a) The frequency of the particle is 0.574 hertz.
b) The spring constant of the particle is 29.777 newtons per meters.
c) The speed at the equilibrium position is approximately 3.328 meters per second.
d) The elastic potential energy is 12.681 joules.
e) The potential energy when the particle is located 36.1 % of the amplitude is 1.651 joules.
The kinetic energy when the particle is located 36.1 % of the amplitude is 11.03 joules.
The speed when the particle is located at 36.1 % of the amplitude is approximately 3.104 meters per second.
The particle experiments a simple harmonic motion, whose angular frequency (
), in radians per second, is described by the following formula:
(1)
Where:
![k](/tpl/images/0601/1848/cc0ac.png)
- Spring constant, in newtons per meter.
![m](/tpl/images/0601/1848/d8537.png)
- Mass, in kilograms.
a) The frequency (
), in hertz, is the number of cycles done in a second, that is to say:
(2)
Where:
![n](/tpl/images/0601/1848/a4ae7.png)
- Number of cycles.
![t](/tpl/images/0601/1848/3d334.png)
- Time associated to a number of cycles.
If we know that
and
, then the frequency is:
![f = 0.574\,hz](/tpl/images/0601/1848/76f83.png)
The frequency of the particle is 0.574 hertz.
b) The angular frequency is also found by the following expression:
![\omega = 2\pi\cdot f](/tpl/images/0601/1848/874f1.png)
![\omega = 2\pi\cdot (0.574\,hz)](/tpl/images/0601/1848/51d4f.png)
![\omega \approx 3.606\,\frac{rad}{s}](/tpl/images/0601/1848/b0647.png)
The spring constant (
), in newtons per meter, is found by (1):
![k = \omega^{2}\cdot m](/tpl/images/0601/1848/cf6da.png)
![k = \left(3.606\,\frac{rad}{s} \right)^{2}\cdot (2.29\,kg)](/tpl/images/0601/1848/44123.png)
![k = 29.777\,\frac{N}{m}](/tpl/images/0601/1848/14b16.png)
The spring constant of the particle is 29.777 newtons per meters.
c) The speed at the equilibrium position (
), in meters per second, occur when translational kinetic energy reaches its maximum. By the principle of energy conservation, such maximum kinetic energy equals a maximum elastic potential energy:
(3)
![v = \sqrt{\frac{k}{m} }\cdot A](/tpl/images/0601/1848/be695.png)
Where
is the amplitude, in meters.
If we know that
,
and
, then the speed at the equilibrium position is:
![v = \sqrt{\frac{29.777\,\frac{N}{m} }{2.29\,kg} }\cdot (0.923\,m)](/tpl/images/0601/1848/ce3fe.png)
![v \approx 3.328\,\frac{m}{s}](/tpl/images/0601/1848/695b8.png)
The speed at the equilibrium position is approximately 3.328 meters per second.
d) The potential energy (
), in joules, at an endpoint is the maximum possible elastic potential energy:
(4)
If we know that
and
, then the potential energy is:
![U = \frac{1}{2}\cdot \left(29.777\,\frac{N}{m} \right)\cdot (0.923\,m)^{2}](/tpl/images/0601/1848/2820b.png)
![U = 12.681\,J](/tpl/images/0601/1848/b74da.png)
The elastic potential energy is 12.681 joules.
e) The potential energy when the particle is located 36.1 % of the amplitude away from the equilibrium position: (
,
)
![U = \frac{1}{2}\cdot \left(29.777\,\frac{N}{m} \right)\cdot (0.333\,m)^{2}](/tpl/images/0601/1848/d1064.png)
![U = 1.651\,J.](/tpl/images/0601/1848/c3dde.png)
The potential energy when the particle is located 36.1 % of the amplitude is 1.651 joules.
By the principle of energy conservation, we derive an expression of the translational kinetic energy:
(5)
Where:
![U_{max}](/tpl/images/0601/1848/61a47.png)
- Maximum potential energy, in joules.
![U](/tpl/images/0601/1848/ac4d7.png)
- Potential energy, in joules.
![K](/tpl/images/0601/1848/fc399.png)
- Kinetic energy, in joules.
If we know that
and
, then the kinetic energy of the particle is:
![K = 12.681\,J-1.651\,J](/tpl/images/0601/1848/fdb49.png)
![K = 11.03\,J](/tpl/images/0601/1848/e1d90.png)
The kinetic energy when the particle is located 36.1 % of the amplitude is 11.03 joules.
And the speed (
), in meters per second, is described by the following formula:
(6)
If we know that
and
, then the speed of the particle is:
![v = \sqrt{\frac{2\cdot (11.03\,J)}{2.29\,kg} }](/tpl/images/0601/1848/8b51a.png)
![v\approx 3.104\,\frac{m}{s}](/tpl/images/0601/1848/4abc4.png)
The speed when the particle is located at 36.1 % of the amplitude is approximately 3.104 meters per second.
We kindly invite to check this question on principle on energy conservation: link