A mass on a spring vibrates in simple harmonic motion at a frequency of 4.0 Hz and an amplitude of 4.0 cm. If a timer is started when its displacement is a maximum (hence x = 4 cm when t = 0), what is the speed of the mass when t = 5 s?
To find the speed of the mass when t = 5 s, we first need to find the equation of motion for simple harmonic motion. The equation is given by:
x = A * cos(ωt + φ)
where:
- x is the displacement from the equilibrium position
- A is the amplitude of the motion
- ω is the angular frequency (2πf)
- t is the time
- φ is the phase constant
In this case, A = 4 cm, f = 4.0 Hz, and we are given that x = 4 cm when t = 0. Therefore, the equation becomes:
4 = 4 * cos(2π * 4 * 0 + φ)
4 = 4 * cos(0 + φ)
4 = 4 * cos(φ)
From this equation, we can solve for the phase constant φ. Taking the inverse cosine of both sides, we get:
cos(φ) = 1
φ = 0
Now that we have the equation of motion, x = 4 * cos(2π * 4 * t), we can differentiate it to find the velocity equation:
v = dx/dt = -4 * 2π * 4 * sin(2π * 4 * t)
To find the velocity when t = 5 s, we substitute t = 5 into the equation:
v = -4 * 2π * 4 * sin(2π * 4 * 5)
v = -4 * 2π * 4 * sin(40π)
v = -4 * 2π * 4 * 0
v = 0
Therefore, the speed of the mass when t = 5 s is 0 cm/s.