Center of Mass of a Rod

The video tutorial explains how to compute the center of mass for a rod with a variable density function, rho(x) = 2x + 3√x, over the interval from 0 to 4. It covers the calculation of the moment about the origin and the mass of the rod using integrals. The center of mass is determined by dividing the moment by the mass, resulting in a balance point at approximately x = 2.53. The tutorial provides a step-by-step guide to solving the problem, including simplifying antiderivatives and evaluating definite integrals.

What is the density function of the rod described in the problem?

• 2x + 3√x

• 3x + 2√x

• x^2 + 3x

• 2x^2 + 3x

What is the interval over which the rod is placed along the x-axis?

• 0 to 4

• 1 to 3

• 0 to 2

• 2 to 4

How is the moment about the origin defined in terms of integrals?

• Integral of x times rho(x) dx

• Integral of rho(x) dx

• Integral of x^2 times rho(x) dx

• Integral of x times x dx

What is the first step in calculating the moment about the origin?

• Finding the antiderivative

• Simplifying the expression

• Distributing the x

• Finding the derivative

What is the antiderivative of 2x squared?

• x^3/3

• 2x^3/3

• 2x^2

• x^2/2

What is the mass of the rod after integration and simplification?

• 16

• 32

• 128

• 64

What is the final expression for the mass of the rod?

• x^2 + 2x^(3/2)

• 2x^2 + 3x^(1/2)

• x^3 + 3x^(1/2)

• 2x^3 + x^(1/2)

How is the center of mass calculated?

• By multiplying the moment by the mass

• By dividing the moment by the mass

• By adding the moment and the mass

• By subtracting the mass from the moment

What is the exact value of the center of mass?

• 15/38

• 32/15

• 38/15

• 15/32

At approximately what x location does the rod balance horizontally?

• 1.53

• 2.53

• 3.53

• 4.53