A 250-g object attached to a spring oscillates on a frictionless horizontal tabl

Question

A 250-g object attached to a spring oscillates on a frictionless horizontal table with a frequency of 3 Hz and an amplitude of 30 cm. Calculate the following. (a) the maximum potential energy of the system (b) the displacement of the object when the potential energy is one-half of the maximum (c) the potential energy when the displacement is 15 cm

Explanation / Answer

m = 0.25 Kg f = 3 Hz =>T = (1/3) seconds A = 30 cm = 0.3 m T = 2pi/w =>w = 2pi/T = 6*(22/7) rad/s w = sqrt(k/m) =>k = m*w^2 = (0.25*((6*(22/7))^2)) x(t) = Asin(wt) a)Potential energy of system = 0.5*k*x^2 = 0.5*k*A^2*sin(wt)^2 Maximum Potential energy of system = 0.5*(0.25*((6*(22/7))^2))*(0.3^2) = 4.00041 J b)Half of Maximum Potential energy of system = (4.000408163/2) =>0.5*k*A^2*sin(wt)^2 = (4.000408163/2) =>sin(wt) = 0.707106781 =>wt = 0.785398163 =>t = 0.041649903 x(0.041649903) = Asin(wt) = 0.3*sin(0.041649903*6*(22/7)) = 0.212132036 m c)Asin(wt) = 0.15 =>sin(wt) = 1/2 =>t = 0.027766602 =>Potential energy of system = 0.5*k*x^2 = 0.5*k*A^2*sin(wt)^2 =>Potential energy of system = 0.5*(0.25*((6*(22/7))^2))*(0.3^2)*(sin(6*(22/7)*0.027766602)^2) = 1.000102058 J

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