Question

Координата колеблющейся материальной точки изменяется по закону Х=А*sin(2ПиVt), где А=4 см, V=2 Гц. Определить скорость и ускорение точки в положении Х=1 смПомогите пожалуйста

Answer 1

Сравнивая данное уравнение с уравнением в общем виде
x=xm*coswt. видим, что амплитуда ( xm=4м ) , циклическая частота (w=2П рад). Из формулы циклической частоты
w=2П / T. выразим период (Т) .
Т=2П / w. подставим
Т=2П / 2П=1с.
ответ А.

Answer 2

Calculation of Velocity and Acceleration of a Oscillating Particle

To determine the velocity and acceleration of a particle at a given position, we can use the equation of motion for simple harmonic motion. The equation for the position of the particle is given as:

X = A * sin(2πVt)

Where: - X is the position of the particle, - A is the amplitude of the oscillation (4 cm in this case), - V is the frequency of the oscillation (2 Hz in this case), - t is the time.

To find the velocity and acceleration of the particle at a specific position, we need to differentiate the equation of motion with respect to time.

Velocity:

The velocity of the particle is given by the derivative of the position equation with respect to time:

V = dX/dt = 2πAV * cos(2πVt)

Where: - V is the velocity of the particle.

To find the velocity when X = 1 cm, we substitute X = 1 cm into the equation and solve for V:

1 = 4 * sin(2π * 2 * t) sin(2π * 2 * t) = 1/4 2π * 2 * t = arcsin(1/4) t = arcsin(1/4) / (2π * 2)

Now we can substitute this value of t into the velocity equation to find the velocity at X = 1 cm.

Acceleration:

The acceleration of the particle is given by the second derivative of the position equation with respect to time:

a = d^2X/dt^2 = -4π^2A * V^2 * sin(2πVt)

Where: - a is the acceleration of the particle.

To find the acceleration when X = 1 cm, we substitute X = 1 cm into the equation and solve for a:

1 = 4 * sin(2π * 2 * t) sin(2π * 2 * t) = 1/4 2π * 2 * t = arcsin(1/4) t = arcsin(1/4) / (2π * 2)

Now we can substitute this value of t into the acceleration equation to find the acceleration at X = 1 cm.

Calculation:

Let's calculate the velocity and acceleration at X = 1 cm using the given values of A = 4 cm and V = 2 Hz.

Velocity: V = 2π * 4 * 2 * cos(2π * 2 * t)

Acceleration: a = -4π^2 * 4 * 2^2 * sin(2π * 2 * t)

Now, substitute the value of t we calculated earlier into these equations to find the velocity and acceleration at X = 1 cm.

Please note that the exact numerical values of velocity and acceleration will depend on the specific value of t calculated using the arcsin function.