Respuesta :
Answer:
y = 5 cos 2πt
Explanation:
We will use the formula for simple harmonic motion curve where;
y = a cos ωt
Where;
a is amplitude
t is period
ω is angular frequency with the formula; ω = 2π/t
We are told that when the spring is compressed, the mass is located 5 cm above its rest position.
Thus;
a = 5 cm
it's highest point is 5 cm, but we are told that after 1/2 second of being released, it reaches its lowest point.
Since highest point is 5, then lowest point will be -5.
The difference in time between the highest and lowest point is ½ s. Which is half of the period.
Thus;
t/2 = ½
Thus, t = 1 s
Now, we know that;
t = 1/f = 2π/ω
Since t = 1, then 1 = 1/f
f = 1
Thus;
2π/ω = 1
Thus, ω = 2π
Thus, the equation is;
y = 5 cos 2πt
The equation that describes the motion of the mass is y = 5 cos 2πt.
The given parameters;
- maximum displacement of the spring, A = 5 cm
- time taken for the mass to reach the lowest point (half period), t = 0.5 s
The general equation of the wave is given as;
[tex]y = A\ cos\ \omega t[/tex]
where;
- A is the amplitude of the vibration
- ω is the angular speed of mass
The angular speed of the mass is calculated as;
[tex]\omega = 2\pi f\\\\[/tex]
The period of the oscillation is calculated as;
[tex]T = 2t \\\\T = 2(0.5 s) = 1 \ s[/tex]
The frequency of the wave is calculated as;
[tex]f = \frac{1}{T} \\\\f = \frac{1}{1} \\\\f = 1\ Hz[/tex]
The equation that describes the motion of the mass is calculated as;
[tex]y = A\ cos \ \omega t\\\\y = A\ cos \ 2\pi ft\\\\y = 5\ cos \ 2\pi (1) t\\\\y = 5 \ cos \ 2\pi t[/tex]
Thus, the equation that describes the motion of the mass is y = 5 cos 2πt.
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