An saw blade starts from 5 rad/s and accelerates at 3 rad/s2 and covers 252 rad. Determine the time it takes.Note, the formula will still have t and t2 so it's a quadratic equation, and we can use the quadratic formula. POSTED IN IMAGES Steps:Simplify as much as possible and arrange in descending order - that is, get everything on one side of the equation with a zero on the other side and rearrange the terms so that t2 comes first, t comes second and the constant term is lastthe quad formula has a, b, and c which are the coefficients of the t2 term, the t term and the constant term including the positive or negative signsexample:If 4 = 2 t + 1/2 (8) t2Simplify 1/2 (8) = 4, subtract 4 from both sides and rearrange the terms to get 0 = 4t2 + 2t - 4now a = 4, b = 2, c = -4plug these into the quadratic formula and get the two solutions, one when we use the + and the other when we use the - in the formula, just before the square root. Usually we want the positive solution(-2+√(22-4(4)(-4))).(2*4) = 0.78(-2-√(22-4(4)(-4))).(2*4) = -1.3 The image displays the quadratic formula for finding the variable \( t \) in a quadratic equation:\[ t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]This formula is used to find the solutions to the quadratic equation \( at^2 + bt + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients of the equation, and the symbol \( \pm \) indicates that there are generally two solutions: one involving addition and the other involving subtraction. The expression under the square root, \( b^2 - 4ac \), is known as the discriminant.

Initial angular velocity (ωi) = 5 radsAngular acceleration (α) = 3 rads2Angular displacement (θ) = 252 rad Using equation of angular motion : θ = ωit + (12αt2) 252 = 5t + (32t2) 1.5t2 + 5t - 252 = 0 Comparing the equation with standard quadratic equation : at2 + bt + c = 0 Here a = 1.5b = 5c = -252Now, using the formula to solve the quadratic equation : t = -5 ±(5)2 - (4×1.5×[-252])2×1.5t = -5 ±39.23t = 11.4 seconds , - 14.74 secHere t cannot be negative , therefore :Thus, time taken = 11.4 seconds

Answered: An saw blade starts from 5 rad/s and…

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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An saw blade starts from 5 rad/s and accelerates at 3 rad/s² and covers 252 rad. Determine the time it takes.

Note, the formula will still have t and t² so it's a quadratic equation, and we can use the quadratic formula. POSTED IN IMAGES

Steps:

Simplify as much as possible and arrange in descending order - that is, get everything on one side of the equation with a zero on the other side and rearrange the terms so that t2 comes first, t comes second and the constant term is last
the quad formula has a, b, and c which are the coefficients of the t² term, the t term and the constant term including the positive or negative signs
example:
- If 4 = 2 t + 1/2 (8) t²
- Simplify 1/2 (8) = 4, subtract 4 from both sides and rearrange the terms to get 0 = 4t² + 2t - 4
- now a = 4, b = 2, c = -4
- plug these into the quadratic formula and get the two solutions, one when we use the + and the other when we use the - in the formula, just before the square root. Usually we want the positive solution
  - (-2+√(2²-4(4)(-4))).(2*4) = 0.78
  - (-2-√(2²-4(4)(-4))).(2*4) = -1.3

The image displays the quadratic formula for finding the variable \( t \) in a quadratic equation:

\[
t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]

This formula is used to find the solutions to the quadratic equation \( at^2 + bt + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients of the equation, and the symbol \( \pm \) indicates that there are generally two solutions: one involving addition and the other involving subtraction. The expression under the square root, \( b^2 - 4ac \), is known as the discriminant.

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