Calculate the angular frequency of the system shown in the figure. Friction is absent and threads are massless? Hint: The system performs SHM. Try not to use the energy conservation method.
#m_A = m_B = m#
1 Answer
Let
#kx_0+m_Agsintheta=T# ..........(1)
#2T=m_Bg# ..........(2)
Rewriting (1) with the help of (2)
#kx_0+m_Agsintheta=(m_Bg)/2# .....(3)
Let the spring be stretched by a displacement
1. Its velocity would be
2. it is lowered by a displacement
Total Energy equation of the system becomes
#TE="PE of spring"+"KE of mass "m_A+"KE of mass "m_B+"PE of "m_A+"PE of mass "m_B#
#=>TE=1/2k(x_0+x)^2+1/2m_Av^2+1/2m_B(v/2)^2+m_Agxsintheta-m_Bgx/2#
#=>TE=1/2k(x_0+x)^2+1/2m_Av^2+1/8m_Bv^2+(m_Agsintheta)x-(1/2m_Bg)x#
Differentiating with respect to time and noting that Law of conservation of energy holds so that
#0=d/dt(1/2k(x_0+x)^2+1/2m_Av^2+1/8m_Bv^2+(m_Agsintheta)x-(1/2m_Bg)x)#
#=>0=k(x_0+x)dotx+m_Avdotv+1/4m_Bvdotv+(m_Agsintheta)dotx-(1/2m_Bg)dotx#
We know that
#0=kx+m_Addotx+1/4m_Bddotx#
Given
#0=kx+mddotx+1/4mddotx#
#=>-kx=5/4mddotx#
#=>-kx=m^'ddotx# .......(4)
where#m^'=5/4m#
(4) is a second-order linear ordinary differential equation for which angular frequency is given as
#omega=sqrt(k/m^')#
Inserting value of
#omega=sqrt(4/5k/m)#