A mass is vibrating at the end of a spring with a spring constant 1.09 N/m. The

Question

A mass is vibrating at the end of a spring with a spring constant 1.09 N/m. The figure shows a graph of its position x (in centimetres) as a function of time t (in seconds). A) At what time between t=0 s and the first maximum after t=0 s is the mass not moving?

B) What is the magnitude of the acceleration of the object at the second maximum in the x-t curve after t = 0 s?

C) What is the mass of the object?

D) How much energy did the system originally contain?

E) How much energy did the system lose between t = 0 s and the third maximum after t = 0 s? Think about where this energy has gone.

Explanation / Answer

this is equation of SHM.
A.) we can observe that at every maxima and minima the velocity of object become zero for some instant.so between t=0 s and the first maximum at t = 0.8 s the object is not moving.

B.)acceleration = d^2 x/dt { double differentiation of x-t graph with respect to t }

x= A cos(2*pi*f *t)

accelaration(a)= A * (2pi*f)^2 { A = amplitude and f is frequency = 1/T = 1/1.6 =0.625 s^-1 }

= 2.3*(2*3.14 *0.625)^2 = 35.46 m/s^2

C.)
T = 2pi * sqrt(m/k).

K= 1.09 N/m , T = 1.6 s

m = 0.0707 kg

D.)
Energy of spring = (1/2)kx^2 at the maximum value of x.

= 0.5 * 1.09 * 6*6 =19.62 J

E.) energy loose  between t = 0 s and the third maximum after t = 0 s

= initial energy - final energy

final energy = 0.5 kx^2

= 0.5 * 1.09 * (1.8)^2

=1.769 J

initial (t=0 s) = 19.62 J

lose energy = 19.62 - 1.769 = 17.851 J

this energy may be lose in friction force air resistance and in entropy etc.

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