A weight on the end of a coiled spring is pulled
10cm
below its resting position. The weight is released at time
=t0
seconds and moves upward, oscillating in simple harmonic motion. It completes one cycle in
3
seconds. (Note that upward is the positive direction.)
Give the equation modeling the displacement
d
as a function of time
t
.
Thank you
To find the equation modeling the displacement (d) as a function of time (t) for simple harmonic motion, we need to consider the amplitude and the period of the motion.
Here, the amplitude (A) is the maximum displacement of the weight from its resting position, which is 10 cm. The period (T) is the time taken for one complete cycle, which is 3 seconds.
The equation for the displacement as a function of time for simple harmonic motion is given by:
d = A * cos((2π/T) * (t - t0))
Plugging in the values we have:
d = 10 cm * cos((2π/3) * (t - t0))
So, the equation modeling the displacement (d) as a function of time (t) is:
d = 10 * cos((2π/3) * (t - t0)) cm
To find the equation modeling the displacement (d) as a function of time (t) for simple harmonic motion, we can use the following formula:
d = A * cos(2πf(t - t₀))
Where:
- d represents the displacement at time t.
- A is the amplitude of the oscillation, which is the maximum displacement from the equilibrium position.
- f is the frequency of the oscillation, which is the number of complete cycles per unit of time.
- t₀ is the initial time when the weight is released.
In this case, we are given the following information:
- The weight is pulled 10 cm below its resting position, which means the amplitude (A) is 10 cm (Note: amplitude is always positive).
- The weight completes one cycle in 3 seconds, so the frequency (f) is 1 cycle/3 seconds.
- The weight is released at time t₀ seconds.
Therefore, the equation modeling the displacement d as a function of time t is:
d = 10 * cos(2π(1/3)(t - t₀))
Note: In this equation, we assume that the equilibrium position is at zero displacement and the positive direction is upwards.
T = 3
so
f = 1/T = .333 Hz
omega = 2 pi f = 2.09 radians/s
at t = t0, d = -10
Amplitude = 10
d = -10 cos [ omega (t - t0) ]
that is -10 when t = t0