One end of a spring with a force constant of k = 10.0 N/m is attached to the end

Question

One end of a spring with a force constant of k = 10.0 N/m is attached to the end of a long horizontal frictionless track and the other end is attached to a mass m = 2.20 kg which glides along the track. After you establish the equilibrium position of the mass-spring system, you move the mass in the negative direction (to the left), compressing the spring 2.48 m. You then release the mass from rest and start your stopwatch, that is x(t = 0) = A, and the mass executes simple harmonic motion about the equilibrium position. Determine the following.

(a) displacement of the mass (magnitude and direction) 1.0 s after it is released magnitude ____m

(b) velocity of the mass (magnitude and direction) 1.0 s after it is released magnitude ____m/s

(c) acceleration of the mass (magnitude and direction) 1.0 s after it is released magnitude ____m/s2

(d) force the spring exerts on the mass (magnitude and direction) 1.0 s after it is released magnitude ____N

(e) How many times does the object oscillate in 12.0 s? (Do not round your answer to an integer.) ____oscillations

Explanation / Answer

A) displacement of the mass 1s after it is released

at t = 0 s and x = -A =-2.48 m

w = sqrt(k/m) = sqrt(10/2.2) = 2.13 rad/sec

x =-A*cos(w*t) = -2.48*cos(2.13*1) = 1.31 m

B) v = A*w*sin(w*t) = 2.48*2.13*sin(2.13*1) = 4.47 m/s

C) a = -w^2*x = -2.13*2.13*1.31 = -5.94 m/s^2

d) F = -k*x = -10*1.31 = -13.1 N

magnitude is F = 13.1 N

e) time period is T = 2*pi*sqrt(m/k) = 2*3.142*sqrt(2.2/10) = 2.94 s

no.of oscillations is 12/2.94 = 4.071 oscillations

so the 4 oscillations

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