Equation for Sinusoidal Function for Mass Above Table Top

In summary, the height of the mass above the table top can be represented by the equation y = A cos(ωt) + h, where A is the amplitude and h is the initial height. The angular frequency ω is equal to 2π divided by the period of 1.2 seconds. This equation can be used to calculate the height of the mass at any time within the first 2.0 seconds of its motion.
  • #1
lomantak
10
0
A mass is supported by a spring so that it rests 0.5 m above a table top. The mass is pulled down 0.4 m and released at time t=0. This creates a periodic up-and-down motion, called simple harmonic motion. It takes 1.2 s for the mass to return to the low position each time.

Could someone please give me an equation of the sinusoidal function when the height of the mass above the table top is a function of time for the first 2.0 s?

Thanks.
 
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  • #2
You know the amplitude and you know the frequency (via the period). What more do you need? :)
 
  • #3
Your amplitude is given to you (the amount pulled down). The period, 1.2s, is the amount of time it will take to go all the way up, and then back down again.

Remember your unit circle? In this case, the low point corresponds to [itex]\cos{\pi}[/itex]. It makes one full revolution around the unit circle in 1.2 seconds. Since the entire circle ([itex]2\pi[/itex]) is traveled once, then [itex]\omega = \frac{2\pi}{1.2sec}[/itex] where [itex]\omega[/itex] is called the angular frequency. Basically, [itex]\omega[/itex] is the angular rate of change in [itex]t[/itex] seconds.

So, if cosine is at a maximum of 1 and a minimum of -1, then you need to multiply it by the amplitude at its maximum and minimum to get the max and min amplitudes, right?

[tex]y = A\cos{(\omega t)}[/tex]
But, it's 0.5m above the table, so everything is shifted up h = 0.5m
[tex]y = A\cos{(\omega t)} + h[/tex]

I'll leave you to do the actual calculations, but hopefully you see how this works. Also note that cosine and sine are the same function, but cosine is shifted by pi/2 radians. You could also write that as y = Asin( wt + pi/2 ) + h
 
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1. What is a sinusoidal function?

A sinusoidal function is a mathematical function that describes a repeating pattern or oscillation. It is commonly represented by the sine or cosine function and is used to model many natural phenomena such as sound waves and electromagnetic waves.

2. How is a sinusoidal function used to model mass above a table top?

The equation for a sinusoidal function can be used to model the vertical displacement of an object above a table top due to its weight and any external forces acting on it. The function takes into account the amplitude, period, and phase shift of the oscillation.

3. What is the equation for a sinusoidal function for mass above a table top?

The general equation for a sinusoidal function for mass above a table top is y = A*sin(Bx + C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical displacement of the object.

4. How does the amplitude affect the sinusoidal function for mass above a table top?

The amplitude of the sinusoidal function represents the maximum displacement of the object above the table top. A larger amplitude means the object will move further away from the table top, while a smaller amplitude means it will oscillate closer to the table top.

5. How can the equation for a sinusoidal function for mass above a table top be applied in real-life situations?

The equation for a sinusoidal function for mass above a table top can be used in various engineering and physics applications, such as predicting the movement of a pendulum, analyzing vibrations in structures, and designing oscillating systems. It can also be used to study the behavior of sound and light waves.

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