(a) The amplitude of the motion is the maximum displacement from the equilibrium position. In the given equation, x = (0.28 m) cos(0.5𝜋t), the amplitude is the coefficient of the cosine function. Therefore, the amplitude is 0.28 m.

(b) The spring constant can be found using the formula for the period of a spring-mass system, T = 2π√(m/k), where m is the mass and k is the spring constant. We can rearrange this to solve for k: k = m(2π/T)^2. The period can be found from the given equation, as T = 2π/ω, where ω is the angular frequency. In the given equation, ω = 0.5𝜋, so T = 2π/(0.5𝜋) = 4 s. Substituting these values in, we get k = 0.30 kg * (2π/4 s)^2 = 0.59 N/m.

(c) The position of the object at t = 0.34 s can be found by substituting t = 0.34 s into the given equation: x = (0.28 m) cos(0.5𝜋*0.34 s) = 0.28 m * cos(0.17𝜋) = 0.13 m.

(d) The object's speed at t = 0.34 s can be found by taking the derivative of the position function to get the velocity function, v = dx/dt = -0.28 m * 0.5𝜋 * sin(0.5𝜋t). Substituting t = 0.34 s into this equation gives v = -0.28 m * 0.5𝜋 * sin(0.17𝜋) = -0.42 m/s. The speed is the absolute value of the velocity, so the speed is 0.42 m/s.

Question

(a) The amplitude of the motion is the maximum displacement from the equilibrium position. In the given equation, x = (0.28 m) cos(0.5𝜋t), the amplitude is the coefficient of the cosine function. Therefore, the amplitude is 0.28 m.

(b) The spring constant can be found using the formula for the period of a spring-mass system, T = 2π√(m/k), where m is the mass and k is the spring constant. We can rearrange this to solve for k: k = m(2π/T)^2. The period can be found from the given equation, as T = 2π/ω, where ω is the angular frequency. In the given equation, ω = 0.5𝜋, so T = 2π/(0.5𝜋) = 4 s. Substituting these values in, we get k = 0.30 kg * (2π/4 s)^2 = 0.59 N/m.

(c) The position of the object at t = 0.34 s can be found by substituting t = 0.34 s into the given equation: x = (0.28 m) cos(0.5𝜋*0.34 s) = 0.28 m * cos(0.17𝜋) = 0.13 m.

(d) The object's speed at t = 0.34 s can be found by taking the derivative of the position function to get the velocity function, v = dx/dt = -0.28 m * 0.5𝜋 * sin(0.5𝜋t). Substituting t = 0.34 s into this equation gives v = -0.28 m * 0.5𝜋 * sin(0.17𝜋) = -0.42 m/s. The speed is the absolute value of the velocity, so the speed is 0.42 m/s.

Knowee AI · Accepted Answer

(a) The amplitude of the motion is the maximum displacement from the equilibrium position. In the given equation, x = (0.28 m) cos(0.5𝜋t), the amplitude is the coefficient of the cosine function. Therefore, the amplitude is 0.28 m.

(b) The spring constant can be found using the formula for the period of a spring-mass system, T = 2π√(m/k), where m is the mass and k is the spring constant. We can rearrange this to solve for k: k = m(2π/T)^2. The period can be found from the given equation, as T = 2π/ω, where ω is the angular frequency. In the given equation, ω = 0.5𝜋, so T = 2π/(0.5𝜋) = 4 s. Substituting these values in, we get k = 0.30 kg * (2π/4 s)^2 = 0.59 N/m.

(c) The position of the object at t = 0.34 s can be found by substituting t = 0.34 s into the given equation: x = (0.28 m) cos(0.5𝜋*0.34 s) = 0.28 m * cos(0.17𝜋) = 0.13 m.

(d) The object's speed at t = 0.34 s can be found by taking the derivative of the position function to get the velocity function, v = dx/dt = -0.28 m * 0.5𝜋 * sin(0.5𝜋t). Substituting t = 0.34 s into this equation gives v = -0.28 m * 0.5𝜋 * sin(0.17𝜋) = -0.42 m/s. The speed is the absolute value of the velocity, so the speed is 0.42 m/s.

The position of a 0.30-kg object attached to a spring is described byx = (0.28 m) cos(0.5𝜋t). (Assume t is in seconds.)(a) Find the amplitude of the motion. m(b) Find the spring constant. N/m(c) Find the position of the object at t = 0.34 s. m(d) Find the object's speed at t = 0.34 s. m/s

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