**Problem Statement:**Find the mass and center of mass of the lamina that occupies the region \( D \) and has the given density function \( \rho \).Region \( D \) is defined as: \[ D = \{(x, y) \mid 1 \leq x \leq 7, 1 \leq y \leq 4\} \]The density function is given by:\[ \rho(x, y) = ky^2 \]**Equations to Calculate:**- **Mass (\( m \)):** \[ m = \int \int_{D} \rho(x, y) \, dA \] (The actual computation will involve evaluating the double integral over the defined region with the given density function.)- **Center of Mass (\( \bar{x}, \bar{y} \)):** \[ \left( \bar{x}, \bar{y} \right) = \left( \frac{1}{m} \int \int_{D} x \rho(x, y) \, dA, \frac{1}{m} \int \int_{D} y \rho(x, y) \, dA \right) \] (These equations represent the coordinates for the center of mass based on the calculated mass.)**Steps to Solve:**1. Compute the mass \( m \) by setting up and evaluating the double integral of the density function over the region \( D \).2. Calculate the \( x \)-coordinate (\( \bar{x} \)) of the center of mass by finding the moment in the \( x \)-direction using integration.3. Calculate the \( y \)-coordinate (\( \bar{y} \)) of the center of mass by finding the moment in the \( y \)-direction using integration.4. Use these calculated values to find the center of mass coordinates \( \left( \bar{x}, \bar{y} \right) \).This problem involves calculus and integration over a defined region with a variable density function, which is typical in physics for determining mass properties of objects.

Given, 1≤x≤7 and 1≤y≤4. Also ρx,y=ky2 The mass can be calculated as, m=∬ρx,ydA Here, we are…

Answered: Find the mass and center of mass of the…

Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D = {(x, y) | 1 < x < 7, 1 < y s 4}; p(x, y) = ky2 m 3D (x, y)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

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Concept explainers

Rigid Body

A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.

Rigid Body Dynamics

Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).

Question

**Problem Statement:**

Find the mass and center of mass of the lamina that occupies the region \( D \) and has the given density function \( \rho \).

Region \( D \) is defined as:
\[ D = \{(x, y) \mid 1 \leq x \leq 7, 1 \leq y \leq 4\} \]

The density function is given by:
\[ \rho(x, y) = ky^2 \]

**Equations to Calculate:**

- **Mass (\( m \)):**
\[ m = \int \int_{D} \rho(x, y) \, dA \]
(The actual computation will involve evaluating the double integral over the defined region with the given density function.)

- **Center of Mass (\( \bar{x}, \bar{y} \)):**
\[
\left( \bar{x}, \bar{y} \right) = \left( \frac{1}{m} \int \int_{D} x \rho(x, y) \, dA, \frac{1}{m} \int \int_{D} y \rho(x, y) \, dA \right)
\]
(These equations represent the coordinates for the center of mass based on the calculated mass.)

**Steps to Solve:**

1. Compute the mass \( m \) by setting up and evaluating the double integral of the density function over the region \( D \).
2. Calculate the \( x \)-coordinate (\( \bar{x} \)) of the center of mass by finding the moment in the \( x \)-direction using integration.
3. Calculate the \( y \)-coordinate (\( \bar{y} \)) of the center of mass by finding the moment in the \( y \)-direction using integration.
4. Use these calculated values to find the center of mass coordinates \( \left( \bar{x}, \bar{y} \right) \).

This problem involves calculus and integration over a defined region with a variable density function, which is typical in physics for determining mass properties of objects.

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