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 know that we are using the maximum velocity, v(max) = ωA, rather than just v, because the particle is oscillating with a maximum displacement, A, from equilibrium. The maximum velocity occurs when the particle is at the equilibrium position and it passes through this position twice during each complete oscillation. At this position, the particle temporarily comes to rest and changes direction.
The equation v(max) = ωA represents the relationship between the angular frequency ω and the amplitude A of the oscillation. It tells us that the maximum velocity of the particle is equal to the product of the angular frequency and the amplitude of the oscillation. Since the particle is hanging motionless relative to the elevator car, it means that the velocity of the particle at the equilibrium position (maximum displacement) is equal to ωA.
b) The equation x = -Asin(ωt) is derived from the displacement of a particle undergoing simple harmonic motion. The negative sign indicates that the particle is moving in the opposite direction to the displacement from equilibrium.
The use of sine function (sin) in the equation is appropriate because it describes the relationship between the displacement of the particle (x) and time (t) during simple harmonic motion. The sine function oscillates between -1 and 1, representing the alternating displacement of the particle from equilibrium.
If we were to use the cosine function (cos) instead, we would be describing a phase-shifted motion where the particle starts at maximum displacement. This would not accurately represent the behavior of a particle in simple harmonic motion that starts from the equilibrium position.
Therefore, using sin(ωt) for the equation x = -Asin(ωt) correctly represents the displacement of the particle as it oscillates back and forth around the equilibrium position.