## Lesson on Acceleration and Coordinate Systems### Part 4: Solve for $ a $ in terms of $ m $, $ g $, $ \theta $, and $ \mu_k $.To solve for the acceleration, you must express $ a $ using the given variables. This involves using the equation editor for Greek symbols and subscripts:\[ a = g(\sin \theta - \mu_k \cos \theta) \]This expression represents the acceleration in terms of gravitational acceleration $ g $, angle $ \theta $, and the coefficient of kinetic friction $ \mu_k $.### Part 5: Does $ a $ depend on $ m $?The answer is **No** (selected from dropdown), indicating that the acceleration does not depend on the mass $ m $.### Cross-Check:Verify that the dimensions of your solution for $ a $ match the dimensional formula for acceleration.### Discussion:Consider analyzing this problem by initially aligning your coordinate axes with the x-axis parallel to the ground and the y-axis perpendicular to it. Here, both the normal force $ N $ and the force of friction $ f $ will show both horizontal and vertical components. This is true for acceleration as well.**Alternative Coordinate System:**Since coordinate axes are arbitrary, you can choose an orientation that simplifies your calculations. For inclined planes, it’s generally easier to rotate the axes: let the x-axis be parallel to the incline and the y-axis perpendicular to it. This setup aids in identifying forces that act purely in one direction and in differentiating single-component vectors from those with two components.**Focus on Vector Decomposition:**By decomposing vectors based on this rotated coordinate system, you can determine which forces are simplified. Pay close attention to the incline angle $ \theta $ and its effect on how forces decompose into components. ### Free-Body Diagram and Newton's Second Law#### Scenario Description:An object of mass $m$ is sliding down an incline making an angle $\theta$ with the horizontal with acceleration $a$. The incline has a coefficient of kinetic friction $\mu_k$.#### Diagram Explanation:- The diagram shows a free-body representation of the forces acting on the object.- The forces include the component of the object's weight in the x-direction ($w_x$), the component in the y-direction ($w_y$), the normal force ($N$), and the friction force ($f$).- Vectors are depicted to indicate direction, with labels for their components.#### Problem Setup:1. **Set-up Newton's Second Law Equations:** - **Along the x-axis:** \[ \Sigma F_x = w_x - f = ma \quad \text{(1)} \] - **Along the y-axis:** \[ \Sigma F_y = N - w_y = 0 \quad \text{(2)} \]2. **Part 1:** - Determine the acceleration components along the x and y directions using the above equations. - **Remember**: - Use subscripts for the weight components ($w_x, w_y$). - Apply correct signs according to the coordinate system shown in the diagram.3. **Part 2:** - Calculate the magnitudes of $w_x$ and $w_y$: - For $w_x$, $w_x = m \cdot g \cdot \sin(\theta)$ - For $w_y$, $w_y = m \cdot g \cdot \cos(\theta)$#### Key Notes:- Adjust vector directions and signs depending on the coordinate system.- Angles and vectors may not be exact and magnitudes are not considered in drawing.This page is designed to help students apply Newton's Second Law to objects on an incline, taking into account frictional forces.

Answered: Image /qna-images/answer/d1a47148-3408-458b-8793-df9b68ca2f12.jpg

Answered: Part 4: Solve for a in terms of m, g,…

Part 4: Solve for a in terms of m, g, 0, and μk. YOU WILL REQUIRE THE EQUATION EDITOR FOR THE GREEK SYMBOLS AND SUBSCRIPTS. Factorize as much as possible. a = g(sine-μcose)

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

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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