A particle moving in simple harmonic motion passes through the equilibrium point (x=0) 9 times per second. At t=0t=0 its velocity at x=−0.01 m is negative. It travels 0.4 m in a complete cycle.
The particle's position as a function of time is described by the following function:
x(t) = ___sin(___t +___) cm
please help
general equation
x = a sin(b(t-c))+d
Since equilibrium is x=0, d=0
x = a sin(b(t-c))
the period is 9, so b = 2π/9
x = a sin(2π/9 (t-c))
If it travels .4 m in a cycle, the amplitude is 0.1, so
x = 0.1 sin(2π/9 (t-c))
Now we have a problem. If x(0) = -0.1, that is its minimum value, so the velocity must be zero, not negative.
Check for a typo somewhere.
Recall that cos(0) is a max, so we can rewrite the above equation as
x = -0.1 cos(2π/9 t)
To find the equation for the particle's position as a function of time, we can use the given information.
1. The particle passes through the equilibrium point 9 times per second, which means it completes 9 cycles per second.
2. It travels 0.4 m in a complete cycle.
We know that the general equation for simple harmonic motion can be written as:
x(t) = A * sin(ωt + φ)
Where:
- x(t) is the position of the particle at time t.
- A is the amplitude of the motion.
- ω is the angular frequency, given by 2πf, where f is the frequency.
- φ is the phase constant.
Using the given information, we can determine these values:
Amplitude:
In one complete cycle, the particle travels 0.4 m. This means the amplitude A is half of the total distance traveled, so A = 0.4 m / 2 = 0.2 m.
Angular frequency:
The frequency is given as 9 cycles per second. The angular frequency is then ω = 2πf = 2π * 9 = 18π rad/s.
Phase constant:
At t = 0, the velocity is negative when x = -0.01 m, indicating a phase shift of π.
Putting it all together, the equation becomes:
x(t) = 0.2 * sin(18πt + π) cm
Please note that the equation is written in cm, as specified in the question.
To find the equation for the particle's position as a function of time, we need to start by identifying the parameters in the given information.
1. The particle passes through the equilibrium point 9 times per second. This tells us the frequency (f) of the oscillation. Frequency is the number of complete cycles in one second, so in this case, f = 9 Hz.
2. At t = 0, the particle's velocity at x = -0.01 m is negative. This information implies that the particle is moving in the negative direction when at x = -0.01 m, which corresponds to a phase shift (ϕ) of 180 degrees or π radians.
3. The particle travels 0.4 m in a complete cycle which corresponds to its amplitude (A).
Now we can construct the equation for the particle's position (x) as a function of time (t):
x(t) = A * sin(2πft + ϕ)
Substituting the known values into the equation:
x(t) = A * sin(2π(9)t + π)
Simplifying the equation:
x(t) = A * sin(18πt + π)
Now we have the equation for the particle's position as a function of time.