Question

Груз массой 4кг совершает гармонические колебания под действием пружины жесткостью 75 Н/м. При каком смещении груза от положения равновесия модуль его скорости равен 5 м/с, если в положении равновесия модуль его скорости равен 10 м/c?

Answer 1

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Answer 2

Calculation of the Displacement of the Mass

To determine the displacement of the mass from the equilibrium position, we can use the equation for the velocity of an object undergoing simple harmonic motion:

v = ω√(A^2 - x^2)

Where: - v is the velocity of the mass, - ω is the angular frequency of the oscillation, - A is the amplitude of the oscillation, and - x is the displacement of the mass from the equilibrium position.

We are given that the velocity of the mass when it is at the equilibrium position is 10 m/s, and we need to find the displacement when the velocity is 5 m/s.

Let's denote the displacement when the velocity is 5 m/s as x'. We can rewrite the equation as follows:

10 = ω√(A^2 - 0^2) (at equilibrium position) 5 = ω√(A^2 - x'^2) (at displacement x')

Dividing the second equation by the first equation, we get:

5/10 = ω√(A^2 - x'^2) / ω√(A^2 - 0^2)

Simplifying the equation, we have:

1/2 = √(A^2 - x'^2) / A

Squaring both sides of the equation, we get:

(1/2)^2 = (√(A^2 - x'^2) / A)^2

Simplifying further, we have:

1/4 = (A^2 - x'^2) / A^2

Multiplying both sides of the equation by A^2, we get:

A^2/4 = A^2 - x'^2

Rearranging the equation, we have:

x'^2 = A^2 - A^2/4

Simplifying, we get:

x'^2 = 3A^2/4

Taking the square root of both sides of the equation, we get:

x' = √(3A^2/4)

Now, we need to find the value of A, the amplitude of the oscillation. Unfortunately, the given information does not provide the value of A. Therefore, we cannot determine the exact displacement of the mass from the equilibrium position when the velocity is 5 m/s without knowing the value of A.

Please provide the value of the amplitude (A) so that we can calculate the displacement accurately.