A mass m oscillates at the end of an ideal, massless spring k_n, passing through

Question

A mass m oscillates at the end of an ideal, massless spring k_n, passing through points PQRSP, and repeating indefinitely.(Neglect all friction, unless told otherwise.) Let = 1.7 kg, k = 8.5 N/m, and maximum amplitude of displacement A = 10.0 cm. a. Hind the period of oscillation. Show your work. b. Hind the maximum speed v_max of the mass. Show your work. At time t = 0, the mass is released at point Q (where x = A and v = 0). You Jo NOT need to show sour work for the following four fill-in-the-blank questions. but be sure to include units, appropriate notequalto signs, and appropriate sig figs! Choose the + x-direction to the right. c. Find the mass's displacement at t = 6 0 s: _____ d. Find the horiz. component of mass's velocity at t = 6.0 s: _____ e. Find the horiz, component of the mass's acceleration at t 6.0 s: ______ f. Find the hot component of net force acting on the mass at t 6.0 s: _____ g. suppose that mild friction and air resistance cause the amplitude of oscillations to diminish gradually over time, shrinking from 10.0 cm to 5.00 cm. Find the total energy dissipated by friction over this time period. Show your work

Explanation / Answer

period of oscillation T = 2*pisqrt(m/k)

T = 2*pi*sqrt(1.7/8.5)

T = 2.81 s <<<--------answer

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(b)

Vmax = A*w = A*2*pi/T = 0.1*2*pi/2.81 = 0.224 m/s

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(c)

x = A*cos(wt)

w = sqrt(k/m) = sqrt(8.5/1.7) = 2.24 /s

x = 10*cos(2.24*6) = 6.42 cm = 0.0642 m

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(d)

v = dx/dt = -A*w*sinwt

v = -10*2.24*sin(2.24*6) = -17.7 cm/s = -0.177 m/s

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e)

ax = dv/dt = -w^2*x = -2.24^2*6.42 = -32.2 m/s^2 = -0.322 m/s^2

(f)

Fnet - k*x = 8.5*0.0642 = 0.546 N

(g)

energy dissipated = (1/2)*k*(A1^2-A2^2)

energy dissipated = (1/2)*8.5*(0.1^2-0.05^2) = 0.32 J

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