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(II) A 0.835 -kg block oscillates on the end of a spring whose spring constant is $k=41.0 \mathrm{N} / \mathrm{m}$ . The mass moves in a fluid which offers a resistive force $F=-b v,$ where $b=0.662 \mathrm{N} \cdot \mathrm{s} / \mathrm{m} .$ (a) What is the period of the motion? (b) What is the fractional decrease in amplitude per cycle? (c) Write the displacement as a function of time if at $t=0, x=0,$ and at $t=1.00 \mathrm{s}, x=0.120 \mathrm{m}$

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a.$$0.898 \mathrm{s}$$b. $$-0.300$$c. $$x=(0.273 \mathrm{m}) e^{-\left(0.396 \mathrm{s}^{-1}\right) \mathrm{v}} \sin [(7.00 \mathrm{rad} / \mathrm{s}) t]$$

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

Sarah-Thérèse S.

May 2, 2022

From where do you have the formula in b)

Rutgers, The State University of New Jersey

Simon Fraser University

McMaster University

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

07:20

(II) A 0.835 -kg block osc…

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A 0.835-kg block oscillat…

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A $1.2-\mathrm{kg}$ block …

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$\bullet$ If an object on …

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A 1.00-kg mass is suspende…

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A 1.00 -kg mass is suspend…

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A block weighing 20 $\math…

02:07

A block of mass $0.773 \ma…

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A simple harmonic oscillat…

04:48

$\mathrm{A} 0.500$ -kg mas…

01:43

So here, for part athe damped, angular velocity is going to be equal to the square root of the spring. Constant divided by the mass, uh, minus B squared over for em. And so this is going to be equal to the square root of 41 Newtons per meter, divided by the mass of 0.835 kilograms. And then this will be minus 0.6662 uh, Newton seconds per meter squared. And then this will be divided by four times the mass of 0.835 kilograms, and we find that the damped angular velocity will be equal to 6.996 radiance per second. Now, for part B, they want to find the fractional change. So the fractional change would be equal to the amplitude times e to the negative gamma t AA minus a the amplitude divided by a the amplitude. And so this is gonna be equal to e to the negative gamma capital T period minus one. Um, and then we can say that this is gonna be equal to e x p e x p just is the exponential function. So whatever is in the parentheses is the power with a base of E, so this would be negative b over to em times t and then close parentheses minus one. We can solve and say that this is gonna be the exponential function of negative Ah, 0.6662 and then this will be divided by two times point 835 kilograms minus one and this is giving us negative 0.300 So I would be your answer for part B the fractional change. And then for part C, they want us to find the, um thea position off of a position function as a position function with respect to time. And uh however, the initial position is 0.120 meters. So the general equation for the damp oscillator will be equal to a times e x p of negative gamma times the period brother times t rather times sign of omega prime times t. And so, for the first part, we have 0.120 meters equals a thanks e x p r. The exponential function of negative 0.6662 times one second, divided by two times the mass of 20.835 A sign of 6.996 radiance. And so this is basically except T equals one second equals 10.120 meters. That's how we're setting this equation off. And solving for a A is simply gonna be equal 2.27 three meters. Ah, you would simply have Thio plug this into your T 80 45 or 89 and soul for a ah, you can use a cell function in order to isolate a so a is gonna be equal 2.273 meters. Gamma is going to be equal to be over to, um and this will be equal to a point 662 divided by two times 20.835 And this is giving us 0.396 per second. And so constructing the position versus time. Rather, position function position as a function of Time X would be equal to X would be equal to 0.273 meters, uh, times the exponent exponential function of negative 0.396 t and then sign of seven t. So this would be the full equation. That is the end of the solution. Thank you for watching

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