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 scenario, we are dealing with a particle that is hanging from a spring in an elevator. As the elevator descends at a constant speed, the particle is motionless relative to the elevator car. This means that the net force acting on the particle is zero, resulting in no acceleration or movement of the particle along the vertical direction.
The gravitational force acting on the particle is equal and opposite to the force exerted by the spring. This can be expressed mathematically as:
mg = kx,
where m is the mass of the particle, g is the acceleration due to gravity, k is the spring constant, and x is the displacement of the particle from its equilibrium position.
Since the particle is motionless, its maximum displacement (amplitude) is equal to its equilibrium position. Therefore, we can write x = A, where A is the amplitude.
To find v(max), the maximum velocity of the particle as the elevator descends, we can differentiate the equation x = A with respect to time:
v = dx/dt = d(A sin(ωt))/dt = Aω cos(ωt),
where ω is the angular frequency.
Since the particle is initially motionless, we can find v(max) by substituting t = 0 into this equation:
v(max) = Aω cos(ω(0)) = Aω cos(0) = Aω.
Therefore, in this case, v(max) is equal to ω times the amplitude (v(max) = ωA).
b) The equation x = -A sin(ωt) represents the displacement of the particle from its equilibrium position as a function of time. The negative sign indicates that the displacement is negative (opposite in direction to the displacement at t=0) when the particle is below its equilibrium position.
The choice of using sin instead of cos in this equation depends on the initial conditions and the phase of the motion. If the particle is at its equilibrium position at t=0, and starts its motion by moving below the equilibrium position, then the negative sign and the sin function accurately describe this behavior.
However, if the particle starts its motion at its equilibrium position or moves above the equilibrium position initially, then we would have to use a different trigonometric function, such as cos.
It is important to consider the initial conditions and the behavior of the system to choose the appropriate trigonometric function in the displacement equation. In this specific scenario described, using sin takes into account the initial conditions and accurately represents the motion of the particle.