Question

Тело массой M под действием пружины совершает гармонические колебания на гладком горизонтальном столе.Когда тело проходило положение равновесия, на него свержу упад и прилип кусочек пластилина массой m. Во сколько раз изменится амплитуда колебаний?

Answer 1

Тело массой M под действием пружины совершает гармонические колебания на гладком горизонтальном столе.Когда тело проходило положение равновесия, на него свержу упад и прилип кусочек пластилина массой m. Во сколько раз изменится амплитуда колебаний?

Пусть A1=A - начальная амплитуда

найдем скорость в точке равновесия
k*A^2/2=M*V^2/2      k*A1^2=M*V^2  (1)
V=А*√k/M
по законоу сохранения импульса M*V= U*(m+M)

U^2=M^2*V^2/(m+M)^2

(m+M)*U^2/2=M^2*(m+M)*V^2/2*(m+M)^2=k*A2^2/2

М^2*V^2/(m+M)=k*A2^2  (2)

Разделим (1) на (2)
A1^2/A2^2=(m+M)/M

Ответ A1/A2=√(m+M)/M

В идее ошибок нет проверь вычисления

Answer 2

Change in Amplitude of Oscillations with the Addition of a Piece of Plasticine

When a body of mass M undergoes harmonic oscillations under the influence of a spring on a smooth horizontal table, the amplitude of the oscillations can change when a piece of plasticine with mass m is added to the body at its equilibrium position.

To determine how the amplitude of the oscillations changes, we need to consider the conservation of mechanical energy. In a harmonic oscillator, the total mechanical energy is the sum of the potential energy and the kinetic energy. The potential energy is given by the equation U = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. The kinetic energy is given by the equation K = (1/2)mv^2, where m is the mass of the body and v is its velocity.

When the body is at its equilibrium position, the potential energy is at its maximum and the kinetic energy is zero. As the body oscillates, the potential energy decreases and the kinetic energy increases, reaching maximum values when the body is at the extreme positions of its oscillation.

When the piece of plasticine is added to the body at its equilibrium position, the total mass of the system becomes M + m. As a result, the spring constant k remains the same, but the equilibrium position of the body changes due to the additional mass. Let's denote the new equilibrium position as x0'.

To find the new amplitude of the oscillations, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system before and after adding the plasticine should be the same.

Before adding the plasticine: Total mechanical energy = Potential energy + Kinetic energy = (1/2)kx0^2

After adding the plasticine: Total mechanical energy = Potential energy + Kinetic energy = (1/2)kx0'^2

Since the total mechanical energy is conserved, we can equate the two expressions:

(1/2)kx0^2 = (1/2)kx0'^2

To simplify the equation, we can divide both sides by (1/2)k:

x0^2 = x0'^2

Taking the square root of both sides, we get:

x0 = x0'

Therefore, the amplitude of the oscillations remains the same after adding the plasticine. The addition of the plasticine does not affect the amplitude of the oscillations.

In conclusion, the amplitude of the oscillations does not change when a piece of plasticine with mass m is added to the body.