When mass of m oscillates with spring of spring constant k1 its frequency is 3hz and with spring of spring comstant k2 its frequency is 6hz .then frequency of oscillations when mass is connected with spring as shown?
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The restoring force F1 acting on the mass m due to the spring having force constant k1 will be ,F1 = k1x1
Here x1 is the elongation.
The restoring force F2 acting on the mass m due to the spring having force constant k2 will be, F2 = k2x2
Here x2 is the elongation.
The tension in the two springs will be same.
So, k1x1 = k2x2
But the total extension is x1 + x2.
Thus the effective spring constant k of the combination will be,
k = F/x
= F/(x1+x2)
= 1/ [x1/F + x2/F ]
= 1/ [1/k1 +1/k2]
= k1k2/ k1+ k2
To find out the frequency of oscillation f, substitute k1k2/ k1+ k2 for the spring constant k in the equation f = 1/2π √k/m, we get,
f = 1/2π √k/m
= 1/2π √k1k2/(k1+k2)m
= 1/2π √1/m/k1 +m/k2
=1/2π √1/1/w12 + 1/w22( Since, w12 = k1/m and w22 = k2/m )
=1/2π √ w12w22/w12 +w22
= f1f2/√f12+f22 (Since, w1 = 2πf1 and w2 = 2πf2)
From the above observation we conclude that, the frequency of oscillation of the block will be f1f2/√f12+f2.
Here x1 is the elongation.
The restoring force F2 acting on the mass m due to the spring having force constant k2 will be, F2 = k2x2
Here x2 is the elongation.
The tension in the two springs will be same.
So, k1x1 = k2x2
But the total extension is x1 + x2.
Thus the effective spring constant k of the combination will be,
k = F/x
= F/(x1+x2)
= 1/ [x1/F + x2/F ]
= 1/ [1/k1 +1/k2]
= k1k2/ k1+ k2
To find out the frequency of oscillation f, substitute k1k2/ k1+ k2 for the spring constant k in the equation f = 1/2π √k/m, we get,
f = 1/2π √k/m
= 1/2π √k1k2/(k1+k2)m
= 1/2π √1/m/k1 +m/k2
=1/2π √1/1/w12 + 1/w22( Since, w12 = k1/m and w22 = k2/m )
=1/2π √ w12w22/w12 +w22
= f1f2/√f12+f22 (Since, w1 = 2πf1 and w2 = 2πf2)
From the above observation we conclude that, the frequency of oscillation of the block will be f1f2/√f12+f2.
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