Question

A mass of 0.420 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = (0.660 m)cos[(16.0 rad/s)t]. Determine the following. (a) amplitude of oscillation for the oscillating mass How does the amplitude of oscillation compare to the magnitude of the maximum displacement from equilibrium? m (b) force constant for the spring How are the angular frequency and force constant of the spring related? N/m (c) position of the mass after it has been oscillating for one half a period How long does it take the oscillating mass to travel from the maximum positive displacement to the maximum negative displacement? m (d) position of the mass five-sixths of a period after it has been released How can you determine the position of the object at a specific time from an expression for the position of the object at any time? m (e) time it takes the mass to get to the position x = -0.300 m after it has been released How can you determine the time for the oscillating mass to get to a specific position from an expression for the position of the object at any time? s

          A mass of 0.420 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = (0.660 m)cos[(16.0 rad/s)t]. Determine the following.
(a) amplitude of oscillation for the oscillating mass
How does the amplitude of oscillation compare to the magnitude of the maximum displacement from equilibrium? m
(b) force constant for the spring
How are the angular frequency and force constant of the spring related? N/m
(c) position of the mass after it has been oscillating for one half a period
How long does it take the oscillating mass to travel from the maximum positive displacement to the maximum negative displacement? m
(d) position of the mass five-sixths of a period after it has been released
How can you determine the position of the object at a specific time from an expression for the position of the object at any time? m
(e) time it takes the mass to get to the position x = -0.300 m after it has been released
How can you determine the time for the oscillating mass to get to a specific position from an expression for the position of the object at any time? s

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A mass of 0.420 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = (0.660 m)cos[(16.0 rad/s)t]. Determine the following.
(a) amplitude of oscillation for the oscillating mass
How does the amplitude of oscillation compare to the magnitude of the maximum displacement from equilibrium? m
(b) force constant for the spring
How are the angular frequency and force constant of the spring related? N/m
(c) position of the mass after it has been oscillating for one half a period
How long does it take the oscillating mass to travel from the maximum positive displacement to the maximum negative displacement? m
(d) position of the mass five-sixths of a period after it has been released
How can you determine the position of the object at a specific time from an expression for the position of the object at any time? m
(e) time it takes the mass to get to the position x = -0.300 m after it has been released
How can you determine the time for the oscillating mass to get to a specific position from an expression for the position of the object at any time? s

Added by Rosario W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Justin Swantek

08/24/2023

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Therefore, the amplitude is \( A = 0.66 \, \text{m} \). Show more…

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A mass of 0.420 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = 0.660 m * cos[(16 rad/s)t]. Determine the following: (a) The amplitude of oscillation for the oscillating mass. (b) The force constant for the spring. (c) The position of the mass after it has been oscillating for one half period. (d) The position of the mass five-sixths of a period after it has been released. (e) The time it takes for the mass to get to the position x = 0.300 m after it has been released.

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Transcript

00:01 All right, hello, in this question we're going to go ahead and look at a mass on a spring.

00:03 We're told that the mass is 0 .520 kilograms and we're told that this is the equation of motion of the mass bobbing on the spring.

00:12 From this we want to determine a few things.

00:15 We want to determine a, the amplitude of oscillation.

00:18 Well, i'm going to go ahead and just start before we actually answer any questions here by writing what this looks like in terms of things we know, in terms of symbols.

00:28 So we're going to have our amplitude times the cosine of our angular frequency times time.

00:33 And for a spring we know that our period is 2 pi times root m over k.

00:40 And we also know that our angular frequency is defined as our 2 pi divided by our period.

00:47 So our angular frequency here is going to be the square root of k over m.

00:52 So with that stuff in mind, we can go ahead and look at our amplitude.

00:56 Well, our amplitude is just going to be the number that's out front.

00:58 So 0 .660 and that's given in meters here.

01:01 So that's our amplitude.

01:03 We're asked to determine the force constant for the spring.

01:06 Well, i just discussed that we have omega is root k over m.

01:10 And we know that in this case that is going to equal a value of 16.

01:15 So we're going to have 16 .0 here.

01:17 If i want to go ahead and solve for k, it's going to be 16 squared times our mass of 0 .520 kilograms.

01:26 That's our spring constant is going to be 133 newtons per meter for this case.

01:32 In part c, we're asked the position of the mass after it's been oscillating for one half of a period.

01:37 So if t equals one half of our period, well, we can answer this conceptually by saying x of one half of a period, a period is if we start on one end, we go all the way over and then we go all the way back.

01:50 That is defined as a one full period there.

01:54 So if we have half a period, we're going to just be on the other end.

01:57 So we're going to be at x equals negative 0 .660 meters...

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