a 2 kg frictionless block is attached to an ideal spring with force constant 300N/m. at t=0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12m/s. find a) the amplitude and b) the phase angle, c) write a equation for the position as a function of time.
You know the total energy (which is the starting KE). AT max stretch, you now know PE, solve for amplitude. Phase angle? if you write the equation as a sine function, you know it is either zero or 180 deg, but since is goes negative at the start, the phase angle must be....
To find the amplitude and phase angle, we can use the equation of motion for simple harmonic motion. The displacement of the block as a function of time is given by:
x(t) = A * cos(ωt + φ)
where:
x(t) is the position of the block at time t,
A is the amplitude of oscillation,
ω is the angular frequency (ω = √(k/m)),
t is the time, and
φ is the phase angle.
Let's solve the problem step by step:
a) Finding the amplitude (A):
To find the amplitude, we need additional information. In this case, since the block is moving in the negative direction at 12 m/s, we know that the maximum displacement occurs when the block momentarily comes to rest and changes direction. At this point, all of the initial kinetic energy of the block is converted into potential energy of the spring.
Using the conservation of mechanical energy, we can equate the initial kinetic energy with the potential energy of the spring:
(1/2)mv² = (1/2)kA²
Substituting the given values:
(1/2) * 2 kg * (-12 m/s)² = (1/2) * 300 N/m * A²
Simplifying:
72 = 150A²
Dividing both sides by 150:
A² = 72/150
A = √(72/150)
A ≈ 0.574 m
Therefore, the amplitude (A) is approximately 0.574 m.
b) Finding the phase angle (φ):
To find the phase angle, we need to determine the initial conditions when t = 0. At t = 0, the block is neither stretched nor compressed, and it is moving in the negative direction. This means that the block is at its maximum displacement in the negative direction, and the phase angle (φ) is 180 degrees or π radians.
Therefore, the phase angle (φ) is π radians.
c) Writing the equation for the position as a function of time:
Using the values found above, we can write the equation for the position of the block as a function of time:
x(t) = A * cos(ωt + φ)
Substituting the values:
x(t) = 0.574 * cos(√(300/2) * t + π)
Simplifying:
x(t) = 0.574 * cos(10√3t + π)
Therefore, the equation for the position of the block as a function of time is x(t) = 0.574 * cos(10√3t + π).