Solved Problem on Work and Energy

A block with a mass equal to 5 kg is released with a constant speed of 0.4 m/s and shocks with a spring of elastic constant 80 N/m, and the surface is frictionless. Determine the maximum compression of the spring.

Problem data:

Mass of block: m = 5 kg;
Initial speed of block: v_i = 0.4 m/s;
Spring constant: k = 80 N/m.

Problem diagram:

Initially, the block has kinetic energy, \( K^{block} \), due to its speed. As the problem gives the surface is frictionless, and there are no dissipative forces acting in the system, then when the spring has the maximum compression, the speed of the block is zero, and all the kinetic energy of the block is transferred to the spring as elastic potential energy, \( U_{s}^{spring} \).

Figure 1

Solution

The kinetic energy of the block is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {K=\frac{mv^2}{2}} \end{gather} \]

The elastic potential energy is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {U_s=\frac{kx^2}{2}} \end{gather} \]

Applying the condition, the kinetic energy of the block and the elastic potential energy of the spring must be equal

\[ \begin{gather} K^{block}=U_{s}^{spring}\\[5pt] \frac{mv_i^2}{\cancel 2}=\frac{kx^2}{\cancel 2}\\[5pt] mv_i^2=kx^2\\[5pt] x^2=\frac{mv_i^2}{k}\\[5pt] x=\sqrt{\frac{mv_i^2\;}{k}}\\[5pt] x=\sqrt{\frac{(5\;\mathrm{kg})\times(0.4\;\mathrm{\frac{m}{s}})^{2}\;}{80\;\mathrm{\frac{N}{m}}}}\\[5pt] x=1\times 10^{-1}\;\mathrm m \end{gather} \]

\[ \bbox[#FFCCCC,10px] {x=0.1\;\mathrm m} \]

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