Respuesta :
Answer:
a) x = 0.33 A , b) K = ¾ Em , c) the kinetic energy increases 9 times
Explanation:
a) In a simple harmonic motion the mechanical energy is conserved and is expressed by the relation
Em = ½ k A²
At all points of movement the mechanical energy and
Em = K + U
remember that potential energy is
U = ½ k x²
they ask us for the point where
U = 1/9 Em
we substitute
½ k x² = 1/9 (½ k A²)
x = √1/9 A
x = 0.33 A
b) At all points the equation for mechanical energy is
Em = K + U
K = Em - U
K = ½ k A² - ½ k x²
K = ½ k (A² -x²)
at point x = ½ A
K = ½ k (A² - ¼ A²)
K = (½ k A²) ¾
K = ¾ Em
U = ½ k (½A) 2
U = ¼ Em
the fraction of energy e
U / K = 1/3
c) Kinetic energy is
K = ½ k v²
the system is described by the expression
x = A cos (wt + Ф)
speed is defined
v = dx / dt
v = - A w sin (wt + Ф)
we substitute
K = ½ k (- A w sin ( wt + Ф))²
We write this equation for the initial amplitude A
K₀ = ½ k (w sin (wt + Ф))² A²
we write it for the new amplitude A´ = 3 A
K = ½ k (w sin (wt +Ф))² (3A)²
the relationship between this energies is
K / K₀ = 9
whereby the kinetic energy increases 9 times
