Question

Уравнение колебаний дано в виде x=0.01sin (2.5Пt). Найти смещение от положения равновесия и скорость точки, находящийся на расстоянии 20м от источника для момента времени 1с . Скорость распросстронения волны равна 100м/с

Answer 1

Циклическая частота:
ω=2,5π (из уравнения колебаний)
Период:
T = 2π / ω = 2π / 2,5π = 2/2,5 = 0,8 c
Длина волны:
λ=V*T = 100*0,8 = 80 м

20/80 = 1/4 длины волны.
В этой точке ГОРБ волны, смещение - максимальное 
Xmax=0,01 м

Скорость - первая производная
v=x' = 2,5π*cos (2,5π*t)
v(1) = 2,5π*cos (2,5 π) = 0

Answer 2

To find the displacement from the equilibrium position and the velocity of a point located 20m away from the source at a given time, we can use the equation of motion for simple harmonic motion. The equation given is x = 0.01sin(2.5πt), where x represents the displacement from the equilibrium position and t represents time.

Finding the Displacement:

To find the displacement at a given time, we substitute the value of t into the equation x = 0.01sin(2.5πt). In this case, we need to find the displacement at t = 1s.

Substituting t = 1s into the equation, we have: x = 0.01sin(2.5π * 1) x = 0.01sin(2.5π) x ≈ 0.01sin(7.85) x ≈ 0.01 * 0.9998 x ≈ 0.009998m

Therefore, the displacement from the equilibrium position at t = 1s is approximately 0.009998m.

Finding the Velocity:

To find the velocity of the point located 20m away from the source at t = 1s, we can differentiate the equation x = 0.01sin(2.5πt) with respect to time.

Differentiating x = 0.01sin(2.5πt) with respect to t, we have: v = dx/dt = 0.01 * 2.5π * cos(2.5πt)

Substituting t = 1s into the equation, we have: v = 0.01 * 2.5π * cos(2.5π * 1) v = 0.01 * 2.5π * cos(2.5π) v ≈ 0.01 * 2.5π * (-1) v ≈ -0.07854π

Given that the velocity of wave propagation is 100m/s, we can find the velocity of the point located 20m away from the source by multiplying the velocity of wave propagation by the ratio of the displacement from the source to the distance from the source.

The ratio of the displacement from the source to the distance from the source is given by: ratio = x / distance = 0.009998m / 20m ratio ≈ 0.0004999

Multiplying the ratio by the velocity of wave propagation, we have: velocity = ratio * velocity of wave propagation velocity ≈ 0.0004999 * 100m/s velocity ≈ 0.04999m/s

Therefore, the velocity of the point located 20m away from the source at t = 1s is approximately 0.04999m/s.

Please note that the values provided are approximate due to rounding.