Center of gravity – problems and solutions

1. Determine the coordinate of the center of gravity of the object as shown in the figure below.

Solution :

Divide the object into three parts.

Area of part 1 (A_{1}) = (2)(6) = 12 cm^{2}

The center point lies on the x axis (x_{1}) = 1/2 (2) = 1 cm

The center point lies on the y axis (y_{1}) = 1/2 (6) = 3 cm

Area of part 2 (A_{2}) = (4)(2) = 8 cm^{2}

The center point lies on the x axis (x_{2}) = 2 + (1/2)(4) = 2 + 2 = 4 cm

The center point lies on the y axis (y_{2}) = 2 + (1/2)(2) = 2 + 1 = 3 cm

Area of part 3 (A_{3}) = (2)(6) = 12

The center point lies on the x axis (x_{3}) = 2 + 4 + (1/2)(2) = 2 + 4 + 1 = 7 cm

The center point lies on the y axis (y_{3}) = 1/2 (6) = 3 cm

Coordinate of the center of gravity at x axis :

Coordinate of the center of gravity at y axis :

Coordinate of the center of gravity of the object is at x axis and y axis (x , y) = (4, 3)

2. Determine the coordinate of the center of gravity of object , about the x axis.

Solutions

Divide the object into three parts, A, B, C, and D.

Calculate the area of each part :

A_{A} = ½ (base)(height) = ½ (1.5)(3) = (1.5)(1.5) = 2.25

A_{B} = (length)(width) = (4.5-1.5)(1) = (3)(1) = 3

A_{C} = ½ (base)(height) = ½ (6-4.5)(3) = (1.5)(1.5) = 2.25

A_{D }= ½ (base)(height) = ½ (4.5-1.5)(6-3) = ½ (3)(3) = (1.5)(3) = 4.5

y_{A} = 1/3 (3) = 1

y_{B} = 1/2 (1) = 0,5

y_{C} = 1/3 (3) = 1

y_{D }= 3 + (1/3)(6-3) = 3 + (1/3)(3) = 3 + 1 = 4

Coordinate of the center of gravity at y axis :

Coordinate of the center of gravity about the x axis is 2 cm.

3. Determine coordinate of the center of gravity of the object, as shown in figure.

Solution :

Divide the object into four parts, A, B, C, and D.

Calculate the area of each part.

A_{A} = (length)(width) = (4)(3) = 12

A_{B} = ½ (base)(height) = ½ (6-4)(3) = ½ (2)(3) = (1)(3) = 3

A_{C} = ½ (base)(height) = ½ (8-6)(3) = ½ (2)(3) = (1)(3) = 3

A_{D }= (length)(width) = (8)(6-3) = (8)(3) = 24

y_{A} = 1/2 (3) = 1.5

y_{B} = 3 – (1/3)(3) = 3 – 1 = 2

y_{C} = 3 – (1/3)(3) = 3 – 1 = 2

y_{D }= 3 + (1/2)(6-3) = 3 + (1/2)(3) = 3 + 1.5 = 4.5

x_{A} = 1/2 (4) = 2

x_{B} = 4 + (1/2)(6-4) = 4 + (1/2)(2) = 4 + 1 = 5

x_{C} = 6 + (1/2)(8-6) = 6 + (1/2)(2) = 6 + 1 = 7

x_{D} = 1/2 (8) = 4

Coordinate of the center of gravity at the x axis :

Coordinate of the center of gravity at y axis :

**What is the center of gravity (COG) of an object?****Answer:**The center of gravity is the point in an object where the weight of the object can be considered to act. It’s the average location of the weight of an object.

**How does the COG differ from the center of mass?****Answer:**While both the COG and the center of mass refer to the average location of weight and mass respectively, they can differ if the gravitational field acting on the object isn’t uniform. However, in most practical situations on Earth, they are effectively the same.

**Why is understanding the COG important in vehicle design?****Answer:**The location of the COG affects the stability of a vehicle. A lower COG can make a vehicle less likely to tip over and generally more stable, especially during turns or sudden maneuvers.

**How does distributing weight differently in a vehicle or structure affect its center of gravity?****Answer:**Adding weight to the top of a structure or vehicle will raise its COG, making it potentially less stable. Conversely, adding weight to the bottom or base can lower the COG, enhancing stability.

**If an object is freely suspended from a point, where will its COG lie?****Answer:**The COG of the suspended object will lie directly beneath the point of suspension, aligning itself vertically with the suspension point due to the force of gravity.

**Can the center of gravity of an object be outside the object’s material boundaries?****Answer:**Yes, the COG can be located outside the actual material of the object. A common example is a ring or a hollow cylinder; its COG is at its center, an area with no material.

**Why do tightrope walkers often use long poles?****Answer:**The long pole extends the effective width of their base and lowers their COG. Both of these effects enhance their stability and balance while walking on the tightrope.

**How does the shape and distribution of mass in an object influence its COG?****Answer:**The COG will always trend toward areas of the object with greater mass. For instance, in an object where more mass is concentrated at one end, the COG will be closer to that end.

**How can you experimentally determine the COG of a flat, irregularly shaped object, like a cardboard cutout?****Answer:**One method is to suspend the object by a point and draw a vertical line downwards from the point of suspension. Then, suspend the object from another point and draw another vertical line. The point where these lines intersect is the COG.

**In human biomechanics, why is the COG significant?**

**Answer:**Understanding the COG in humans is crucial for activities like walking, running, and jumping. It affects our balance, stability, and movement mechanics. The human COG is typically located around the region of the navel, but this can change with body posture and movement.