Question

A straight wire 7 units long has density \(\delta(x)=\sqrt{x}\) at a point \(x\) units from one end. Find the distance from this end to the center of mass.

Short answer

The center of mass is approximately 4.22 units from one end of the wire.

Step-by-step solution

Understanding the Problem

We are given a wire of length 7 units, and the density at any point \( x \) on the wire is given by \( \delta(x) = \sqrt{x} \). We need to find the position of the center of mass from one end of the wire.

Setting Up the Integral for Mass

The total mass \( M \) of the wire can be found by integrating the density function over the interval \([0, 7]\). The mass is given by: \[ M = \int_0^7 \sqrt{x} \, dx \]

Solving the Integral for Mass

To find \( M \), integrate \( \sqrt{x} \): \[\int \sqrt{x} \, dx = \frac{2}{3}x^{3/2} + C \] Evaluating from 0 to 7, \[ M = \left[ \frac{2}{3}(7)^{3/2} \right] - \left[ \frac{2}{3}(0)^{3/2} \right] = \frac{2}{3}(7^{3/2}) \]

Approximating 7 to Its Power

Calculate \( 7^{3/2} \):\[ 7^{3/2} = (7^1)^{3/2} = 7^{1.5} = \sqrt{343} \approx 18.52 \]Then, find the total mass: \[ M = \frac{2}{3} \times 18.52 \approx 12.35 \]

Deriving the Moment Function

The moment \( M_x \) about the origin is given by: \[ M_x = \int_0^7 x \cdot \sqrt{x} \, dx = \int_0^7 x^{3/2} \, dx \]

Solving the Integral for the Moment

Integrate \( x^{3/2} \): \[\int x^{3/2} \, dx = \frac{2}{5}x^{5/2} + C \]Evaluating from 0 to 7, \[ M_x = \left[ \frac{2}{5}(7)^{5/2} \right] - \left[ \frac{2}{5}(0)^{5/2} \right] = \frac{2}{5}(7^{5/2}) \]

Approximating 7 to Power 5/2

Calculate \( 7^{5/2} \):\[ 7^{5/2} = (7^2)^{5/4} \approx 130.32 \]Then, find the moment:\[ M_x = \frac{2}{5} \times 130.32 \approx 52.13 \]

Calculating the Center of Mass Position

The position of the center of mass \( x_{cm} \) is given by the ratio of the moment to the mass:\[ x_{cm} = \frac{M_x}{M} \approx \frac{52.13}{12.35} \approx 4.22 \]

Conclusion

The distance from one end of the wire to the center of mass is approximately 4.22 units.

Key concepts

Integration

Integration is a fundamental concept in calculus, helping us find values over continuous domains. Imagine you have a density function describing how dense an object is at any point along its length. To find the total mass of the object we need to integrate this function over its entire length.
The integration sums up infinitely small contributions (slices or pieces) of the function along an interval. In our case of a wire with density \(\delta(x) = \sqrt{x}\), the function should be integrated along the wire from 0 to 7 units.
Key Points to Remember:

• Integration finds total quantities like mass, area, or volume.
• The limits of integration define the interval over which you are summing.
• Always ensure the function you are integrating is expressed correctly and well-defined inside the given domain.

Using integration, we've calculated the total mass \(M\) of the wire, helping us move towards other physical calculations like the center of mass.

Center of Mass

The center of mass, sometimes known as the centroid, is the point where the entire mass of a body or system is thought to be concentrated. It helps in understanding how the mass is distributed along the object. For the straight wire with varying density, we're tasked with finding where this balance point lies.
This position is crucial when dealing with physical systems as it influences stability and motion behaviors. For a linear object like a wire, the center of mass \(x_{cm}\) can be calculated using the formula:
\[x_{cm} = \frac{M_x}{M}\] Where \(M_x\) is the moment about the origin and \(M\) is the total mass. In practice, knowing the center of mass helps predict how an object might behave if spun or tilted around different points.

• The center of mass depends significantly on how mass is distributed, not just the shape.
• For symmetric bodies with uniform density, the center of mass is at the geometric center, but for varying densities, calculations are needed.
• Understanding center of mass is essential in fields like engineering and physics for designing stable structures and devices.

Density Function

A density function explains how mass is distributed over an object or a region. Often denoted as \(\delta(x)\), it determines the amount of weight contained in an infinitesimally small portion at any point on the object. In the problem at hand, density \(\sqrt{x}\) means that the wire's density increases with \(x\).
This concept is vital because it can vary across different objects, influencing how their mass behaves. Even when total mass might be the same, changing the distribution affects where the center of mass will be located.
Density functions are not limited to linear objects. You can define them over surfaces and volumes, making them massively useful for analyzing complex systems.

• Density functions can vary with position, impacting stability and balance.
• They help in understanding real-world objects which rarely have uniform properties.
• Altering density distribution can have significant outcomes in design and manufacturing.

Mastering how to work with density functions allows deeper insights into mechanical and structural systems.

Moment of Inertia

Although not directly required to solve the presented problem, moment of inertia is another important application of calculus when dealing with objects and their mass distributions. It is the rotational equivalent of mass for linear motion and reflects how difficult it is to change an object's rotational motion about an axis.
The moment of inertia \(I\) depends on how the mass is spread concerning the axis of rotation:\[I = \int x^2 \delta(x) \, dx\]Here, the role of density \(\delta(x)\) becomes crucial again in determining moment aspects for more intricate rotational dynamics. Understanding this helps engineers and physicists design systems that move or rotate efficiently.

• Moment of inertia is heavily dependent on the square of the distance from the axis, emphasizing the distribution of mass further than the axis.
• It is a critical component in designing anything from wheels to entire vehicles.
• Calculating moment of inertia involves integrating the point masses of objects over their entire volume or area, considering their distances.

A good grasp of the moment of inertia ties into the comprehensive understanding of how objects balance and rotate, offering essential insights for practical applications.