An object moves in simple harmonic motion with period 7 seconds and amplitude 5cm. At time t=0 seconds, its displacement d from rest is 0cm, and initially it moves in a positive direction.
Give the equation modeling the displacement d as a function of time t.
Since we know d = 0 when t = 0, a sine function would work nicely here
So we know a = 5
period = 2π/k
k = 2π/7
d = 5sin (2π/7 t)
To model the displacement d of an object in simple harmonic motion as a function of time t, we can use the equation:
d(t) = A * cos(2πt / T)
Where:
- d(t) represents the displacement at time t.
- A is the amplitude of the motion (in this case, 5 cm).
- T is the period of the motion (in this case, 7 seconds).
- cos is the cosine function.
Substituting the given values into the equation, we get:
d(t) = 5 * cos((2π / 7) * t)
Therefore, the equation modeling the displacement d as a function of time t is:
d(t) = 5 * cos((2π / 7) * t)
To model the displacement of an object in simple harmonic motion as a function of time, we can use the equation:
d(t) = A * sin((2π/T) * t + φ),
where:
- d(t) represents the displacement of the object at time t,
- A is the amplitude of the motion,
- T is the period of the motion, and
- φ is the phase constant (representing the initial phase of the motion).
In this case, we are given that the period T is 7 seconds, and the amplitude A is 5 cm. We are also given that at time t = 0 seconds, the displacement d is 0 cm, and that the object initially moves in the positive direction.
The equation for the displacement in this situation would be:
d(t) = 5 * sin((2π/7) * t + φ).
To find the value of the phase constant φ, we can use the given information that initially the object moves in the positive direction. At time t = 0 seconds, the displacement is 0 cm, which means the object is at its equilibrium position. This implies that the phase constant φ is 0, as sin(0) = 0.
Therefore, the equation modeling the displacement d as a function of time t would be:
d(t) = 5 * sin((2π/7) * t).