? asked in Science & MathematicsPhysics · 7 years ago

An object with mass 2.5 kg is attached to a spring with spring stiffness constant k = 280 N/m and is executing?

An object with mass 2.5 kg is attached to a spring with spring stiffness constant k = 280 N/m and is executing simple harmonic motion. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.55 m/s.

(a) Calculate the amplitude of the motion.

(b) Calculate the maximum velocity attained by the object. [Hint: Use conservation of energy.]

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  • 7 years ago
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    The natural frequency, w, is w = sqrt(k/m)

    w = sqrt(280 N/m / 2.5 kg) = 10.58 rad/s

    Equations for SHM

    x = A*sin(w*t)

    v = w*A*cos(w*t)

    a) Now, use the intial conditions to find the time and amplitude

    0.020 m = A*sin(10.58 rad/s * t)

    0.55 m/s = 10.58 rad/s * A * cos ( 10.58 rad/s * t)

    Solve the upper equation for A and plug into the lower equation to solve for t. Once we have a number, then we can use either equation to solve for the value of A

    A = 0.020 m / (sin(10.58 * t))

    0.55 m/s = 10.58 rad/s * 0.020 m / (sin(10.58 * t)) * cos(10.58 rad/s * t)

    0.55 m/s = 10.58 rad/s * 0.020 m * (cos (10.58 rad/s * t)) / (sin(10.58 rad/s * t)

    0.55 m/s = 10.58 rad/s * 0.020 m * cot(10.58 rad/s * t)

    cot(10.58 rad/s * t) = 2.60

    cot = 1/tan so...

    tan(10.58 rad/s * t ) = 1/2.60

    10.58 rad/s * t = atan (1/2.60)

    10.58 rad/s * t = 0.367

    t = 0.367 / 10.58 rad/s = 0.0347 s

    Now that we know time...

    x = A*sin(w*t)

    A = x / sin(w*t) = 0.020 m / sin(10.58 rad/s * 0.0347 s) = 0.0557 m

    b) You could use conservation of energy, but I'll show you a quick way...

    v = w*A*cos(w*t)

    The maximum velocity occurs when the cos(w*t) term is equal to 1

    v_max = w*A*1

    v_max = w*A = 10.58 rad/s * 0.0557 m = 0.589 m/s

    To show it with energy....

    Max Spring Energy = 1/2*k*x^2, where x is the maximum displacement or amplitude of the motion.

    That all gets transferred into Kinetic Energy

    Max KE = Max Spring = 1/2*m*v^2

    1/2*m*v^2 = 1/2*k*x^2

    solve for v

    v = sqrt(k/m*x^2)

    v = x*sqrt(k/m)

    Remember, sqrt(k/m) = w and x = amplitude

    v = w*A

    Just like I said, the max velocity occurs when the cos(w*t) term is equal to 1.

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  • sallas
    Lv 4
    3 years ago

    The elastic means power in a spring relies upon basically on the spring consistent, ok, and the area stretched or compressed, x. meaning you do no longer could desire to comprehend the object's mass. the quantity of power is E = 0.5*kx^2. you're able to be able to desire to jot down ok in Newtons in step with meter and x in meters as a fashion to get E in joules.

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