find the center of gravity relative to O. (Hind: the weight, of each rectangular part is proportional to its area).

Example

You are asked to design the decorative mobile shown in Fig. The strings and rods have negligible weight, and the rods are to hang horizontally. (a) Draw a free-body diagram for each rod. (h) Find the weights of the balls A, B, and C. Find the tensions in the strings S1, S2 , and S3 . (c) What can you say about the horizontal location of the mobile's center of gravity? Explain(wA = 3 N, wB = 15 N, wC = 8 N; S1 = 32 N, S2 = 24 N, S3 = 9 N).

The center of mass

The center of mass of a body or system of bodies is defined as that point that moves in the same way that a single particle of the same mass would move when acted on by the same forces. Hence the point reacts as if all the mass of the body were concentrated at that point.

Example

Three masses, m1 = 20 g, m2 = 40 g, and m3 = 5 g are located on the x-axis at 10, 20 and 25 cm, respectively, as shown in figure. Find the center of mass of the system of three masses.

Example

A 59-kg woman and a 71-kg man sit on a seesaw, 3.5 m long. Where is their center of mass?

Example

Find the center of mass of the uniform sheet of plywood shown in Figure.

The Crane Boom

A large uniform boom is connected to the mast by a hinge pin at the point A in figure. A load wl is to be supported at the other end B. A cable is also tied to B and connected to the mast at C to give additional support to the boom. Let us now determine the forces V, H and T acting on the system when θ = 300, wB = 270 N, wl = 900 N and the length of the boom, l = 6 m.

Example

A 50.0-N (11-lb) weight is held in a person’s hand with the forearm horizontal, as in Figure. The biceps muscle is attached 0.030 0 m from the joint, and the weight is 0.350 m from the joint. Find the upward force F exerted by the biceps on the forearm (the ulna) and the downward force R exerted by the humerus on the forearm, acting at the joint. Neglect the weight of the forearm.

The ladder

A uniform ladder 10.0 m long and weighing 50.0 N rests against a smooth vertical wall as in Figure. If the ladder is just on the verge of slipping when it makes a 500 angle with the ground, find the coefficient of static friction between the ladder and ground.

Example

A ladder of length is placed against a wall, as shown in figure. A person, of weight wp, ascends the ladder until the person is located a distance d from the top of the ladder. Determine all the forces that are acting on the ladder.

Thank you for your kind attentions