## Related questions with answers

When a 3.0-kg block is pushed against a massless spring of force constant 4.5 × 10³ N/m, the spring is compressed 8.0 cm. The block is released, and it slides 2.0 m (from the point at which it is released) across a horizontal surface before friction stops it. What is the coefficient of kinetic friction between the block and the surface?

Solution

VerifiedIn this problem we consider a block of mass $m=3$ kg which compresses a horizontal spring of constant $k=4.5\cdot10^3$ N/m by $x=8$ cm. The initial energy stored in the spring is equal to

$\begin{align*} E_{s}&=\frac{1}{2}kx^2\\ &=\frac{1}{2}(4.5\cdot10^3\text{ N/m})\cdot(0.08\text{ m})^2\\ &=14.4\text{ J}. \end{align*}$

This is the total energy of the system, and the friction does work to stop the block after $d=2$ m. The work done is therefore equal to the initial energy in the spring, which gives the friction coefficient

$\begin{align*} W_{fric}&=E_s\\ F_{fric}d&=E_s\\ \mu_Kmgd&=E_s\\ \mu_K&=\frac{E_s}{mgd}\\ &=\frac{14.4\text{ J}}{(3\text{ kg})\cdot(9.81\text{ m/s}^2)\cdot(2\text{ m})}\\ &=\boxed{\color{#c34632}0.24.} \end{align*}$

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