Question

Calculate the mass (m) and determine the center of mass for a linear wire distributed over a specified interval with a given density function. Describe the density function and the interval, and explain the method or mathematical approach employed to find both the total mass and the location of the center of mass. Provide a step-by-step analysis, considering the principles of calculus and physics involved in this calculation, and discuss any implications or insights derived from the results.

Answer 1

To calculate the mass (m) and determine the center of mass for a linear wire distributed over the interval [a, b] with a given density function
`\( \rho(x) \)`, we use the formulas
`\(m = \int_(a)^(b) \rho(x) \, dx\)` for the total mass and
`\( \bar{x} = (1)/(m) \int_(a)^(b) x \cdot \rho(x) \, dx\)` for the center of mass. The density function
`\( \rho(x) \)`represents mass per unit length, and the interval
`\([a, b]\)` specifies the range of the wire.

Step-by-step explanation:

The total mass (m) of the wire is obtained by integrating the density function
`\( \rho(x) \)` over the given interval
`\([a, b]\)` using the formula
`\(m = \int_(a)^(b) \rho(x) \, dx\)`. This integral represents the accumulation of infinitesimally small masses along the length of the wire. The density function provides the mass per unit length, and integrating it over the interval gives the overall mass.

To determine the center of mass
`\(\bar{x}\)`, we use the formula
`\( \bar{x} = (1)/(m) \int_(a)^(b) x \cdot \rho(x) \, dx\)`. This involves multiplying the position (x) by the density function
`\( \rho(x) \)` and integrating the result over the interval. The ratio of this integral to the total mass (m) gives the center of mass. Physically, the center of mass is the weighted average position of the wire.

These calculations involve principles of calculus, where integration is used to sum up infinitesimally small quantities. The results provide valuable insights into the distribution of mass along the wire and its balance point. Understanding the center of mass is crucial for various applications, such as ensuring stability in structures and predicting the behavior of physical systems.

Answer 2

The mass and center of mass of a linear wire are calculated using integration, where the total mass is the integral of the density function over the wire's length, and the center of mass is the ratio of the integral of mass times position over the entire mass.

Step-by-step explanation:

The calculation of the mass and the center of mass of a linear wire with a given density function requires the application of calculus, specifically integration, to determine these quantities based on the wire's mass distribution.

For a wire bent into a semicircular shape, the center of mass will lie along the vertical axis of symmetry due to the uniform distribution of mass.

For a rod with a non-uniform mass density that changes quadratically from one end to the other, the density function p(x) = Po + (P1 − Po)(x/L)² is used.

The total mass can be found by integrating the density function over the length of the rod.

The center of mass can be determined by finding the weighted average position of the mass along the rod, which involves the integration of the product of mass and position over the total mass.

To carry out these calculations for a wire or a rod, we employ the principles of calculus.

We define an infinitesimal element of mass dm as the product of linear mass density λ (lambda) and an infinitesimal length dx. Then, we integrate this dm over the entire length to find the total mass m.

For the center of mass, we calculate the moment arm (x for horizontal distribution, y for vertical distribution), multiply it by dm, and integrate over the length.

This integral is then divided by the total mass m, yielding the position of the center of mass.

The implication of such calculations is that they enable engineers and scientists to determine the behavior of objects under various forces and conditions, such as in designing stable structures or understanding the dynamics of moving objects.