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Chapter 9: Problem 109

Where is the center of mass of a semicircular wire of radius \(R\) that is centered on the origin, begins and ends on the \(x\) axis, and lies in the \(x, y\) plane?

Short Answer

Expert verified
The center of mass of the semicircular wire is located at the point \((0, \frac{2R}{\pi})\).

Step by step solution

01

Finding the parametric equations of the semicircular wire.

To proceed, we need to know the position of a point on the semicircular arc. We start by defining the angle \(\theta\) such that \(\theta\) varies between \(0\) and \(\pi\). The wire's parametric equations are: \[x(\theta) = R\cos(\theta)\] \[y(\theta) = R\sin(\theta)\]
02

Finding the mass density function, dm.

Let the mass per unit length of the wire be represented as \(\mu\), which is constant throughout the wire. We can find the mass of a small piece of the wire, dm, by multiplying \(\mu\) with the length ds of the small piece. To find ds, we need to find the arc length between two consecutive points on the wire with angle \(\Delta \theta\). We know that \(ds = R\Delta\theta\), thus \$dm = \mu R \Delta \theta\$.
03

Finding the center of mass along the y-axis, \( \bar{y} \).

The center of mass along the y-axis, \(\bar{y}\), can be found using the equation: \[\bar{y} = \frac{\int y dm}{\int dm}\] Substitute values of y and dm in this equation to find the center of mass along the y-axis: \[\bar{y} = \frac{\int_0^{\pi}(R\sin(\theta))(\mu R d\theta)}{\int_{0}^{\pi}(\mu R d\theta)}\]
04

Evaluating the integrals.

Evaluate the integral numerators and denominators separately: \[\int_{0}^{\pi}(R\sin(\theta))(\mu R d\theta) = \mu R^2\int_{0}^{\pi}\sin(\theta) d\theta = \mu R^2[-\cos(\pi) + \cos(0)] = 2\mu R^2\] \[\int_{0}^{\pi}(\mu R d\theta) = \mu R [\theta]_0^{\pi} = \mu R \pi\] Now, plug the calculated values of integrals back into the equation for \(\bar{y}\): \[\bar{y} = \frac{2\mu R^2}{\mu R \pi}\]
05

Simplifying the result.

Simplify to get the value of \(\bar{y}\): \[\bar{y} = \frac{2R}{\pi}\] The center of mass of the semicircular wire is located at the point \((0, \frac{2R}{\pi})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations
Parametric equations are a set of equations that express a group of quantities as explicit functions of a number of independent variables, known as parameters. In physics and engineering, they are a convenient way to describe curves, such as the path of a semicircular object.
Mass Density Function
In the context of a semicircular wire with mass distributed along its length, the mass density function describes how mass is distributed along the wire. If the wire has constant density, denoted by the Greek letter \(\mu\), then each small segment of the wire's length \(ds\) will have a corresponding small mass \(dm = \mu ds\). This concept is crucial for finding the center of mass of an object.
Integrals in Physics
Integrals are fundamental in physics for determining quantities like center of mass. When combined with the mass density function, integrals allow us to sum up the contributions of infinitesimally small pieces of an object to find properties such as the object's average position, or center of mass. The integral of the mass distribution \(dm\) over the entire object, compared against the integral of each point's position weighed by its mass, yields the location of the center of mass. The solution relies on evaluating these integrals carefully to incorporate the object's shape and mass distribution.
Semicircular Object Properties
In our case, the semicircular wire has unique geometric properties that influence its center of mass calculation. Specifically, its symmetry about the y-axis simplifies the problem, as the center of mass must lie on this axis. The position of the wire can be described using parametric equations that account for its curved shape. By integrating over the semicircular arc using these parametric equations, we can find the center of mass accurately, taking advantage of the object's symmetry and geometric properties.

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Most popular questions from this chapter

Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of \(1.50 \times 10^{5} \mathrm{kg}\) and a velocity of \((0.30 \mathrm{m} / \mathrm{s}) \hat{\mathrm{i}},\) and the second having a mass of \(1.10 \times 10^{5} \mathrm{kg}\) and a velocity of \(-(0.12 \mathrm{m} / \mathrm{s}) \hat{\mathrm{i}} .\) What is their final velocity?

Find the center of mass of a one-meter long rod, made of \(50 \mathrm{cm}\) of iron (density \(8 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\) ) and \(50 \mathrm{cm}\) of aluminum (density \(2.7 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\) ).

Find the center of mass of a rectangular block of length \(a\) and width \(b\) that has a nonuniform density such that when the rectangle is placed in the \(x, y\) -plane with one corner at the origin and the block placed in the first quadrant with the two edges along the \(x\) - and \(y\) -axes, the density is given by \(\rho(x, y)=\rho_{0} x,\) where \(\rho_{0}\) is a constant.

You are coasting on your 10-kg bicycle at 15 m/s and a 5.0-g bug splatters on your helmet. The bug was initially moving at \(2.0 \mathrm{m} / \mathrm{s}\) in the same direction as you. If your mass is \(60 \mathrm{kg},\) (a) what is the initial momentum of you plus your bicycle? (b) What is the initial momentum of the bug? (c) What is your change in velocity due to the collision with the bug? (d) What would the change in velocity have been if the bug were traveling in the opposite direction?

A family is skating. The father (75 kg) skates at 8.2 \(\mathrm{m} / \mathrm{s}\) and collides and sticks to the mother \((50 \mathrm{kg}),\) who was initially moving at \(3.3 \mathrm{m} / \mathrm{s}\) and at \(45^{\circ}\) with respect to the father's velocity. The pair then collides with their daughter \((30 \mathrm{kg}),\) who was stationary, and the three slide off together. What is their final velocity?
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