Question #218661

Two blocks of masses m1=1.7 kg and m2=3.7 kg are put on a frictionless horizontal surface. A spring of stiffness k=970 N/m and negligible mass is placed between the blocks and they are pushed together to compress the spring by distance d from its equilibrium length. The configuration is fixed by a cord that holds the blocks. When the cord is burned, the first block moves to the left with a speed of v1=1.9 m/s. Calculate the initial compression of the spring, d


1
Expert's answer
2021-07-19T09:50:45-0400

m1=1.7kg;m2=3.7kg;k=970Nmm_1 = 1.7kg;m_2=3.7kg;k= 970\frac{N}{m}

Spring is compressed:\text{Spring is compressed:}

v1=0;v2=0v'_1= 0;v'_2= 0

Δx=d\Delta x'= d

E=Ek+EpE = E'_k+E'_p

Ek=m1v122+m2v222=0E'_k=\frac{m_1v_1'^2}{2}+\frac{m_2v_2'^2}{2}=0

Ep=kΔx22=kd22E_p'=\frac{k\Delta x'^2}{2}= \frac{kd^2}{2}

E=kd22E=\frac{kd^2}{2}

Spring is released:\text{Spring is released:}

v1=1.9ms;v2?v_1=1.9\frac{m}{s} ;v_2-?

Δx=0\Delta x= 0

E=Ek+EpE = E_k+E_p

Ep=kΔx22=0E_p=\frac{k\Delta x^2}{2}= 0

Ek=m1v122+m2v222E_k=\frac{m_1v_1^2}{2}+\frac{m_2v_2^2}{2}

by the law of conservation of momentum\text{by the law of conservation of momentum}

m1v1+m2v2=m1v1+m2v2m_1v_1+m_2v_2 = m_1v'_1+m_2v'_2

m1v1+m2v2=0m_1v_1+m_2v_2 = 0

v2=m1v1m2=1.71.93.7=0.87v_2 = -\frac{m_1v_1}{m_2}= -\frac{1.7*1.9}{3.7}=-0.87

Ek=1.71.922+3.7(0.87)22=4.47JE_k=\frac{1.7*1.9^2}{2}+\frac{3.7*(-0.87)^2}{2}=4.47J

E=Ep+Ek=4.47E = E_p+E_k = 4.47

E=kd22E=\frac{kd^2}{2}

d=2Ek=24.479700.07md = \sqrt{\frac{2E}{k}}= \sqrt{\frac{2*4.47}{970}}\approx 0.07m


Answer: initial compression of the spring, d = 0.07m\text{Answer: initial compression of the spring, d = 0.07m}


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