Answer:
A) L = 0.496 m, B) the movement of the elevator upwards decreases the angular velocity of the pendulum
Explanation:
A) The motion of a simple pendulum is a harmonic motion with angular velocity
w² = g /L
angular velocity and frequency are related
w = 2π f
we substitute
4π² f² = g /L
L = [tex]\frac{g}{4\pi ^2 f}[/tex]
let's calculate
L = 9.8 / 4 pi² 0.5
L = 0.496 m
B) To see the effect of the elevator acceleration (aₐ), let's use Newton's second law.
At the acceleration from the vertical direction upwards, let's decompose it is a component parallel to the movement and another perpendicular
sin θ = a_parallel / aₐ
a_parallel = aₐ sin θ
this component of the acceleration is in the opposite direction to the movement of the system, so it must be negative
- W sin θ = m (a - a_parallel)
- mg sin θ = m ([tex]\frac{d^2 s}{dt^2} - a_a sin \theta[/tex])
all angles are measured in radians, therefore the angular displacement is
s = L θ
We solve the system for small angles
sin θ = θ
we substitute
- mg θ + m aₐ θ = m L [tex]\frac{d^2 \theta}{dt^2 }[/tex]
[tex]- ( \frac{g- a_a}{L} ) \ \theta = \frac{d^2 \theta}{dT^2 }[/tex]
this is the same equation of the simple pendulum therefore the angular velocity is
w² =[tex]\frac{g-a_a}{L}[/tex]
When analyzing this expression, we see that the movement of the elevator upwards decreases the angular velocity of the pendulum
