A particle that hangs from a spring oscillates with an angular frequency ω. The spring-particle system is suspended from the ceiling of an elevator car and hangs motionless (relative to the elevator car) as the car descends at a constant speed v. The car then stops suddenly. (Use omega for ω, v, and t as appropriate in the following.)
a) how do we know in this case that we are using v(max)=wA instead of v.
b)x=-Asinwt, I'm confused why cant we use cos instead of sin
a) In this case, we are using the equation v(max) = ωA instead of v, because the maximum velocity of the oscillating particle depends on the angular frequency (ω) and the amplitude of the motion (A). This equation relates the maximum velocity of the particle to the angular frequency and amplitude of oscillation, and it holds true for this scenario.
b) The equation x = -Asin(ωt) represents the displacement of the particle from its equilibrium position over time. The use of sine instead of cosine in this equation is due to the starting point of the oscillation. When the particle is at its equilibrium position, the displacement is zero, and as it moves away from that position, its displacement takes negative values. The sine function captures this behavior accurately.
If we were to use the cosine function instead, the equation would be x = -Acos(ωt). However, cosine assumes a displacement of zero at t = 0, whereas the particle is actually at its maximum displacement (A) at that instant. Therefore, using cosine would not accurately represent the motion of the particle.