A graph of displacement versus time is shown for four different, horizontal osci

Question

A graph of displacement versus time is shown for four different, horizontal oscillators below The total mechanical energy of each oscillator is the same, but the mass and spring may be different. Rank the maximum speed of each of the oscillators, greatest first. Briefly explain your reasoning. Rank the mass of each of the oscillators, greatest first. Briefly explain your reasoning. Rank the spring constant for each of the oscillators, greatest first. Briefly explain your reasoning. Rank the maximum acceleration of each of the oscillators, greatest first. Briefly explain your reasoning

Explanation / Answer

in case A)

Time period, T = 9s

Apmlitude, A = 20 cm = 0.2 m

angular frequency, w = 2*pi/T

= 2*pi/9

= 2*pi/9

= 0.698 rad/s

Vmax = A*w

= 0.2*0.698

= 0.1396 m/s

a_max = A*w^2 = 0.2*0.698^2 = 0.097 m/s^2

in case B)

Time period, T = 4s

Apmlitude, A = 10 cm = 0.1 m

angular frequency, w = 2*pi/T

= 2*pi/4

= 2*pi/4

= 1.57 rad/s

Vmax = A*w

= 0.1*1.57

= 0.0157 m/s

a_max = A*w^2 = 0.1*1.57^2 = 0.246 m/s^2

in case C)

Time period, T = 2s

Apmlitude, A = 5 cm = 0.05 m

angular frequency, w = 2*pi/T

= 2*pi/2

= pi

= 3.14 rad/s

Vmax = A*w

= 0.05*3.14

= 0.157 m/s

a_max = A*w^2 = 0.05*3.14^2 = 0.493 m/s^2

in case D)

Time period, T = 4s

Apmlitude, A = 20 cm = 0.2 m

angular frequency, w = 2*pi/T

= 2*pi/4

= 1.57 rad/s

Vmax = A*w

= 0.2*1.57

= 0.314 m/s

a_max =A*w^2 = 0.2*1.57^2 = 0.493 m/s^2

a) greatest Vmax

D > C > A > B

b) greatest mass

B > A > C > D

here maximum kinetic energy is same in all the cases.

so, the which has low Vmax will have more mass.

c) greatest spring constant.

C > B > A = D

d) greatest a_max

C = D > B > A

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