Center of Gravity and Centroid

Under what condition does the center of gravity of a body coincide with its centroid?

The coordinates of the center of gravity, G, are determined by equating the moment of the total weight W about an axis to what?

If a body is made from a homogeneous material with constant density rho, how is the centroid of its volume determined?

For a triangular area with base 'b' and height 'h', where is the centroid located with respect to the base?

A rod is bent into the shape of a parabolic arc defined by x = y^2 from y=0 to y=1 m. What is the x-coordinate of its centroid?

What are the coordinates (x_bar, y_bar) of the centroid of a quarter-circular wire segment of radius R, located in the first quadrant with its center at the origin?

If a volume possesses two planes of symmetry, where must its centroid lie?

What is the key difference in methodology between finding the centroid of a body by integration and finding the centroid of a composite body?

How is a hole or a region with no material handled when using the composite body method to find a centroid?

According to the first theorem of Pappus and Guldinus, the surface area (A) of a body of revolution is the product of what two quantities?

What does the second theorem of Pappus and Guldinus state about the volume of a body of revolution?

A semi-elliptical area is defined by (x^2)/4 + y^2 = 1 for y >= 0. The total width is 4 ft and the height is 1 ft. What is the y-coordinate of the centroid of this area?

For a paraboloid of revolution generated by revolving the curve z^2 = 100y about the y-axis, from y=0 to y=100 mm, what is the y-coordinate of the centroid?

A cylinder of radius 0.5 m and height 1 m has a density that varies with its height z from the base, given by rho = 200z kg/m^3. What is the location of the center of mass (z_bar)?

What is the primary assumption about the gravitational field when applying the formulas for the center of gravity in this chapter?

In some cases, the centroid is located at a point that is not on the object itself. Which of the following is an example of this?

What is the x-coordinate of the centroid for the area bounded by the curve y=x^2, the line x=1 m, and the x-axis?

Using the first theorem of Pappus and Guldinus, what is the surface area of a sphere of radius R?

Using the second theorem of Pappus and Guldinus, what is the volume of a sphere of radius R?

An assembly consists of a conical frustum (density 8 Mg/m^3) and a hemisphere (density 4 Mg/m^3). A cylindrical hole is bored through the frustum. In the composite method analysis shown in Example 9.11, how are the frustum and the hole treated in the summation?

What is the formula for the x-coordinate of the centroid of a line segment described by a curve y = f(x)?

For the composite wire shown in Example 9.9, which consists of a semicircular arc and two straight segments, what is the calculated x-coordinate of the centroid?

The plate in Example 9.10 is a composite area made of a large square, a triangle, and a rectangular hole. What is the calculated y-coordinate of the centroid?

The length of a differential element of a curve y(x) is given by the Pythagorean theorem, dL = sqrt((dx)^2 + (dy)^2). How can this be rewritten for integration with respect to x?

For the shaded area in Problem F9-1, bounded by y=x^3, y=1 m, and the y-axis, determine the x-coordinate of the centroid (x_bar).

For the shaded area in Problem F9-2, bounded by y=x^3, the x-axis, and the line x=1m, what is the y-coordinate of the centroid?

A straight rod of length L has a mass per unit length given by m = m0*(1 + x^2/L^2). Where is its center of mass, x_bar, measured from x=0?

A straight rod of length L has a mass per unit length given by m = m0*(1 + x^2/L^2). Where is its center of mass, x_bar?

A homogeneous solid is formed by revolving the shaded area from Problem F9-5 (z^2 = y/4, from z=0 to z=0.5m) about the y-axis. What is the y-coordinate of the centroid?

A homogeneous solid is formed by revolving the shaded area from Problem F9-6 (a triangle with vertices at (0,0), (1.5, 2), and (1.5, 0)) about the z-axis. What is the z-coordinate of the centroid?

What is the formula for the z-coordinate of the center of gravity, z_bar, for a general body?

The centroid of an area is being determined using a rectangular strip for the differential area element. If a vertical strip is used (thickness dx, height y), where is the centroid (x_tilde, y_tilde) of this differential element located?

A wire is bent into a U-shape defined by y=x^2 from x=-2 ft to x=2 ft. The wire is symmetric about the y-axis. What is the x-coordinate of the centroid?

A uniform parabolic-shaped rod is defined by the equation y^2 = 4x, extending from x=0, y=0 to x=4 m, y=4 m. The mass per unit length is 2 kg/m. What is the total mass of the rod?

Determine the area of the shaded region bounded by the parabola y^2 = 4x and the vertical line x = 4 m, as depicted in Problem 9-8.

A plate has a thickness of 0.5 in. and is made of steel with a specific weight of 490 lb/ft^3. It is shaped like the area in Problem 9-20, a quarter circle of radius 3 ft. What is the total weight of the plate?

The z-coordinate of the centroid of a semicircular wire of radius 'r' and angle alpha (as shown in Problem 9-7), when the wire is in the y-z plane and symmetric about the z-axis, is given by z_bar = r*sin(alpha)/alpha. What is the x_bar coordinate?

What is the x-coordinate of the centroid for the area bounded by the curves y^2=x and y=x^2, as shown in Problem 9-26/27?

What is the y-coordinate of the centroid of the area bounded by the curves y=x and y=(1/9)x^3, for x>=0, as shown in Problem 9-25? The intersection point is at x=3, y=3.

Determine the y-coordinate of the centroid for the T-shaped beam's cross-sectional area shown in F9-8.

Determine the y-coordinate of the centroid for the T-shaped beam's cross-sectional area shown in F9-8, which consists of a 300mm by 50mm flange on top of a 300mm by 50mm web.

An L-shaped cross-sectional area is shown in F9-10. It consists of a 4in by 0.5in vertical rectangle and a 2.5in by 0.5in horizontal rectangle. What is the x-coordinate of the centroid?

A homogeneous solid block has the shape shown in F9-11. It is a composite solid made of a rectangular block (A) and a triangular prism (B). Block A is 2ft x 4ft x 3ft. Block B has a triangular base (width 2ft, height 3ft) and length 4ft. What is the x-coordinate of the center of mass?

What does the integral of x_tilde * dA represent in the context of finding centroids?

A wire is bent in the shape shown in Problem 9-44, consisting of three straight segments. The top horizontal segment is 100mm, the vertical segment is 150mm, and the bottom horizontal segment is 50mm. What is the x-coordinate of the centroid?

An area is shaped like a C-channel, as in Problem 9-53/54. If the flanges are 12in wide and 2in thick, and the web is 2in thick and has a total height of 4in between the flanges, what is the y-coordinate of the centroid, measured from the bottom surface?

What is the key idea behind the 'Procedure for Analysis' for finding the centroid of a shape by single integration?

A semi-circular area of radius 'a' has its centroid at y_bar = 4a/(3*pi). If this area is revolved 360 degrees (2*pi radians) about its base (the x-axis), what is the volume of the resulting sphere?

Determine the x-coordinate of the centroid of the shaded area shown in Problem 9-9. The area is bounded by the curve y^2=x^3, the line x=1m, and the x-axis.

For the composite body in Example 9.10, the area of the triangular part (1) is 4.5, the area of the square part (2) is 9, and the area of the hole (3) is -2. The respective x-coordinates of their centroids are 1, -1.5, and -2.5. What is the overall x-coordinate?

When can a single integration be used to find the centroid of an area defined by y=f(x)?

What is the centroid (x_bar, y_bar) of the area of a quarter circle of radius R in the first quadrant?

The center of mass is derived from the center of gravity equations by substituting dW with what expression?

A plate is made in the shape of the shaded area in Problem 9-17, bounded by y = h - (h/a^2)*x^2 and the x-axis. The shape is symmetric about the y-axis. What can be concluded about the x-coordinate of its centroid?

For the cardioid r = a(1 - cos(theta)) shown in Problem 9-19, what method would be most appropriate for finding the centroid?

To determine the center of gravity of the water tank shown on the chapter's title page (and in Problem 9-81), what is the most practical first step?

The composite body equations for the centroid (e.g., x_bar = Sigma(x_tilde*A)/Sigma(A)) are analogous to the integral equations (e.g., x_bar = integral(x_tilde*dA)/integral(dA)). What does the summation operator (Sigma) in the composite equations replace from the integral equations?