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(I) A 1.15 -kg mass oscillates according to the equation $x=0.650 \cos 7.40 t$ where $x$ is in meters and $t$ in seconds. Determine $(a)$ the amplitude, $(b)$ the frequency, $(c)$ the total energy, and $(d)$ the kinetic energy and potential energy when $x=0.260 \mathrm{m} .$

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a. The amplitude $(A)=0.65 \mathrm{m}$b. $$1.18 \mathrm{Hz}$$c. $$13.3 \mathrm{J}$$d. $$2.1 \mathrm{J}$$

05:11

Shital Rijal

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

Cornell University

Rutgers, The State University of New Jersey

University of Washington

University of Sheffield

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.

03:20

A 1.15-kg mass oscillates…

05:31

A 1.15-kg mass oscillates …

02:56

03:49

A $0.650-\mathrm{kg}$ mass…

03:33

A 0.650-kg mass oscillates…

04:45

A 1.25 kg mass oscillates …

04:22

A $2.00-\mathrm{kg}$ mass …

04:00

(II) A $0.60-\mathrm{kg}$ …

05:42

At $t = 0$, an 885-g mass …

02:42

A body of mass $1.80 \math…

02:46

A 507 g mass oscillates wi…

03:45

(II) A 300 -g mass vibrate…

08:15

(II) At $t=0,$ a $785-\mat…

A $200 \mathrm{g}$ mass at…

01:58

A simple harmonic oscillat…

02:09

01:43

A 0.85-kg mass attached to…

For this problem on the topic of oscillators were given the equation that governs the oscillation of a 1.15 Kg mass. And we want to determine the amplitude, the frequency, the total energy, as well as the kinetic and potential energies for this oscillating mass when X is equal to 0.26 m. So we are given the equation X is equal to 0.65 co signed 7.4 times T. And we know the general form of this equation is a co sign omega. T. Where is the amplitude, omega is the angular frequency and T. Is the time. So from this equation we can immediately see that for a we can read off the amplitude, which is the maximum value of X. From the equation and the aptitude A. Is simply zero .65. And we are told that this is in m, so that the amplitude of the military motion, now in part B, we want to find the frequency. We know the frequency F. Is related to the angular frequency by F. Is equal to omega over two pi. And again we can read omega From this equation and Omega Assembly 7.4. That's 7.4 radiance a second divided by two pi radiance. This gives us the frequency of oscillations to be one 0.18 huts. Next in part C. We want to calculate the total energy of this absolutely motion. And this total energy is simply the maximum potential energy at the embassy of motion. So this is a half K A squared. And Kay is the wave number which we can write as a half times M omega squared. And that multiplied by a squared, so that's a half times the mass. That is oscillating. 1.15 Kg. I'm the angular frequency of 7.4 radiance per second squared Times the square of the amplitude, which is zero 65 m squared. And so calculating we get the maximum energy To be 13.3, or 23 significant figures 13.3 jules. So that's the maximum energy that this oscillating mass can have Now at The position X is equal to 0.26 m. Who want to calculate the the potential energy as well as the kinetic energy and the potential energy you is simply a half K X squared. Again. We can write this as a half m Omega squared x squared where X is not the amplitude in this case, but 0.26 m. So that's a half times 1.15 Kg. And the angular frequency 7.4 radiance per second squared an X. In this case 0.26 m two. And so calculating, we get the potential potential energy at this point to be too 0.1 jules. So if we know the potential energy, we can use this information to calculate the kinetic energy. The kinetic energy is what's remaining essentially the total energy minus this potential energy. You This is 13 0.3 jewels minus 2.1 Jules, Which means the rest remaining energy is kinetic energy, and this is 11.2 jewels.

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